1.1 Introduction to Electricity and Magnetism

Electricity and magnetism are closely related branches of physics concerned with electric charges, electric and magnetic fields, electric currents, electromagnetic induction, and electromagnetic radiation. Together, they form the subject known as electromagnetism.

Electrical and magnetic phenomena are encountered in both nature and technology. Lightning is an electrical discharge in the atmosphere. Electric motors convert electrical energy into mechanical energy through magnetic interactions. Generators produce electrical energy through electromagnetic induction, while communication systems depend heavily on electromagnetic waves.

The study of electricity begins with the concept of electric charge. Charges at rest produce electrostatic effects, whereas charges in motion constitute electric currents and are associated with magnetic fields.

Although electrostatics is introduced separately from magnetism, later chapters will show that electricity and magnetism are different aspects of a unified electromagnetic interaction.

1.2 Fundamental Forces in Nature

Interactions between physical objects are described in modern physics in terms of four fundamental forces or interactions. Many forces encountered in everyday mechanics can ultimately be traced to one or more of these fundamental interactions.

The electromagnetic interaction is particularly important in this course. It is responsible for electric forces between charged particles and magnetic effects associated with moving charges.

1.3 Electric Charge

Electric charge is represented by the symbol q or Q. Its SI unit is the coulomb (C).

At the atomic level, ordinary matter contains positively charged protons, negatively charged electrons and electrically neutral neutrons. The magnitudes of the charges of the proton and electron are equal:

Thus, approximately,

and

An electrically neutral body contains equal amounts of positive and negative charge. A charged body has an imbalance between these charges.

1.4 Positive and Negative Charges

There are two kinds of electric charge: positive and negative.

Their interaction follows a simple qualitative rule:

Thus, two positively charged bodies repel each other, and two negatively charged bodies also repel each other. A positively charged body and a negatively charged body attract each other.

A body becomes negatively charged when it gains excess electrons. It becomes positively charged when it loses electrons.

1.5 Fundamental Properties of Electric Charge

Electric charge possesses several important properties that form the foundation of electrostatics.

Additivity of Charge:

Electric charge is additive. The total charge of a system is the algebraic sum of the individual charges:

The signs of the charges must be included when calculating the net charge.

Conservation of Charge:

The total electric charge of an isolated system remains constant. Charge may be transferred between bodies, but the net amount of charge is conserved.

Quantisation of Charge:

Observable charge occurs in discrete multiples of the elementary charge e. For ordinary bodies,

where n is an integer.

1.6 Quantisation of Electric Charge

where:

• Q = total charge,
• n = integer number of elementary charges,
• e=1.602×10^-19C.

1.7 Conservation of Electric Charge

If two initially isolated bodies interact and exchange electrons, the charge lost by one body is gained by the other. The distribution of charge changes, but the total charge does not.

1.8 Conductors and Insulators

Materials differ greatly in the ease with which electric charges can move through them. A basic distinction is made between conductors and insulators.

Metals such as copper, aluminium and silver are good conductors because some of their electrons are mobile throughout the material.

Examples include glass, dry rubber, plastics and ceramics.

Semiconductors:

Semiconductors have electrical behaviour intermediate between good conductors and good insulators, and their conductivity can be strongly controlled by temperature, impurities, light and electric fields. Silicon is an important example.

1.9 Charging by Friction

When two different materials are rubbed together, electrons may transfer from one material to the other because the materials have different tendencies to retain electrons.

The material that gains electrons becomes negatively charged, while the material that loses electrons becomes positively charged.

Friction does not create electric charge. It facilitates the transfer of electrons already present in the materials.

1.10 Charging by Conduction

Suppose a negatively charged conductor touches an initially neutral conductor. Some excess electrons can move onto the neutral conductor. After separation, the second conductor may retain a negative charge.

Similarly, contact with a positively charged conductor can remove electrons from another conductor.

1.11 Charging by Induction

Consider a negatively charged rod brought close to an initially neutral conductor. The rod repels mobile electrons within the conductor toward the farther side. The near side therefore has an induced positive region, while the far side has an excess of electrons.

If the conductor is then connected temporarily to the Earth while the charged rod remains nearby, some electrons may flow to Earth. If the Earth connection is removed first and the charged rod is subsequently taken away, the conductor is left with a net positive charge.

Using a positively charged inducing body reverses the directions of charge movement and can leave the conductor negatively charged.

Grounding:

Because the Earth acts as an enormous charge reservoir, it can accept or supply electrons with negligible change in its overall electrical state.

1.12 Polarisation

A charged object can attract a neutral body even when there is no net transfer of charge. This is explained by polarisation.

In a neutral atom or molecule, the centres of positive and negative charge may shift slightly in opposite directions when an external electric field is applied. The material remains electrically neutral overall, but its charge distribution becomes polarised.

Because the oppositely charged region can be closer to the external charge than the like-charged region, the attractive interaction may exceed the repulsive interaction, producing a net attraction.

1.13 Distribution of Charge on Conductors

When a conductor reaches electrostatic equilibrium, its free charges are no longer undergoing net motion. Several important properties then follow.

• Excess charge on an isolated conductor resides on its surface.
• The electric field inside the conducting material is zero under electrostatic equilibrium.
• The conductor is an equipotential body.
• The electric field immediately outside the surface is perpendicular to the surface.
• Surface charge density tends to be greater at regions of greater curvature, such as sharp points.

The concentration of charge near sharp points can produce very strong local electric fields and can lead to ionisation of the surrounding air.

1.14 Electrostatic Discharge

A small spark produced after walking across a dry surface and touching a metal object is an everyday example. Lightning is a much larger atmospheric electrical discharge.

Electrostatic discharge can be useful in some technologies but may also damage sensitive electronic components. Proper grounding and antistatic procedures are therefore important in electronic environments.

1.15 Applications of Electrostatics

Electrostatic principles have important scientific, industrial and technological applications.

Electrostatic Precipitators:

Electrostatic precipitators remove fine particles from industrial exhaust gases. Particles are electrically charged and attracted toward oppositely charged or grounded collecting surfaces.

Photocopiers and Laser Printers:

Controlled electrostatic charge patterns are used to attract toner particles to selected regions before the toner is transferred and fixed onto paper.

Electrostatic Painting:

Paint droplets can be electrically charged so that they are attracted toward an oppositely charged or grounded object. This can improve coverage and reduce paint wastage.

Lightning Protection:

Lightning protection systems provide a low-resistance conducting path through which large electrical currents can be directed safely toward the ground.

Chapter Summary

2.1 Introduction to Electrostatic Force

Charged bodies can exert forces on one another even when they are not in physical contact. These electrostatic forces are manifestations of the electromagnetic interaction.

The magnitude of the force between two stationary point charges depends on the magnitudes of the charges, their separation and the electrical properties of the surrounding medium.

For point charges in vacuum, this relationship is expressed quantitatively by Coulomb's law.

2.2 Coulomb's Law

For point charges q₁ and q₂ separated by distance r in vacuum,

where

The electrostatic constant is approximately

and the permittivity of free space is

Thus Coulomb's law may also be written as

2.3 Direction and Vector Nature of Coulomb Force

Electric force is a vector quantity. The force between two point charges acts along the straight line joining them.

A vector form of Coulomb's law may be written as

where r̂₁₂ is a unit vector specifying the chosen direction from one charge toward the other.

The sign of q₁q₂ determines whether the force follows or opposes the chosen unit-vector direction.

The forces exerted by two charges on one another satisfy Newton's third law:

2.4 Comparison of Electric and Gravitational Forces

Coulomb's law and Newton's law of universal gravitation have similar mathematical forms.

Despite their mathematical similarity, the two interactions have very different physical properties. In ordinary matter, positive and negative charges can largely cancel, whereas gravitational mass does not ordinarily occur in positive and negative forms that cancel one another.

2.5 Principle of Superposition

When more than two charges are present, the net electric force on any one charge is found by calculating the force due to each of the other charges separately and then adding the individual force vectors.

Because force is a vector, forces acting in different directions must be resolved into components before they are combined.

2.6 Electric Field

The idea of an electric field provides a convenient way of describing how a source charge influences the space around it.

The SI unit of electric field is

It is also equivalent to volts per metre, V/m, a relationship that becomes clearer when electric potential is studied.

The direction of the electric field at a point is defined as the direction of the force that would act on a positive test charge placed there.

2.7 Electric Field Due to a Point Charge

Consider a source charge Q and a small positive test charge q₀ at distance r.

From Coulomb's law,

Since

the magnitude of the field is

Equivalently,

For a positive source charge, the field points radially outward. For a negative source charge, the field points radially inward.

2.8 Superposition of Electric Fields

Electric fields obey the principle of superposition. If several charges produce electric fields at the same point, the resultant field is the vector sum of the individual fields.

For fields along the same straight line, signs may be assigned according to a chosen positive direction. For two-dimensional arrangements, the fields should be resolved into Cartesian components.

2.9 Electric Field Lines

Electric field diagrams follow several useful rules:

• Field lines originate on positive charges and terminate on negative charges, or extend to or from infinity when appropriate.
• The tangent to a field line gives the local field direction.
• Field lines are more densely packed where the field is stronger.
• Electric field lines never cross.
• For a point charge, the lines are radial.
• In electrostatic equilibrium, field lines meet a conductor's surface perpendicularly.

Positive Point Charge:

The field lines of an isolated positive point charge extend radially outward.

Negative Point Charge:

The field lines of an isolated negative point charge point radially inward.

2.10 Electric Field of Two-Charge Systems

The electric field of two charges is found by applying the superposition principle at every point of interest.

Two Like Charges:

For two charges of the same sign, field lines bend away from the region between the charges because each charge repels a positive test charge.

If the two charges are equal, the electric field is zero at the midpoint because the two field contributions are equal and opposite.

Two Unlike Charges:

For a positive and a negative charge, field lines are directed from the positive charge toward the negative charge.

The detailed properties of electric dipoles, including dipole moment, torque and potential energy, will be developed in Chapter 4.

2.11 Continuous Charge Distributions

Coulomb's law is naturally expressed for point charges. However, many physical objects contain charge distributed continuously along a line, over a surface or throughout a volume.

A continuous distribution can be treated as being composed of very small charge elements dq. Each element produces a small electric field contribution dE, and the total field is obtained by integration.

The mathematical difficulty depends strongly on the geometry and symmetry of the charge distribution.

2.12 Linear, Surface and Volume Charge Densities

Charge density describes how electric charge is distributed in space.

Linear Charge Density:

For charge distributed along a line,

For a uniform distribution over length L,

The SI unit is C/m.

Surface Charge Density:

For charge distributed over a surface,

For a uniform distribution over area A,

The SI unit is C/m².

Volume Charge Density:

For charge distributed throughout a volume,

For a uniform distribution throughout volume V,

The SI unit is C/m³.

2.13 Electric Field Due to Continuous Charge Distributions

For a small charge element dq, the magnitude of its electric field contribution at a point a distance r away is

Because electric field is a vector, the direction of each contribution must also be considered:

The total field is then

Depending on the distribution,

or

Symmetry can greatly simplify such calculations. Chapter 3 will develop Gauss's law, which provides a particularly efficient method for highly symmetric charge distributions.

2.14 Motion of a Charged Particle in an Electric Field

When a particle of charge q is placed in an electric field E, it experiences force

If the electric force is the only significant force, Newton's second law gives

Therefore,

In a uniform electric field, the force is constant, so the charged particle undergoes constant acceleration. A positive charge accelerates in the direction of the field, while a negative charge accelerates in the opposite direction.

2.15 Applications of Electric Fields

Electric fields provide a powerful framework for understanding and controlling charged particles. They are important in many physical systems and technologies.

Charged-Particle Control:

Electric fields can accelerate or deflect charged particles. This principle is used in particle-beam systems and many laboratory instruments.

Electrostatic Precipitation:

Electric fields can exert forces on charged pollutant particles and drive them toward collecting surfaces.

Printing and Imaging:

Controlled electric fields guide charged toner particles toward selected regions in electrostatic printing processes.

Capacitive Systems:

Electric fields between conductors are central to the operation of capacitors, which store electrical energy. Capacitance and energy storage will be developed in Chapter 4.

Chapter Summary

3.1 Introduction to Electric Flux

In the previous chapter, electric field was introduced as a vector quantity that describes the force per unit positive test charge at a point. For many charge distributions, especially those possessing a high degree of symmetry, calculating the electric field directly from Coulomb's law can become unnecessarily difficult.

A powerful alternative is obtained by considering how much electric field passes through a surface. This idea is described by electric flux. Electric flux, together with Gauss's law, provides one of the most important methods for calculating electric fields in electrostatics.

Electric flux depends on the strength of the electric field, the area of the surface and the orientation of the surface relative to the field.

3.2 Area Vectors and Electric Flux

To describe the orientation of a surface, an area vector is introduced. For a flat surface of area A, the area vector A has magnitude equal to the area and is directed perpendicular to the surface.

For an open surface, one of the two possible normal directions is chosen according to the problem. For a closed surface, the conventional area vector points outward from the enclosed volume.

where n̂ is a unit vector normal to the surface.

3.3 Electric Flux Through a Flat Surface

For a uniform electric field passing through a flat surface, the electric flux is

or

where θ is the angle between the electric field vector and the area vector.

Three important cases follow immediately.

The SI unit of electric flux is

3.4 Flux Through Closed Surfaces

For a curved or closed surface, the electric field and surface orientation may vary from point to point. The surface is therefore divided conceptually into very small area elements dA.

The infinitesimal flux through one element is

The total flux through a closed surface is

The circle on the integral sign indicates integration over a closed surface.

Flux leaving a closed surface is taken as positive, while flux entering the surface is negative because the outward area vector points opposite to the incoming field direction.

3.5 Gauss's Law

The charge Q_enc is the algebraic sum of all charges enclosed by the Gaussian surface.

Charges outside the surface may produce electric fields at points on the surface, but their total contribution to the net flux through the closed surface is zero.

3.6 Gaussian Surfaces and Symmetry

The most useful Gaussian surfaces correspond to the symmetry of the physical charge distribution:

• spherical surfaces for spherical symmetry;
• cylindrical surfaces for cylindrical symmetry;
• pillbox-shaped surfaces for planar symmetry.

A suitable Gaussian surface allows the magnitude of E to be constant over relevant portions of the surface and makes the angle between E and dA simple.

3.7 Electric Field of a Spherically Symmetric Charge Distribution

Consider a point charge Q at the centre of a spherical Gaussian surface of radius r.

Because of spherical symmetry, the electric field has the same magnitude everywhere on the sphere and is radial. Thus,

Since the surface area of a sphere is 4πr²,

Hence,

This is the same result obtained from Coulomb's law.

Field of a Charged Conducting Sphere:

For a conducting sphere in electrostatic equilibrium, all excess charge resides on the outer surface. Therefore:

• inside the conducting material, E=0;
• outside the sphere, the field is the same as that of a point charge located at the centre.

Thus for r>R,

while for points inside the conductor,

3.8 Electric Field of an Infinite Line of Charge

Consider an infinitely long straight line carrying uniform linear charge density λ.

By cylindrical symmetry, the electric field points radially outward or inward and has constant magnitude at every point on a cylindrical Gaussian surface of radius r and length L.

Flux passes only through the curved surface because the field is parallel to the flat end surfaces.

The enclosed charge is

From Gauss's law,

Therefore,

3.9 Electric Field of an Infinite Plane Sheet of Charge

Consider an infinite plane sheet with uniform surface charge density σ. By symmetry, the electric field is perpendicular to the plane and has equal magnitude on both sides.

Choose a cylindrical pillbox Gaussian surface passing through the plane. Flux passes through the two flat ends but not through the curved side.

The enclosed charge is

Using Gauss's law,

Thus,

An important feature is that, for an ideal infinite sheet, the field magnitude does not depend on distance from the sheet.

3.10 Conductors in Electrostatic Equilibrium

Gauss's law helps explain important properties of conductors under electrostatic equilibrium.

Electric Field Inside a Conductor:

If a non-zero electric field existed inside the conducting material, free charges would continue to move. Therefore, at electrostatic equilibrium,

within the conductor.

Excess Charge Resides on the Surface:

Consider a Gaussian surface entirely within the conducting material. Since E=0 everywhere on the Gaussian surface,

Hence, by Gauss's law,

Thus there can be no net excess charge in the bulk of the conductor; excess charge resides on its surface.

Electric Field Just Outside a Conductor:

Immediately outside the surface of a conductor in electrostatic equilibrium, the field is perpendicular to the surface. For a surface charge density σ,

just outside the surface in vacuum, provided the local surface can be treated appropriately.

3.11 Electric Potential and Potential Difference

Electric field is a vector quantity. In many problems, a scalar quantity called electric potential provides a simpler way to describe electrostatic systems.

The SI unit is the volt:

For motion from point A to point B,

The negative sign indicates that electric potential decreases in the direction of the electric field.

3.12 Potential Due to a Point Charge

Taking the electric potential to be zero at infinity, the potential at distance r from a point charge Q is

Equivalently,

Unlike electric field, electric potential is a scalar. Therefore, the sign of Q directly determines the sign of the potential.

• A positive point charge produces positive potential.
• A negative point charge produces negative potential.

3.13 Electric Potential Due to Several Charges

Because electric potential is scalar, the total potential due to several point charges is obtained by ordinary algebraic addition:

This is often easier than adding electric field vectors.

3.14 Relationship Between Electric Field and Potential

The electric field points in the direction in which electric potential decreases most rapidly.

In one dimension,

In vector form,

For a uniform electric field along the x-direction,

when displacement is taken along the field direction.

3.15 Equipotential Surfaces

No net work is required to move a test charge along an equipotential surface because the potential difference is zero:

Hence,

Electric field lines intersect equipotential surfaces at right angles. If there were a component of electric field along the surface, a charge moving along that surface would experience work, contradicting the definition of an equipotential.

For a point charge, equipotential surfaces are concentric spheres centred on the charge.

3.16 Potential Energy of Electric Charges

The potential energy of a charge q placed at a point of electric potential V is

For two point charges,

The sign of the potential energy depends on the signs of the charges.

• Like charges give positive potential energy.
• Unlike charges give negative potential energy.

3.17 Applications of Electric Potential

Electric potential and potential difference are central to the behaviour of electrical systems. Potential difference provides the energy-per-unit-charge description that is used in batteries, circuits, capacitors and electric power systems.

A battery maintains a potential difference between its terminals. When connected in a complete circuit, electric charges can move through components and exchange energy.

Electric potential is also useful in accelerating charged particles. If a particle of charge q moves through a potential difference ΔV, the change in electric potential energy is

When electric potential energy decreases and no other energy-transfer mechanisms are important, kinetic energy increases.

Chapter Summary

4.1 Introduction to Electric Dipoles

An electric dipole is one of the simplest charge systems beyond a single point charge. It consists of equal positive and negative charges separated by a finite distance. Dipoles are important because many molecules and electrical systems possess dipole-like behaviour.

If the charges are +q and -q, separated by a displacement vector from the negative charge toward the positive charge, the system is characterized by its electric dipole moment.

4.2 Electric Dipole Moment

Its magnitude is

The SI unit of dipole moment is

The direction of p is from the negative charge toward the positive charge.

4.3 Electric Field of a Dipole

The electric field of a dipole is obtained by vector addition of the fields due to the positive and negative charges.

At points very far from the dipole compared with the charge separation, useful approximate expressions can be obtained.

Field on the Axial Line:

For a point far from the dipole on its axis,

The field is directed along the dipole moment on the side beyond the positive charge.

Field on the Equatorial Line:

For a point far from the dipole on the perpendicular bisector,

Its direction is opposite to the dipole moment.

4.4 Torque on a Dipole in an Electric Field

When an electric dipole is placed in a uniform electric field, the positive and negative charges experience forces equal in magnitude and opposite in direction.

The resultant force is zero in a perfectly uniform field, but the two forces form a couple that can rotate the dipole.

where θ is the angle between the dipole moment and the electric field.

In vector form,

The torque tends to rotate the dipole until its dipole moment aligns with the field.

4.5 Potential Energy of an Electric Dipole

The potential energy of a dipole in a uniform electric field is

Thus,

The potential energy is minimum when the dipole aligns with the field:

Then,

The potential energy is maximum when the dipole is opposite to the field:

giving

4.6 Introduction to Capacitance

A capacitor is an electrical device designed to store separated electric charge and electrical energy. It usually consists of two conductors separated by an insulating material or vacuum.

The SI unit of capacitance is the farad:

Common practical units include:

4.7 The Parallel-Plate Capacitor

A parallel-plate capacitor consists of two large conducting plates of area A separated by a small distance d.

If the plates carry equal and opposite surface charge densities +σ and -σ, the electric field between them, neglecting edge effects, is

Since

then

The potential difference between the plates is

Therefore,

Since C=Q/V,

4.8 Factors Affecting Capacitance

For a parallel-plate capacitor,

This expression shows that capacitance depends on geometry and the material between the plates.

Plate Area:

Capacitance is directly proportional to plate area:

Increasing the plate area increases the amount of charge that can be stored for a given potential difference.

Plate Separation:

Capacitance is inversely proportional to separation:

Reducing the separation increases capacitance.

Dielectric Material:

Introducing an insulating dielectric between the plates generally increases the capacitance. This effect will be examined later in the chapter.

4.9 Combination of Capacitors in Parallel

Capacitors are connected in parallel when both terminals of each capacitor are connected to the same pair of nodes.

Therefore, each capacitor has the same potential difference:

The total charge supplied is

Since Q_i=C_iV,

Thus,

4.10 Combination of Capacitors in Series

Capacitors are connected in series when they are joined end-to-end so that the same magnitude of charge appears on each capacitor.

The total potential difference is

Since

then

Therefore,

For two capacitors,

4.11 Energy Stored in a Capacitor

Work must be done to transfer charge from one capacitor plate to the other. This work is stored as electric potential energy.

The energy stored in a capacitor is

Using Q=CV, this becomes

Using V=Q/C, it may also be written as

4.12 Energy Density in an Electric Field

The energy stored by a capacitor may be regarded as energy stored in its electric field.

For a parallel-plate capacitor in vacuum,

and

Then,

Thus,

Since Ad is the volume between the plates,

The SI unit of energy density is J/m³.

4.13 Dielectrics

Examples include glass, mica, paper, ceramics, certain plastics and air under ordinary conditions.

Although free charges do not move through a dielectric as readily as they do in a conductor, the positive and negative charges within atoms or molecules can shift slightly under an external electric field.

The response of a dielectric is described in part by its relative permittivity, also called dielectric constant:

where ε is the permittivity of the material.

4.14 Polarisation of Dielectrics

When a dielectric is placed in an external electric field, bound positive and negative charges within the material shift slightly in opposite directions, or permanent molecular dipoles tend to align partially with the field.

This creates induced surface charges called bound charges. Their electric field generally opposes the applied electric field.

Polarisation therefore reduces the net electric field inside the dielectric compared with the field that would exist in vacuum for the same free-charge arrangement.

4.15 Capacitors with Dielectrics

If the space between the plates of a capacitor is completely filled with a dielectric of relative permittivity κ, the capacitance becomes

where C₀ is the capacitance without the dielectric.

For a parallel-plate capacitor,

Isolated Capacitor with Dielectric:

If a charged capacitor is disconnected from the battery before a dielectric is inserted, the charge remains constant.

Since C increases by a factor κ,

decreases by the same factor:

The electric field between the plates also decreases:

The stored energy becomes

so it decreases by a factor κ.

Battery-Connected Capacitor with Dielectric:

If the battery remains connected during insertion, the potential difference remains constant:

Because C increases,

also increases by a factor κ.

The stored energy

also increases by the factor κ.

4.16 Dielectric Strength and Breakdown

Every insulating material can withstand only a limited electric field before it becomes electrically conducting.

If the electric field exceeds this value, the material may ionise or otherwise lose its insulating behaviour. A sudden current can then flow through the material.

Air breakdown is responsible for sparks and contributes to lightning formation when the local electric field becomes sufficiently large.

In capacitor design, the applied voltage must remain safely below the breakdown voltage:

4.17 Applications of Capacitors and Dielectrics

Capacitors are widely used because they can store electric charge and energy, respond to changing voltages and influence the timing and frequency characteristics of electrical circuits.

Energy Storage:

Capacitors can store energy temporarily and release it rapidly. This principle is used in systems requiring short bursts of energy.

Electronic Timing:

Capacitors combined with resistors can produce characteristic charging and discharging times. These properties are important in timing and signal-processing circuits.

Filtering and Smoothing:

Capacitors are used in power supplies and electronic circuits to reduce unwanted voltage fluctuations and help produce smoother electrical signals.

Signal Coupling:

Capacitors can allow changing signals to pass between sections of a circuit while blocking steady direct-current components under appropriate operating conditions.

Sensing:

Capacitance changes when plate area, separation or dielectric properties change. This principle is used in capacitive touch sensors, proximity detectors and other measurement systems.

Chapter Summary

5.1 Introduction to Electric Current

Electrostatics deals primarily with electric charges at rest. Many practical electrical systems, however, depend on the controlled motion of charge. When electric charges move through a conductor or another medium, they constitute an electric current.

Electric current is central to the operation of electrical circuits, electronic devices, heating systems, batteries, motors and communication equipment. To analyse such systems quantitatively, we must understand current, potential difference, resistance, electromotive force, electrical power and the relationships among them.

5.2 Electric Current and Charge Flow

If a charge ΔQ passes through a cross-section in time Δt, the average current is

The instantaneous current is

The SI unit of current is the ampere.

5.3 Conventional Current and Electron Flow

The conventional direction of current is defined as the direction in which positive charge would move. This convention was established before the microscopic nature of electric conduction was fully understood.

In metallic conductors, the mobile charge carriers are primarily electrons. Since electrons are negatively charged, their average drift is opposite to the direction of conventional current.

Circuit analysis normally uses conventional current. Unless otherwise stated, all current directions in this course follow that convention.

5.4 Current Density and Drift Velocity

Current describes the total rate of charge flow through a conductor. Current density describes how that current is distributed over cross-sectional area.

For a uniform current distribution,

The SI unit is

In vector form, J points in the direction of conventional current.

Drift Velocity:

Charge carriers in a conductor already possess microscopic motion, but an applied electric field produces a small average directed velocity called the drift velocity.

For charge carriers of number density n, charge magnitude q, cross-sectional area A, and drift speed v_d,

Therefore,

5.5 Potential Difference and Electromotive Force

Potential difference describes energy transferred per unit charge between two points:

A source such as a cell or battery supplies energy to charges and maintains a potential difference between its terminals.

Although called an electromotive force, emf is not a force. It is an energy-per-charge quantity and is measured in volts.

5.6 Electrical Resistance

For a component with potential difference V and current I, resistance is defined under the relevant operating condition as

The SI unit of resistance is the ohm:

A larger resistance generally means that a smaller current flows for a given potential difference.

5.7 Ohm's Law

Hence,

An ohmic conductor has an approximately straight-line current-voltage characteristic through the origin over the range in which its resistance remains constant.

Non-Ohmic Devices:

Not all electrical devices obey Ohm's law. A device is non-ohmic when the ratio V/I is not constant over the operating range.

Examples include semiconductor diodes and filament lamps whose resistance changes significantly with operating conditions.

5.8 Resistivity and Conductivity

Resistance depends partly on the dimensions of a conductor, while resistivity characterizes the electrical nature of the material itself.

For a uniform conductor,

so

The SI unit of resistivity is

Its SI unit is siemens per metre:

5.9 Dependence of Resistance on Dimensions

For a uniform wire,

This relation reveals two important geometric dependences:

and

A longer wire has greater resistance because charge carriers must move through more material. A thicker wire has lower resistance because the larger cross-sectional area provides more conducting pathways.

5.10 Temperature Dependence of Resistance

The resistance of a material generally depends on temperature. For many metals over a moderate temperature range,

where R₀ is the resistance at reference temperature T₀ and α is the temperature coefficient of resistance.

For ordinary metallic conductors, resistance generally increases with increasing temperature because lattice vibrations increase and impede electron motion.

5.11 Resistors in Series

Resistors are in series when they are connected end-to-end so that the same current passes through each resistor.

The total potential difference is

Using V=IR,

Therefore,

5.12 Resistors in Parallel

Resistors are in parallel when their corresponding terminals are connected to the same pair of nodes. Each branch therefore has the same potential difference.

The total current is the sum of the branch currents:

Using Ohm's law,

Hence,

For two resistors,

5.13 Mixed Resistor Networks

Many practical circuits contain combinations of series and parallel resistors. Such circuits are simplified in stages by identifying groups that are unmistakably in series or parallel.

5.14 Electrical Power

If a charge Q moves through potential difference V, the energy transferred is

Since I=Q/t,

For a resistor obeying Ohm's law,

5.15 Electrical Energy and Heating Effect

When current flows through a resistance, electrical energy is commonly converted into thermal energy. For a constant current,

Therefore,

For a resistor,

or

This conversion of electrical energy into thermal energy is known as Joule heating.

Kilowatt-Hour:

Electrical energy supplied to homes and industries is often measured in kilowatt-hours rather than joules.

5.16 Cells, Batteries and Internal Resistance

A real cell or battery does not behave exactly like an ideal source. The materials within the source oppose current flow and can be represented by an internal resistance.

A practical cell may be modelled as an ideal emf ε in series with internal resistance r.

If the external load resistance is R, the total circuit resistance is

The current is therefore

5.17 Terminal Potential Difference

When a cell supplies current, some potential is lost across its internal resistance. The potential difference measured across its external terminals is therefore

This quantity is called the terminal potential difference.

Since the external load has resistance R,

Thus,

5.18 Measuring Current and Voltage

Ammeter:

An ammeter measures current and is connected in series with the component whose current is required.

An ideal ammeter has zero resistance so that its insertion does not significantly alter the circuit current.

Voltmeter:

A voltmeter measures potential difference and is connected in parallel across the two points of interest.

An ideal voltmeter has infinite resistance so that it draws no current from the circuit.

5.19 Applications of DC Circuits

Direct-current circuits are fundamental in battery-powered electronics, automotive systems, control circuits, portable devices and many laboratory instruments.

Understanding current, resistance and power allows engineers and scientists to select conductors appropriately, control current levels, divide voltages, estimate energy consumption and protect equipment against excessive heating.

Chapter Summary

6.1 Introduction to Circuit Analysis

Simple circuits can often be reduced by combining resistors in series and parallel. More complicated networks cannot always be simplified by these methods alone. In such cases, a systematic analysis can be performed using Kirchhoff's laws.

Kirchhoff's circuit laws are direct consequences of two fundamental conservation principles. The junction rule follows from conservation of electric charge, while the loop rule follows from conservation of energy.

These laws make it possible to determine unknown currents and potential differences in circuits containing several branches, resistors and sources.

6.2 Nodes, Branches and Loops

Before applying Kirchhoff's laws, three circuit terms should be clearly understood.

Correct identification of branches and independent loops is essential in multi-loop circuit analysis.

6.3 Kirchhoff's Junction Rule

Equivalently, if currents entering a junction are treated as positive and currents leaving as negative,

6.4 Conservation of Charge in Circuits

The junction rule follows from conservation of electric charge. In a steady-state circuit, charge does not continuously accumulate at an ordinary junction. Therefore, whatever amount of charge arrives per unit time must leave per unit time.

Since current is charge flow rate,

the conservation of charge requires that the algebraic sum of currents at a junction be zero.

6.5 Kirchhoff's Loop Rule

As one moves around a loop, the potential may rise across a source and fall across resistive elements. After returning to the starting point, the net change in electric potential must be zero.

6.6 Conservation of Energy in Circuits

The loop rule is a consequence of energy conservation. A charge returning to its original point after travelling around a complete circuit loop must return to the same electric potential.

A source supplies energy per unit charge, while resistors convert electrical energy into thermal energy or other forms. Thus, in a simple loop,

For a source of emf ε and a resistor R carrying current I,

Therefore,

for an ideal source and single external resistor.

6.7 Sign Conventions for Loop Equations

Correct signs are essential when applying Kirchhoff's loop rule. The following conventions provide a consistent method.

Moving Through a Resistor:

If the chosen loop direction moves through a resistor in the same direction as the assumed current, the potential change is

If the loop direction moves through the resistor opposite to the assumed current, the potential change is

Moving Through a Source:

Moving from the negative terminal to the positive terminal of an ideal source gives a potential rise:

Moving from the positive terminal to the negative terminal gives a potential fall:

6.8 Single-Loop Circuits

Consider a simple loop containing an ideal source ε and two series resistors R₁ and R₂.

Applying the loop rule,

Therefore,

6.9 Multi-Loop Circuit Analysis

A multi-loop circuit usually contains more than one source or more than one branch, making simple series-parallel reduction insufficient.

A systematic approach is:

• assign current variables to the required branches;
• choose convenient current directions;
• apply the junction rule where branches meet;
• select enough independent loops;
• write a loop equation for each independent loop;
• solve the resulting simultaneous equations.

The number of independent equations must equal the number of unknown currents.

6.10 Two-Loop Networks

Consider a circuit with two loops sharing a resistor. Let current I₁ circulate in the left loop and current I₂ circulate in the right loop. If both mesh currents pass through the shared resistor in opposite directions, the actual current through that resistor is the difference between the mesh currents.

for the chosen orientation.

The current in the shared resistor is

6.11 Multiple Sources in a Circuit

When more than one source appears in a loop, the algebraic contribution of each emf depends on the direction in which the loop passes through that source.

6.12 Cells Connected in Series

Cells may be connected in series to increase the total emf available to a circuit.

For n identical cells, each of emf ε and internal resistance r, connected in series aiding,

and

If connected to an external resistance R, the current is

6.13 Cells Connected in Parallel

Identical cells connected in parallel have the same equivalent emf as one cell, but their equivalent internal resistance is reduced.

For n identical cells,

and

This arrangement can be advantageous when supplying a relatively large current to a low-resistance load.

6.14 Potential Dividers

A potential divider uses resistors in series to obtain a desired fraction of a supply voltage.

For two resistors R₁ and R₂ in series across a supply V, the current is

The voltage across R₂ is

Therefore,

6.15 Current Division

In a parallel network, total current divides among branches. The branch with lower resistance carries the larger current.

For two parallel resistors,

Because both branches have the same voltage,

Thus,

and

6.16 Wheatstone Bridge

The Wheatstone bridge is a circuit arrangement used for comparing resistances and determining an unknown resistance.

A basic bridge contains four resistors arranged in a diamond-shaped network and a detector connected between two intermediate junctions.

At balance, no current flows through the detector. If the four arms have resistances R₁, R₂, R₃ and R₄, the balance condition is

Equivalently,

6.17 RC Circuits

A circuit containing both a resistor and capacitor is called an RC circuit. Unlike a purely resistive DC circuit, an RC circuit exhibits time-dependent behaviour while the capacitor charges or discharges.

When an initially uncharged capacitor is connected to a DC source through a resistor, current initially flows strongly. As charge accumulates on the capacitor, its voltage increases and the current gradually decreases.

When a charged capacitor is disconnected from its source and allowed to discharge through a resistor, both the capacitor charge and current decrease exponentially with time.

6.18 Charging of a Capacitor

Consider a series circuit containing a source of emf ε, a resistor R and a capacitor C.

Kirchhoff's loop rule gives

Since

the charging equation is

Its solution for an initially uncharged capacitor is

Since the final charge is

the charging equation may be written as

The capacitor voltage is

The charging current is

At t=0,

while after a very long time,

and

6.19 Discharging of a Capacitor

Suppose a capacitor initially carries charge Q₀ and is then connected across a resistor with no external source.

The charge decreases according to

The capacitor voltage is

The magnitude of the discharge current is

where

6.20 Time Constant

The SI unit of RC is the second.

During charging, after one time constant,

Thus the capacitor has reached approximately 63.2% of its final charge.

During discharging,

Thus approximately 36.8% of the initial charge remains after one time constant.

6.21 Power and Energy in DC Networks

Kirchhoff's laws can also be checked through energy conservation. In a steady DC network, the total power supplied by sources equals the total power absorbed or dissipated by circuit elements, provided all energy transfers are included.

For a resistor,

For a source delivering current out of its positive terminal, the rate at which it supplies electrical energy is

6.22 Practical Circuit Analysis

Successful circuit analysis requires more than memorising formulas. A circuit should first be examined to determine its actual topology: which elements share the same current, which elements share the same voltage, and which nodes and independent loops are present.

For complex DC circuits, a reliable approach is to identify known quantities, assign current directions, simplify any obvious series or parallel groups, and then apply Kirchhoff's laws to the remaining network.

Chapter Summary

7.1 Introduction to Magnetism

Magnetism is closely connected with moving electric charges. A stationary electric charge produces an electric field, whereas a moving charge produces both electric and magnetic effects. Electric currents, which consist of moving charges, therefore generate magnetic fields.

Magnetic phenomena occur in permanent magnets, current-carrying conductors, electric motors, generators, transformers, loudspeakers, particle accelerators and many other devices. At the microscopic level, magnetic behaviour in matter is associated principally with the magnetic moments of electrons arising from their orbital motion and intrinsic spin.

The central quantity used to describe magnetic interactions is the magnetic field, represented by the vector B.

7.2 Magnetic Field

The SI unit of magnetic field is the tesla (T). From the magnetic force relation,

one tesla may be expressed as

Because magnetic field is a vector quantity, both magnitude and direction must be specified.

7.3 Magnetic Field Lines

Magnetic field lines provide a graphical representation of a magnetic field. The tangent to a field line at any point gives the direction of B at that point, while closer spacing of the lines represents a stronger field.

Outside a bar magnet, magnetic field lines are conventionally drawn from the north pole toward the south pole. Within the magnet, they continue from south to north, forming continuous closed loops.

Unlike electric field lines, magnetic field lines do not begin or terminate on isolated magnetic charges. No isolated magnetic monopole has been experimentally established in ordinary classical electromagnetism.

7.4 Magnetic Flux

For a uniform magnetic field passing through a flat surface,

where θ is the angle between the magnetic field and the normal to the surface.

More generally,

The SI unit of magnetic flux is the weber:

7.5 Magnetic Force on a Moving Charge

A charged particle moving through a magnetic field experiences a force when its velocity has a component perpendicular to the field.

The vector expression is

The magnitude is

Here θ is the angle between the velocity and magnetic field vectors.

Three important cases follow immediately:

Direction of Magnetic Force:

For a positive charge, the direction of v×B is determined using the right-hand rule. Point the fingers of the right hand in the direction of the velocity and curl them toward the magnetic field; the thumb indicates the direction of force for a positive charge.

For a negative charge, the force is opposite to the direction obtained from the right-hand rule.

7.6 Lorentz Force

A charged particle may simultaneously experience electric and magnetic fields. The total electromagnetic force is called the Lorentz force.

The electric part, qE, can act on a charge whether it is moving or stationary. The magnetic part requires motion and depends on the orientation of the velocity relative to the magnetic field.

The electric force can do work on a charged particle and change its speed. A purely magnetic force is always perpendicular to the instantaneous velocity and therefore does no work on the particle.

Consequently, a magnetic field alone can change the direction of a particle's velocity but not its kinetic energy or speed.

7.7 Motion of Charged Particles in Magnetic Fields

The path of a charged particle in a magnetic field depends on the orientation of its velocity.

If the velocity is parallel to the magnetic field, no magnetic force acts and the particle continues in a straight line at constant speed, provided no other force is present.

If the velocity is perpendicular to the magnetic field, the magnetic force is always perpendicular to the motion and acts as a centripetal force. The particle therefore follows a circular path.

If the velocity has both parallel and perpendicular components, the perpendicular component produces circular motion while the parallel component remains unchanged. The resulting trajectory is a helix.

7.8 Circular Motion of Charged Particles

For a charged particle moving perpendicular to a uniform magnetic field,

The centripetal force required for circular motion is

Equating the two forces,

Therefore,

The radius increases with mass and speed but decreases as the charge magnitude or magnetic field strength increases.

Period and Frequency:

The period of the circular motion is

Substituting r=mv/(|q|B),

The frequency is therefore

In the non-relativistic approximation, the period and frequency are independent of the particle's speed.

7.9 Velocity Selector

A velocity selector uses mutually perpendicular electric and magnetic fields to allow charged particles of a particular speed to pass through without deflection.

For the selected particle, the electric and magnetic forces have equal magnitudes and opposite directions:

Hence,

7.10 Magnetic Force on a Current-Carrying Conductor

A current-carrying wire placed in a magnetic field experiences a force because the moving charge carriers within the wire experience magnetic forces.

For a straight conductor of vector length L,

The magnitude is

Here θ is the angle between the direction of conventional current and the magnetic field.

7.11 Force Between Parallel Current-Carrying Conductors

A current in a wire produces a magnetic field. Consequently, when two parallel wires carry currents, each wire lies in the magnetic field produced by the other and experiences a magnetic force.

For two long, straight parallel conductors separated by distance d, the force per unit length is

where

Parallel currents in the same direction attract, whereas currents in opposite directions repel.

7.12 Torque on a Current Loop

A current loop placed in a uniform magnetic field can experience a torque. This principle is fundamental to electric motors and moving-coil instruments.

For a rectangular coil containing N turns, each of area A, carrying current I,

where θ is the angle between the normal to the plane of the coil and the magnetic field.

The torque is maximum when θ=90° and zero when the magnetic dipole moment is parallel or antiparallel to the field.

7.13 Magnetic Dipole Moment

For a coil of N turns,

and its magnitude is

The direction of μ is perpendicular to the plane of the loop and is determined by the right-hand rule.

The torque may therefore be written compactly as

7.14 Magnetic Potential Energy

A magnetic dipole in an external magnetic field possesses orientation-dependent potential energy:

For a uniform field,

The potential energy is minimum when the magnetic dipole moment is aligned with the field and maximum when it is oppositely directed.

7.15 Applications of Magnetic Forces

The principles developed in this chapter have numerous applications. Electric motors use torque on current-carrying coils to produce rotational motion. Charged-particle beams are controlled by magnetic fields in accelerators and analytical instruments. Velocity selectors separate particles according to speed, while related arrangements allow mass-to-charge ratios to be measured.

Magnetic forces also underlie the operation of loudspeakers, moving-coil meters and many electromechanical systems. The important point is that magnetic fields interact with moving charges and currents in predictable ways governed by vector relationships.

Chapter Summary

8.1 Magnetic Fields Produced by Electric Currents

The preceding chapter considered how magnetic fields exert forces on moving charges and currents. We now examine the complementary question: how are magnetic fields produced?

An electric current produces a magnetic field in the surrounding space. Since electric current is an organized flow of charge, this observation establishes the fundamental connection between moving electric charges and magnetism.

Two important laws are used to calculate magnetic fields generated by steady currents: the Biot-Savart law and Ampère's circuital law. The Biot-Savart law is generally applicable to steady-current distributions, whereas Ampère's law is especially convenient when the geometry possesses sufficient symmetry.

8.2 Biot-Savart Law

In vector form,

The magnitude is

where:

• I is the current;
• dℓ is the length of the current element;
• r is the distance from the current element to the observation point;
• θ is the angle between dℓ and the displacement toward the observation point;
• r̂ is a unit vector directed from the current element to the observation point.

The total field is obtained by integrating over the complete current distribution:

The direction of the field contribution follows from the cross product and can be found using the right-hand rule.

8.3 Magnetic Field of a Long Straight Conductor

For a very long straight wire carrying current I, integration of the Biot-Savart law gives the magnetic field at perpendicular distance r:

The field lines are concentric circles centred on the wire.

Direction of the Field:

The direction is obtained using the right-hand grip rule. Point the thumb of the right hand in the direction of conventional current; the curled fingers show the direction of the magnetic field around the wire.

8.4 Magnetic Field of a Circular Current Loop

A circular loop carrying current produces a magnetic field whose direction along the axis of the loop follows the right-hand rule.

At the centre of a circular loop of radius R, the Biot-Savart law gives

For a closely wound coil containing N turns,

8.5 Magnetic Field on the Axis of a Circular Loop

The magnetic field at a point on the axis of a circular loop is also obtained from the Biot-Savart law.

For a loop of radius R, carrying current I, at an axial distance x from the centre,

For N closely wound turns,

At the centre, x=0, and the expression reduces to

8.6 Magnetic Field of a Current Arc

For a circular arc of radius R carrying current I and subtending an angle φ in radians at its centre, the magnetic field at the centre is

For a complete circle, φ=2π, giving

as expected.

8.7 Superposition of Magnetic Fields

When several currents contribute to the magnetic field at the same point, the resultant field is the vector sum of the individual fields:

The directions must be considered carefully. Fields in the same direction add; fields in opposite directions subtract. For non-collinear fields, vector addition is required.

8.8 Ampère's Circuital Law

Ampère's law is always valid for magnetostatic situations, but it is most useful computationally when symmetry allows the magnitude of B to be treated as constant along a suitable integration path.

Common high-symmetry applications include:

• a long straight conductor;
• a long solenoid;
• a toroid;
• highly symmetric current distributions.

8.9 Ampère's Law for a Long Straight Wire

Consider a long straight wire carrying current I. Choose a circular Amperian path of radius r centred on the wire.

By symmetry, B has the same magnitude everywhere on this circle and is tangent to the path. Therefore,

Since the circumference is 2πr,

Hence,

This agrees with the result obtained using the Biot-Savart law.

8.10 Magnetic Field Inside a Long Solenoid

For an ideal long solenoid, the magnetic field inside is approximately uniform and parallel to the axis, while the external field is comparatively weak.

If the solenoid contains N turns over length L, the number of turns per unit length is

Ampère's law gives the field inside an ideal long solenoid as

Equivalently,

8.11 Magnetic Field of a Toroid

For an ideal toroid containing N turns carrying current I, choose a circular Amperian path of radius r inside the windings.

Ampère's law gives

Therefore,

Unlike the ideal long solenoid, the magnetic field inside a toroid varies with radial distance r.

8.12 Magnetic Dipoles and Current Loops

A small current loop behaves as a magnetic dipole. Its magnetic dipole moment is

This provides a useful bridge between current-generated magnetic fields and the behaviour of magnetic materials. At distances much larger than the dimensions of the loop, the field pattern resembles that of a small bar magnet.

The magnetic dipole model is also useful in describing microscopic magnetic effects in matter, although atomic magnetism ultimately requires quantum mechanics for a complete description.

8.13 Magnetic Properties of Matter

Materials respond differently when placed in external magnetic fields. Their behaviour depends on atomic and molecular magnetic moments and on how these moments respond collectively to an applied field.

At introductory level, magnetic materials are commonly classified as:

• diamagnetic;
• paramagnetic;
• ferromagnetic.

These categories describe substantially different responses to an applied magnetic field.

8.14 Diamagnetic, Paramagnetic and Ferromagnetic Materials

Diamagnetic Materials:

Diamagnetic materials develop an induced magnetic response opposite to the applied field and are therefore weakly repelled from regions of stronger magnetic field.

Examples include bismuth, copper, silver and water.

Paramagnetic Materials:

Paramagnetic substances possess microscopic magnetic moments that tend to align partially with an applied magnetic field. Thermal agitation prevents complete alignment under ordinary conditions, so the attraction is usually weak.

Examples include aluminium and oxygen.

Ferromagnetic Materials:

Ferromagnetic materials exhibit strong collective alignment of magnetic moments. They can become strongly magnetized and may retain significant magnetization after the external field is removed.

Iron, cobalt and nickel are familiar examples.

8.15 Magnetic Energy Density

Energy can be stored in a magnetic field. In vacuum, the magnetic energy density is

This quantity gives the magnetic energy stored per unit volume of space.

8.16 Applications of Current-Produced Magnetic Fields

Current-generated magnetic fields form the operating basis of electromagnets, relays, motors, magnetic actuators, inductors, transformers and numerous measurement instruments.

A solenoid provides a useful means of producing an approximately uniform magnetic field within a cylindrical region. Introducing a suitable magnetic core can greatly modify and often strengthen the field. Toroidal windings are valuable where magnetic flux is intended to remain largely confined to a closed magnetic path.

The Biot-Savart law and Ampère's law are complementary rather than competing laws. The Biot-Savart law is particularly useful when the detailed contribution from current elements must be integrated, whereas Ampère's law provides a much shorter route for highly symmetric current distributions.

Chapter Summary

9.1 Introduction to Electromagnetic Induction

A magnetic field can exert forces on moving charges, but the relationship between electricity and magnetism goes further. A changing magnetic environment can produce an electric effect even when no battery or electrostatic source is present. This phenomenon is known as electromagnetic induction.

The induced electromotive force may result from a changing magnetic field, motion of a conductor through a magnetic field, rotation of a coil, or any process that changes the magnetic flux linkage.

Electromagnetic induction is the operating principle behind electric generators, transformers, induction cookers, many sensors, wireless charging systems and numerous electrical machines.

9.2 Magnetic Flux and Flux Linkage

Magnetic flux through a surface was introduced earlier as

For a uniform magnetic field passing through a flat surface,

where θ is the angle between the magnetic field and the normal to the surface.

For a coil containing N turns, each turn links magnetic flux Φ_B. The total flux linkage is

9.3 Faraday's Law of Electromagnetic Induction

For a coil of N turns,

For an average change over a finite time interval,

The negative sign represents Lenz's law and determines the direction of the induced emf.

Ways of Changing Magnetic Flux:

Since

magnetic flux can change by changing:

• the magnetic field strength B;
• the area A enclosed by the circuit;
• the angle θ between the field and the area vector;
• the position of the coil in a non-uniform magnetic field.

9.4 Lenz's Law

The word change is essential. The induced field does not necessarily oppose the original magnetic field. It opposes the increase or decrease in magnetic flux.

For example, if magnetic flux directed into a loop increases, the induced current produces a field directed out of the loop. If the inward flux decreases, the induced current produces an inward field in an attempt to maintain the original flux.

9.5 Direction of Induced Current

The direction of induced current can be found systematically:

• determine the direction of the original magnetic flux;
• determine whether that flux is increasing or decreasing;
• choose the direction of the induced magnetic field required to oppose the change;
• use the right-hand grip rule to determine the corresponding current direction.

9.6 Motional Electromotive Force

An emf may be induced when a conductor moves through a magnetic field. The charge carriers within the moving conductor experience the magnetic force

Positive and negative charges are driven toward opposite ends of the conductor, producing charge separation and therefore a potential difference.

For a straight rod of length ℓ moving with speed v perpendicular both to its length and to a uniform magnetic field B,

More generally, if the relevant angle is θ,

9.7 Moving Conducting Rod in a Magnetic Field

Consider a conducting rod sliding along conducting rails in a uniform magnetic field perpendicular to the plane of the rails. If the rails form a closed circuit with total resistance R, the induced current is

Using ε=Bℓv,

The induced current in the rod experiences a magnetic force

Substituting for I,

The force acts opposite to the motion, in agreement with Lenz's law.

9.8 Induced Electric Fields

Faraday's law has a deeper interpretation than the production of current in wires. A changing magnetic field produces an electric field in the surrounding space.

The electric field produced by changing magnetic flux is generally non-conservative and forms closed loops. Faraday's law may therefore be written as

This equation shows that a changing magnetic field can create an electric field even where no conducting wire is present.

This idea becomes one of the fundamental components of Maxwell's equations.

9.9 Rotating Coil and Alternating Emf

Consider a coil of N turns, each of area A, rotating with constant angular speed ω in a uniform magnetic field B.

If the angle between the field and the area vector is

then the flux through one turn is

The total flux linkage is

From Faraday's law,

Therefore,

The maximum emf is

Thus,

This is a sinusoidally alternating emf.

9.10 Electric Generator Principle

In a simple generator, a coil rotates in a magnetic field. Because the magnetic flux through the coil changes continuously, an emf is induced.

A basic alternating-current generator produces an emf of the form

Mechanical work must be supplied to keep the coil rotating because the induced current produces magnetic effects that oppose the rotation. This is an application of Lenz's law and energy conservation.

9.11 Eddy Currents

Eddy currents are not confined to a single wire path. They circulate through the conducting material itself.

Like all induced currents, their direction is determined by Lenz's law. They produce magnetic fields that oppose the change responsible for their creation.

9.12 Applications of Eddy Currents

Eddy currents may be useful or undesirable depending on the application.

Induction Heating:

Alternating magnetic fields induce strong eddy currents in conducting materials. The electrical resistance of the material converts the induced electrical energy into thermal energy.

Electromagnetic Damping:

Eddy currents can oppose the motion of conducting components and reduce unwanted oscillations in measuring instruments and mechanical systems.

Metal Detection:

Changing magnetic fields induce currents in nearby conductive objects. The magnetic fields produced by these currents can be detected electronically.

Energy Losses:

In transformer and machine cores, eddy currents can produce unwanted heating. These losses can be reduced by constructing cores from thin insulated laminations rather than solid blocks.

9.13 Electromagnetic Braking

Electromagnetic braking is a practical application of Lenz's law. When a conducting disc or plate moves through a magnetic field, eddy currents are induced. Their magnetic interaction with the applied field produces a force opposing the motion.

Because there is no requirement for mechanical contact between rubbing surfaces, electromagnetic braking can reduce mechanical wear.

However, the lost mechanical energy is converted primarily into thermal energy in the conductor.

9.14 Energy Conservation in Electromagnetic Induction

When a conductor moves through a magnetic field and an induced current flows, the induced magnetic force opposes the motion. External work must therefore be done to maintain constant speed.

For a rod of length ℓ moving with speed v through magnetic field B in a circuit of resistance R,

and

The magnetic force opposing the motion is

Therefore,

The mechanical power required to maintain the motion is

Hence,

Electrical power dissipated is

Substituting I=Bℓv/R,

Thus,

The mechanical energy supplied is converted into electrical and ultimately thermal energy.

9.15 Practical Applications of Electromagnetic Induction

Electromagnetic induction is central to modern electrical technology. Generators convert mechanical energy to electrical energy. Transformers use changing magnetic flux to transfer electrical energy between circuits. Induction cookers produce heating directly in conducting cookware. Dynamic microphones and magnetic pickups convert mechanical vibrations into electrical signals.

Inductive sensors detect position, speed, metal objects and changes in magnetic environments. Wireless power transfer systems use changing magnetic fields to transfer energy between nearby coils without direct electrical contact.

Chapter Summary

10.1 Introduction to Inductance

When current flows through a circuit, it produces a magnetic field. If the current changes, the magnetic field changes, and therefore the magnetic flux linked with the circuit also changes. According to Faraday's law, this changing flux produces an induced emf.

This effect gives rise to the important property known as inductance. Inductance describes how strongly a circuit responds electromagnetically to changes in current.

10.2 Self-Induction

Suppose current through a coil increases. The increasing current strengthens the coil's magnetic field and increases the magnetic flux through its turns. The resulting induced emf acts in a direction that opposes the increase in current.

If the current decreases, the induced emf acts in a direction that tends to maintain the current.

For this reason, an inductor tends to resist sudden changes in current.

10.3 Self-Inductance

For a coil in which the magnetic flux linkage is proportional to current,

where L is the self-inductance.

Therefore,

The SI unit of inductance is the henry (H).

From the induced-emf relation,

10.4 Inductance of a Solenoid

For a long air-core solenoid of length ℓ, cross-sectional area A, and N turns, the magnetic field inside is approximately

The magnetic flux through one turn is

Therefore,

The flux linkage is

Since NΦ_B=LI,

If a magnetic material is present and behaves approximately linearly, μ₀ is replaced by the permeability μ of the material.

10.5 Induced Emf in an Inductor

From Faraday's law and the definition of inductance,

The induced emf is proportional to the rate of change of current, not simply to the current itself.

A rapidly changing current therefore produces a larger self-induced emf than a slowly changing current.

10.6 RL Circuits

An RL circuit contains resistance R and inductance L. When such a circuit is connected to a DC source, current does not immediately reach its final steady value because the inductor produces a self-induced emf opposing the increase.

For a series RL circuit connected to a source of emf ε, Kirchhoff's loop rule gives

Equivalently,

This differential equation describes the growth of current.

10.7 Growth of Current in an RL Circuit

If the current is initially zero and the switch is closed at t=0, the current rises according to

where

and

Immediately after the switch is closed, the current is zero because the inductor initially opposes the sudden current change.

After a long time, dI/dt approaches zero, so the ideal inductor behaves like a conducting wire in steady DC operation and the current approaches

10.8 Decay of Current in an RL Circuit

When the source is removed from an RL circuit while a conducting path remains, the inductor attempts to maintain the existing current.

The current decays exponentially according to

where

10.9 RL Time Constant

Its SI unit is seconds.

During current growth, after one time constant,

During decay, after one time constant,

10.10 Energy Stored in an Inductor

An inductor stores energy in its magnetic field while current flows through it.

The energy stored in an inductor is

The energy is zero when the current is zero and increases with the square of the current.

10.11 Energy Density in a Magnetic Field

The energy stored in an inductor may be regarded as energy stored throughout its magnetic field.

For a magnetic field in vacuum, the energy density is

This expression was introduced earlier and now receives a direct interpretation through the energy stored in an inductor.

For a linear magnetic medium of permeability μ, a corresponding expression is

10.12 Mutual Induction

A changing current in one circuit can induce an emf in a nearby second circuit because the changing current produces a changing magnetic field and therefore changing magnetic flux through the second circuit.

If changing current I₁ in circuit 1 produces flux linkage in circuit 2, the induced emf in circuit 2 is

where M is the mutual inductance.

10.13 Mutual Inductance

If current I₁ produces flux Φ₂₁ through each of N₂ turns of a second coil, then

The SI unit of mutual inductance is also the henry.

10.14 The Transformer Principle

A basic transformer consists of two coils wound around a common magnetic core:

• the primary coil, connected to the input supply;
• the secondary coil, connected to the output circuit.

Alternating current in the primary produces a changing magnetic flux in the core. This changing flux links the secondary coil and induces an alternating emf according to Faraday's law.

10.15 Ideal Transformer Equations

For an ideal transformer, assume that all magnetic flux produced by the primary links the secondary and that there are no energy losses.

The rms or instantaneous voltage ratio follows the turns ratio:

Equivalently,

Here:

• V_p = primary voltage;
• V_s = secondary voltage;
• N_p = primary turns;
• N_s = secondary turns.

10.16 Step-Up Transformers

For a step-up transformer,

and therefore

In an ideal transformer, increasing voltage is accompanied by a corresponding decrease in current.

10.17 Step-Down Transformers

For a step-down transformer,

and therefore

10.18 Current and Power Relations in Transformers

For an ideal transformer, input power equals output power:

Therefore,

Combining this with the turns ratio gives

Thus voltage and current ratios are reciprocal in an ideal transformer.

10.19 Transformer Efficiency and Energy Losses

Real transformers are not perfectly efficient. Their efficiency is

Since AC electrical power depends on the nature of the load, a simple resistive-load expression is

Important transformer losses include:

Copper Losses:

The windings possess electrical resistance and dissipate energy at a rate approximately

Eddy-Current Losses:

Changing magnetic flux induces currents in the iron core. Laminating the core into insulated sheets reduces the available paths for large circulating currents.

Hysteresis Losses:

Repeated magnetization and demagnetization of magnetic core material dissipates energy. Appropriate magnetic materials are chosen to reduce this loss.

Flux Leakage:

Not all magnetic flux produced by the primary necessarily links the secondary. Good core design improves magnetic coupling between the coils.

Transformers in Power Transmission:

Electrical power transmitted over long distances may be written approximately as

For a fixed transmitted power, increasing voltage allows current to be reduced:

Power lost as heat in transmission lines is

Reducing current therefore greatly reduces line losses. This is why transformers are used to step voltage up for long-distance transmission and step it down again for distribution and use.

10.20 Applications of Inductors and Transformers

Inductors are used whenever circuits must oppose rapid current changes, store magnetic energy or interact with alternating signals. They occur in filters, switching power supplies, electromagnetic actuators, radio-frequency circuits and energy-storage systems.

Transformers are essential in power transmission, electronic power supplies, electrical isolation systems, impedance matching and signal coupling. Their operation is one of the clearest practical demonstrations of Faraday's law and mutual induction.

Inductance also forms the magnetic counterpart to capacitance. A capacitor stores energy in an electric field, while an inductor stores energy in a magnetic field. This parallel becomes especially important in the study of electromagnetic oscillations and alternating-current circuits.

Chapter Summary

11.1 From Electricity and Magnetism to Electromagnetism

The study of electricity and magnetism begins with phenomena that may initially appear separate. Electric charges produce electric fields, electric currents produce magnetic fields, magnetic fields exert forces on moving charges, and changing magnetic fields produce electric fields. When these ideas are considered together, they form a unified physical theory known as electromagnetism.

The mathematical framework that brings the principal classical laws of electricity and magnetism together is expressed through Maxwell's equations. These equations describe how electric charges and currents produce electromagnetic fields and how changing electric and magnetic fields are related to one another.

One of the most important consequences of Maxwell's theory is that changing electric and magnetic fields can sustain each other and propagate through space as waves. These are electromagnetic waves. Visible light is one example, but radio waves, microwaves, infrared radiation, ultraviolet radiation, X-rays and gamma rays belong to the same physical family.

11.2 Maxwell's Equations: An Overview

In vacuum, their integral forms may be written as follows:

These four relations contain much of the classical theory of electric and magnetic fields. The purpose at this level is not to develop their full mathematical treatment, but to understand their physical content and use their simpler consequences.

11.3 Gauss's Law for Electricity

Gauss's law relates the electric flux through a closed surface to the net electric charge enclosed by that surface:

The quantity on the left is the total electric flux through a closed surface, commonly called a Gaussian surface. The quantity Q_enc is the algebraic sum of the charges enclosed by the surface.

The law shows that electric charge is a source of electric field. Positive charges produce outward electric flux, whereas negative charges produce inward electric flux.

11.4 Gauss's Law for Magnetism

The magnetic counterpart of Gauss's law states that

The net magnetic flux through every closed surface is therefore zero.

This result expresses a fundamental difference between electric and magnetic fields. Electric field lines may originate on positive charges and terminate on negative charges. Magnetic field lines, by contrast, form continuous closed loops.

If a closed surface surrounds part of a magnet, every magnetic field line entering the surface eventually leaves it. The inward and outward contributions therefore give zero net magnetic flux.

11.5 Faraday's Law in Integral Form

Faraday's law states that a changing magnetic flux produces a circulating electric field:

The left side represents the circulation of the electric field around a closed path. The right side describes the rate at which magnetic flux through the surface bounded by that path changes.

This formulation extends the concept of electromagnetic induction beyond conducting wires. A changing magnetic field creates an electric field in space even if no conductor is present.

The negative sign expresses Lenz's law: the induced effect is directed so as to oppose the change in magnetic flux responsible for it.

11.6 Ampère-Maxwell Law

Ampère's circuital law for steady currents was introduced as

For situations involving changing electric fields, this relation requires an additional term. The more general Ampère-Maxwell law is

The first term represents the magnetic field associated with ordinary conduction current. The second term represents the magnetic effect of a changing electric flux.

Maxwell's addition of the changing-electric-flux term is crucial because it establishes a symmetry between changing electric and magnetic fields: a changing magnetic field produces an electric field, and a changing electric field produces a magnetic field.

11.7 Displacement Current

The quantity

is called the displacement current.

It is important to understand that displacement current does not necessarily represent physical charges crossing the region in question. A familiar example occurs between the plates of a charging capacitor.

Conduction current flows through the connecting wires, but charges do not cross the dielectric gap between the capacitor plates. Nevertheless, the electric field between the plates changes as the capacitor charges. This changing electric field contributes to the magnetic field through the displacement-current term.

11.8 Physical Interpretation of Maxwell's Equations

Maxwell's equations may be understood conceptually through four statements.

The last two relations are especially important for electromagnetic waves. They show that time-varying electric and magnetic fields are dynamically coupled.

11.9 Production of Electromagnetic Waves

A stationary charge produces an electrostatic field. A charge moving steadily produces electric and magnetic fields, but electromagnetic radiation is associated with accelerating charges.

Suppose charges oscillate back and forth in a transmitting antenna. Their changing motion produces time-varying electric and magnetic fields. A changing electric field contributes to a magnetic field through the Ampère-Maxwell law, while a changing magnetic field produces an electric field through Faraday's law.

The coupled disturbance can detach from the source region and propagate through space. This propagating combination of electric and magnetic fields is an electromagnetic wave.

11.10 Properties of Electromagnetic Waves

Electromagnetic waves have several important properties:

• they can propagate through vacuum and therefore do not require a material medium;
• in vacuum, they travel at the speed of light;
• they are transverse waves;
• their electric and magnetic fields are mutually perpendicular;
• both fields are perpendicular to the direction of propagation;
• they carry energy and momentum;
• they can undergo reflection, refraction, interference, diffraction and polarization;
• they are produced by accelerated electric charges.

For a plane electromagnetic wave travelling in a particular direction, the electric field E, magnetic field B and propagation direction are mutually perpendicular.

11.11 Speed of Electromagnetic Waves

Maxwell's equations predict that electromagnetic waves travel through vacuum with speed

Using

and

we obtain approximately

This is the measured speed of light in vacuum. The agreement revealed that light itself is an electromagnetic wave.

11.12 Frequency, Wavelength and Wave Speed

Like other periodic waves, electromagnetic waves satisfy

where v is wave speed, f is frequency and λ is wavelength.

In vacuum, v=c, so

Therefore,

and

Frequency and wavelength are inversely related when wave speed is fixed. Increasing frequency therefore decreases wavelength.

11.13 Electric and Magnetic Fields in an Electromagnetic Wave

For a plane electromagnetic wave travelling through vacuum, the electric and magnetic fields oscillate in phase. Their magnitudes are related by

Therefore,

Although the numerical values of E and B have different units, their amplitudes are directly related through the speed of light.

11.14 Energy Transport in Electromagnetic Waves

Electromagnetic waves transport energy. The flow of electromagnetic energy is described by the Poynting vector,

The direction of S gives the direction in which electromagnetic energy propagates.

For perpendicular electric and magnetic fields, the instantaneous magnitude is

For a sinusoidal plane electromagnetic wave in vacuum, the average intensity may be expressed as

or equivalently,

Here E₀ and B₀ are the field amplitudes.

11.15 Electromagnetic Spectrum

The electromagnetic spectrum is the complete range of electromagnetic radiation arranged according to frequency or wavelength.

The boundaries between neighbouring regions of the electromagnetic spectrum are conventional rather than perfectly sharp. All regions are governed by the same electromagnetic principles.

As frequency increases, wavelength decreases. The photon energy associated with electromagnetic radiation also increases with frequency according to E=hf, although a detailed treatment of photon physics belongs to quantum physics.

11.16 Applications of Electromagnetic Waves

Electromagnetic waves are central to communication, medicine, scientific measurement, navigation, remote sensing and modern information technology.

Radio-frequency electromagnetic waves carry broadcasting and communication signals. Microwaves are used in radar, satellite links and wireless systems. Infrared radiation is useful for thermal imaging and sensing. Visible light carries visual information and is used extensively in optical communication and instrumentation.

Higher-frequency ultraviolet radiation has applications in sterilization and fluorescence. X-rays are widely used in medical and industrial imaging because of their penetrating ability, while gamma radiation has important applications in medicine, nuclear science and astrophysics.

Chapter Summary

12.1 Introduction to Alternating Current

Direct current flows in one direction, although its magnitude need not always be perfectly constant. In contrast, an alternating current changes direction periodically and usually changes magnitude continuously with time.

The most important practical form of alternating current is sinusoidal. It is naturally produced when a coil rotates uniformly in a magnetic field, as discussed in electromagnetic induction.

A sinusoidal current may be written as

where I₀ is the peak current, ω is the angular frequency and φ is the phase constant.

12.2 Sinusoidal Alternating Voltage and Current

A sinusoidal alternating voltage may be represented by

where V₀ is the maximum or peak voltage.

When the phase constant is zero, the equations reduce to

and

During one complete cycle, a sinusoidal current passes through positive and negative values. Its average value over a complete cycle is zero. This does not mean that AC cannot transfer energy. The heating and power effects are described using root-mean-square values.

12.3 Period, Frequency, Phase and Angular Frequency

The period, T, is the time required for one complete cycle. The frequency, f, is the number of cycles completed per second.

Frequency is measured in hertz.

Angular frequency is related to frequency by

Therefore,

12.4 Peak and Root-Mean-Square Values

Because the instantaneous value of alternating current changes continuously, a useful equivalent value is required for power calculations. This is the root-mean-square, or rms, value.

For a sinusoidal current,

Similarly,

Hence,

and

12.5 Alternating Current Through a Resistor

Consider a pure resistor connected to a sinusoidal AC source. Ohm's law applies instantaneously:

If

then

Since

we obtain

Therefore, voltage and current in a pure resistor are in phase. They reach zero, positive maximum and negative maximum at the same times.

Using rms quantities,

12.6 Alternating Current Through an Inductor

For an ideal inductor, the voltage is related to the rate of change of current:

If the current is sinusoidal,

then

Therefore,

Since

the voltage leads the current by 90°. Equivalently, the current lags the voltage by 90°.

12.7 Inductive Reactance

An inductor opposes alternating current through a quantity known as inductive reactance.

Since ω=2πf,

The SI unit of reactance is the ohm.

For a pure inductor,

Inductive reactance increases as frequency increases. An inductor therefore offers greater opposition to higher-frequency alternating current.

12.8 Alternating Current Through a Capacitor

For a capacitor,

Since current is the rate of flow of charge,

If

then

and

Thus the current leads the voltage by 90°. Equivalently, the voltage lags the current by 90°.

12.9 Capacitive Reactance

The opposition offered by a capacitor to alternating current is described by capacitive reactance:

Since ω=2πf,

For a pure capacitor,

Capacitive reactance decreases as frequency increases. A capacitor therefore offers greater opposition to low-frequency AC and less opposition to high-frequency AC.

12.10 Comparison of R, L and C in AC Circuits

These distinctions are essential when analysing circuits containing combinations of resistors, inductors and capacitors.

12.11 Series RLC Circuit

A series RLC circuit contains a resistor, inductor and capacitor connected in series to an AC source. Because the components are in series, the same instantaneous current flows through all three components.

However, the voltage drops across the components are not generally in phase:

• V_R is in phase with current;
• V_L leads current by 90°;
• V_C lags current by 90°.

Consequently, the voltages cannot generally be added as ordinary scalar magnitudes. Their phase relationships must be taken into account.

12.12 Impedance

For a series RLC circuit, the magnitude of the impedance is

The rms current is therefore

The impedance is measured in ohms.

12.13 Phase Angle and Phasor Relationships

The phase angle φ between the source voltage and current in a series RLC circuit satisfies

The cosine of the phase angle is

Three cases should be distinguished.

12.14 Resonance in a Series RLC Circuit

Therefore,

Multiplying by ωC,

Hence the resonant angular frequency is

Since ω=2πf, the resonant frequency is

At resonance,

and therefore the current reaches its maximum value for a fixed source voltage:

The source voltage and current are also in phase at resonance.

12.15 Power in Alternating Current Circuits

In a purely resistive circuit, voltage and current are in phase, and the average power is

In a general AC circuit where voltage and current have phase difference φ, the average power is

For a series RLC circuit, average power is dissipated in the resistor:

An ideal inductor and ideal capacitor do not consume net energy over a complete cycle. They temporarily store energy and return it to the circuit.

12.16 Power Factor

For a series RLC circuit,

A power factor of unity means that voltage and current are in phase. For a given real power and supply voltage, a lower power factor requires a larger current.

12.17 Electromagnetic Oscillations in LC Circuits

An ideal LC circuit contains a capacitor of capacitance C and an inductor of inductance L. Such a circuit can sustain electromagnetic oscillations in which energy is transferred repeatedly between the electric field of the capacitor and the magnetic field of the inductor.

Suppose a capacitor is initially charged and then connected to an ideal inductor. The capacitor begins to discharge, producing current through the inductor. As the current increases, magnetic energy accumulates in the inductor.

When the capacitor becomes momentarily uncharged, the current reaches its maximum value and the energy is entirely magnetic. The inductor then maintains the current, charging the capacitor with opposite polarity. The process repeats.

The natural angular frequency of an ideal LC oscillator is

The corresponding frequency is

This is the same natural frequency that appears as the resonant frequency of an ideal series RLC circuit.

12.18 Energy Exchange in an LC Circuit

The electric energy stored in a capacitor is

The magnetic energy stored in an inductor is

In an ideal LC circuit with no resistance, the total energy remains constant:

Thus,

At maximum capacitor charge, the current is zero and all the energy is electric. At maximum current, the capacitor is momentarily uncharged and all the energy is magnetic.

12.19 Applications of AC and Resonant Circuits

Alternating-current circuits are fundamental to electrical power systems because AC voltage levels can be changed efficiently using transformers. This allows electrical energy to be transmitted at high voltage and comparatively low current, reducing resistive losses in transmission lines.

Resonant circuits are important in frequency selection. A circuit can be designed so that its response becomes particularly strong near a selected frequency. This principle is used in radio receivers, transmitters, filters, communication equipment and many electronic systems.

Inductors and capacitors also play important roles in power supplies, signal processing, filtering and energy storage. Their contrasting dependence on frequency makes them especially useful for controlling alternating signals.

12.20 Course Integration

The topics developed throughout this course form a connected description of electricity and magnetism. Electric charge produces electric fields, and Coulomb's law describes the force between stationary point charges. Gauss's law provides a powerful field method for symmetric charge distributions, while electric potential and capacitance introduce the concepts of electrical energy and energy storage.

When charges move collectively, they form electric current. Current, resistance, electromotive force and circuit laws provide the foundation for DC circuit analysis. Moving charges also generate magnetic fields and experience magnetic forces. The Biot-Savart and Ampère laws relate currents to the magnetic fields they produce.

When magnetic flux changes, Faraday's law predicts an induced electric field or emf, while Lenz's law determines the direction of the induced effect. Self-inductance describes the response of a circuit to changes in its own current, and mutual inductance explains electromagnetic coupling between circuits and the operation of transformers.

Maxwell's equations complete the classical unification by showing that electric and magnetic fields are dynamically connected. Changing magnetic fields produce electric fields, while changing electric fields contribute to magnetic fields. Electromagnetic waves emerge naturally from this coupling and propagate through vacuum at the speed of light.

Alternating-current circuits provide a major practical application of these principles. Resistors dissipate electrical energy, while capacitors and inductors alternately store and return energy in electric and magnetic fields. In RLC circuits, their combined behaviour leads to impedance, phase differences and resonance.

Chapter Summary