Chapter 1 — The Electromagnetic Model

Key Theory — Chapter 1

Condensed from Cheng, Field and Wave Electromagnetics, §1-1 through §1-3. Read this before attempting the problems below.

1-1   What is Electromagnetics?

Electromagnetics is the study of the effects of electric charges at rest and in motion. There are two kinds of charges — positive and negative — and both are sources of an electric field. Moving charges constitute a current, which gives rise to a magnetic field.

A field is a spatial distribution of a quantity, which may or may not be a function of time. A time-varying electric field is always accompanied by a magnetic field, and vice versa — the two are coupled, forming an electromagnetic field. Under certain conditions, time-dependent electromagnetic fields produce waves that radiate from the source.

Why fields, not just circuits? Circuit theory is a restricted special case of electromagnetic theory, valid only when the dimensions of a network are much smaller than the wavelength (quasi-static regime). Two situations where circuit theory fails:

1. A monopole antenna on a walkie-talkie appears as an open circuit to circuit theory, yet it clearly radiates — only a non-uniform current flowing along the open-ended conductor (predicted by field theory) explains transmission.
2. An electromagnetic wave incident on a conducting wall with a small aperture produces fields on the far side, even at points not directly behind the hole — a diffraction/shielding problem circuit theory cannot describe.

1-2.1   Deductive vs. Inductive Approach

Two approaches exist in developing a scientific subject:

• Inductive: follow the historical path — observe experiments, infer laws and theorems. Reasoning moves from particular phenomena to general principles.
• Deductive (axiomatic): postulate a few fundamental relations as axioms, then derive particular laws as consequences. The model's validity is verified by predictions that agree with experiment.

Cheng uses the deductive approach because it is more elegant and develops the subject in an orderly way.

Three steps to build any theory on an idealized model

1. Define the basic quantities germane to the subject.
2. Specify the rules of operation (the mathematics) for those quantities.
3. Postulate the fundamental relations (axioms/laws), based on experimental observation.

Familiar example — circuit theory: basic quantities are \(V, I, R, L, C\); rules of operation are algebra, ODEs, and Laplace transforms; fundamental postulates are Kirchhoff's voltage and current laws.

For electromagnetics: Chapter 1 covers step 1; Chapter 2 covers step 2 (vector algebra & calculus); step 3 is introduced in three substeps — Chapters 3 (electrostatics), 6 (magnetostatics), and 7 (time-varying fields/Maxwell's equations).

1-2.2   Source Quantities

Quantities in the electromagnetic model fall into two categories: source quantities (the charges and currents that produce fields) and field quantities (the fields themselves).

Electric charge and the charge of an electron

Electric charge is a fundamental property of matter and exists only in positive or negative integer multiples of the elementary charge \(e\):

Principle of conservation of charge — like conservation of momentum, this is a fundamental postulate of physics: charge can neither be created nor destroyed. The algebraic sum of positive and negative charges in an isolated system remains unchanged. This principle is expressed mathematically by the equation of continuity (§5-4), and Kirchhoff's current law is simply an assertion of it applied to a junction.

Volume, surface, and line charge densities

Although charge is discrete microscopically, electromagnetic effects of large aggregates are well described by smoothed-out density functions defined as point functions of space coordinates:

"Small enough" to give accurate variation yet "large enough" to contain many discrete charges — e.g. a cube of side 1 μm has volume \(10^{-18}\ \text{m}^3\) and still contains about \(10^{11}\) atoms.

Current and current density

Current is the rate of change of charge with respect to time:

Current flows through a finite area, so it is not a point function. Two vector point functions are defined to describe current density:

• Volume current density \(\mathbf{J}\) — current per unit area normal to flow direction, units A/m²; direction is the direction of current flow.
• Surface current density \(\mathbf{J}_s\) — used for currents confined to a thin surface layer (e.g. high-frequency currents on good conductors); units A/m.

1-2.3   The Four Fundamental Field Quantities

Electromagnetics has four fundamental vector field quantities, all point functions of space (and time, in the dynamic case):

Which field do you need, and when?

• \(\mathbf{E}\) — the only vector needed for electrostatics in free space; defined as the electric force on a unit test charge.
• \(\mathbf{D}\) — useful when a material medium is present (Ch. 3).
• \(\mathbf{B}\) — the only vector needed for magnetostatics in free space; related to the magnetic force on a moving charge.
• \(\mathbf{H}\) — useful when a magnetic medium is present (Ch. 6).

Static vs. dynamic coupling: under static, steady, or stationary conditions the two pairs \(\{\mathbf{E},\mathbf{D}\}\) and \(\{\mathbf{B},\mathbf{H}\}\) are independent. In time-dependent cases they are coupled — a time-varying \(\mathbf{E}\)/\(\mathbf{D}\) produces \(\mathbf{B}\)/\(\mathbf{H}\), and vice versa.

Constitutive relations

Material (medium) properties determine the relations between \(\mathbf{E}\) and \(\mathbf{D}\), and between \(\mathbf{B}\) and \(\mathbf{H}\). These relations are called the constitutive relations of the medium. In free space (see §1-3):

1-3.1   The SI System

SI (Système International d'Unités) is a rationalized MKSA system — four base units, from which every other unit in electromagnetics is derived:

All units in Table 1-1 are derived:

• Coulomb (C) = A·s
• Volt per meter (V/m) = kg·m/(A·s³)
• Tesla (T) = kg/(A·s²) = V·s/m² = Wb/m²

"Rationalized" means the factor \(4\pi\) has been absorbed into the constitutive constants so that it does not appear in Maxwell's equations themselves (though it does appear in many derived relations such as Coulomb's law).

1-3.2   The Three Universal Constants

Free space (vacuum) is characterized by three universal constants, which are not independent of one another:

In SI, \(\mu_0\) is defined exactly as \(4\pi\times 10^{-7}\) H/m (a choice of the unit system, not an experimentally measured quantity). The speed of light is fixed by the modern definition of the meter. The permittivity \(\epsilon_0\) is then derived from the master identity

which links electromagnetism to optics — the velocity of light is a consequence of the electric and magnetic properties of vacuum.

Chapter 1 at a Glance

• Source quantities: \(\rho,\ \rho_s,\ \rho_\ell\) (charge densities), \(I,\ \mathbf{J},\ \mathbf{J}_s\) (current and current densities).
• Field quantities: \(\mathbf{E},\ \mathbf{D}\) (electric) and \(\mathbf{B},\ \mathbf{H}\) (magnetic); 4 vectors total.
• Constitutive relations (free space): \(\mathbf{D}=\epsilon_0\mathbf{E}\), \(\mathbf{B}=\mu_0\mathbf{H}\).
• Universal constants: \(c\), \(\mu_0\), \(\epsilon_0\), with \(c = 1/\sqrt{\epsilon_0\mu_0}\).
• Approach: deductive/axiomatic — define quantities, specify rules, postulate fundamental relations; postulates introduced in Chapters 3, 6, and 7.
• Why not just circuit theory: circuit theory is a quasi-static special case; radiation, diffraction and wave phenomena require the full electromagnetic model.

Problems and Solutions

Review Questions (Tier 1)

Chapter 2 — Vector Analysis

Key Theory — Chapter 2

Condensed from Cheng, Field and Wave Electromagnetics, §2-1 through §2-12. Read this before attempting the problems below.

2-1   Introduction — Why Vector Analysis?

Electromagnetic quantities are either scalars (charge, energy, potential) or vectors (\(\mathbf{E}\), \(\mathbf{B}\), current density). In three-dimensional space a vector relation is three scalar relations, so an efficient vector calculus is essential.

Three pillars of this chapter:

1. Vector algebra — addition, dot and cross products.
2. Orthogonal coordinate systems — Cartesian, cylindrical, spherical.
3. Vector calculus — gradient, divergence, curl, and the integral theorems that tie them together.

The physical laws themselves are invariant under choice of coordinates; coordinates are chosen only to match the geometry of a given problem.

2-2   Vector Addition and Subtraction

A vector has magnitude and direction; a unit vector points without magnitude:

Addition is commutative and associative; subtraction is \(\mathbf{A}-\mathbf{B} = \mathbf{A}+(-\mathbf{B})\).

2-3   Products of Vectors

Dot (scalar) product — yields a scalar:

Zero iff vectors are orthogonal; equals the projection of one onto the other times the other's magnitude. Commutative and distributive; not associative (triple dot is meaningless).

Cross (vector) product — yields a vector perpendicular to the plane of the operands, magnitude equals the area of their parallelogram:

Direction \(\mathbf{a}_n\) by right-hand rule. Cross product is anti-commutative, distributive, and not associative.

Scalar triple product — cyclic permutation leaves it unchanged (geometrically, volume of the parallelepiped):

Vector triple product — the BAC-CAB rule:

Parentheses matter — \((\mathbf{A}\times\mathbf{B})\times\mathbf{C}\) is a different vector. Division by a vector is undefined.

2-4   Orthogonal Coordinate Systems

A right-handed orthogonal system has base vectors \(\mathbf{a}_{u_1},\mathbf{a}_{u_2},\mathbf{a}_{u_3}\) satisfying:

Coordinates \(u_i\) may not themselves be lengths (angles, for instance). The metric coefficient \(h_i\) converts a coordinate change to a length:

Cartesian (§2-4.1):

Cylindrical (§2-4.2): \((r,\phi,z)\). Useful for line charges, long wires, coax. \(\mathbf{a}_r\times\mathbf{a}_\phi=\mathbf{a}_z\).

Spherical (§2-4.3): \((R,\theta,\phi)\). \(R\) is distance from origin, \(\theta\) is polar angle from \(+z\), \(\phi\) is azimuth. Ideal for point sources and far-field antenna problems. \(\mathbf{a}_R\times\mathbf{a}_\theta=\mathbf{a}_\phi\).

Caution: in curvilinear systems the base vectors depend on position (\(\partial\mathbf{a}_r/\partial\phi = \mathbf{a}_\phi\), \(\partial\mathbf{a}_\phi/\partial\phi = -\mathbf{a}_r\)). Differentiation of a vector in such systems must account for this.

2-5   Line, Surface, Volume Integrals

The differential area is treated as a vector \(d\mathbf{s} = \mathbf{a}_n\,ds\) (outward normal for a closed surface, right-hand-rule normal for an open surface bounded by a contour).

2-6   Gradient of a Scalar Field

The gradient is the vector pointing in the direction of the maximum spatial rate of increase of a scalar \(V\), with magnitude equal to that rate:

General orthogonal curvilinear form:

Cartesian form — also defines the del operator:

\(\nabla V\) is always perpendicular to surfaces of constant \(V\). In electrostatics we use \(\mathbf{E}=-\nabla V\) — the minus sign makes \(V\) increase opposite to \(\mathbf{E}\).

2-7   Divergence of a Vector Field

Divergence measures the net outward flux of \(\mathbf{A}\) per unit volume — the flow source density:

Coordinate forms:

A field with \(\nabla\cdot\mathbf{A}=0\) everywhere is solenoidal — flux lines close on themselves, no sources or sinks (e.g., magnetic flux density \(\mathbf{B}\)).

2-8   Divergence Theorem

Also known as Gauss's theorem:

Converts volume integral of a divergence into a closed surface integral of the vector — the workhorse behind integral forms of Gauss's law and charge conservation.

2-9   Curl of a Vector Field

Circulation is the line integral of \(\mathbf{A}\) around a closed contour. Curl is the vortex-source density:

The direction of \(\mathbf{a}_n\) is that which maximizes the circulation per unit area (right-hand rule relative to \(d\mathbf{\ell}\)).

Determinant form in Cartesian coordinates:

General orthogonal curvilinear form:

A field with \(\nabla\times\mathbf{A}=0\) everywhere is irrotational (or conservative). A static electrostatic field is always irrotational.

2-10   Stokes's Theorem

Converts surface integral of a curl into a closed line integral around the bounding contour. The directions of \(d\mathbf{\ell}\) and \(d\mathbf{s}=\mathbf{a}_n\,ds\) obey the right-hand rule.

Dictionary. The two integral theorems convert dimensionality:

2-11   Two Null Identities

Repeated del operations give two identities of deep importance for potential theory:

Converse statements (crucial for EM):

• If \(\nabla\times\mathbf{E}=0\), then \(\mathbf{E}\) can be written as the gradient of a scalar: \(\mathbf{E} = -\nabla V\) (2-148). This is why electric potential exists in electrostatics.
• If \(\nabla\cdot\mathbf{B}=0\), then \(\mathbf{B}\) can be written as the curl of a vector: \(\mathbf{B} = \nabla\times\mathbf{A}\) (2-152). This is why magnetic vector potential exists.

2-12   Helmholtz's Theorem

A vector field is determined — to within an additive constant — if both its divergence and its curl are specified everywhere, and the field vanishes at infinity.

Classification of vector fields:

Any such field decomposes into an irrotational part and a solenoidal part:

This is the axiomatic foundation for all of Cheng's subsequent chapters — specifying both div and curl of \(\mathbf{E}\) (Chapter 3) or of \(\mathbf{B}\) (Chapter 6) fixes the entire field. That is why Cheng postulates div and curl rather than starting from Coulomb's law.

Chapter 2 at a Glance

• Algebra: \(\mathbf{A}\cdot\mathbf{B}=AB\cos\theta\); \(\mathbf{A}\times\mathbf{B}=\mathbf{a}_n AB\sin\theta\) (right-hand rule). Cyclic scalar triple product; BAC-CAB rule.
• Three coordinate systems: Cartesian \((x,y,z)\), cylindrical \((r,\phi,z)\), spherical \((R,\theta,\phi)\); Table 2-1 has their base vectors, \(h_i\), and \(dv\).
• Del operator \(\nabla\) in Cartesian: \(\mathbf{a}_x\partial_x + \mathbf{a}_y\partial_y + \mathbf{a}_z\partial_z\).
• Gradient \(\nabla V\) — maximum rate of increase; \(dV=\nabla V\cdot d\mathbf{\ell}\); perpendicular to level surfaces.
• Divergence \(\nabla\cdot\mathbf{A}\) — flow-source density; zero \(\Rightarrow\) solenoidal.
• Curl \(\nabla\times\mathbf{A}\) — vortex-source density; zero \(\Rightarrow\) irrotational/conservative.
• Divergence theorem: \(\int_V \nabla\cdot\mathbf{A}\,dv = \oint_S \mathbf{A}\cdot d\mathbf{s}\).
• Stokes's theorem: \(\int_S (\nabla\times\mathbf{A})\cdot d\mathbf{s} = \oint_C \mathbf{A}\cdot d\mathbf{\ell}\).
• Null identities: \(\nabla\times(\nabla V)=0\), \(\nabla\cdot(\nabla\times\mathbf{A})=0\) — allow definition of scalar potential \(V\) and vector potential \(\mathbf{A}\).
• Helmholtz: \(\mathbf{F}=-\nabla V+\nabla\times\mathbf{A}\); divergence + curl uniquely determine a well-behaved field.

Problems and Solutions

Review Questions (Tier 1)

Textbook Practice Problems

Textbook Practice — Approach Hints

Chapter 3 — Static Electric Fields

Key Theory — Chapter 3

Condensed from Cheng, Field and Wave Electromagnetics, §3-1 through §3-11. Read this before attempting the problems below.

3-1   Introduction — Why Postulate, Not Derive?

Elementary texts start from Coulomb's experimental law and build upward. Cheng instead postulates the divergence and curl of \(\mathbf{E}\) in free space, then derives Coulomb's and Gauss's laws from them. Justification:

• Coulomb measured the inverse-square power to limited accuracy; assuming it is exactly 2 is already a postulate.
• By Helmholtz's theorem (§2-12) a vector field in a bounded region is uniquely determined by its divergence and curl. Specifying both of \(\mathbf{E}\) therefore fixes the entire electrostatic theory.
• Coulomb's law, Gauss's law, and scalar potential then follow as consequences, not independent postulates — the axiomatic approach.

In electrostatics, charges are at rest, fields do not change with time, and no magnetic field is present. Only one of the four field quantities — \(\mathbf{E}\) — and one of the three universal constants — \(\epsilon_0\) — are needed in free space.

3-2   Fundamental Postulates of Electrostatics in Free Space

Electric field intensity is defined as the force per unit charge on a stationary test charge:

The entire theory of electrostatics in free space rests on two postulates — the divergence and the curl of \(\mathbf{E}\):

Integral forms follow by the divergence theorem and Stokes's theorem respectively:

Physical reading:

• (3-4) — a static \(\mathbf{E}\) field is not solenoidal unless \(\rho=0\); it has sources (positive charge) and sinks (negative charge).
• (3-5) — a static \(\mathbf{E}\) field is irrotational; its line integral around any closed path is zero. Kirchhoff's voltage law is an assertion of (3-8).

3-3   Coulomb's Law (Derived)

Apply (3-7) to a spherical Gaussian surface of radius \(R\) centered on a point charge \(q\) in free space. Symmetry forces \(\mathbf{E}=\mathbf{a}_R E_R\), giving \(E_R(4\pi R^2)=q/\epsilon_0\), hence:

For a system of discrete charges, superposition gives \(\mathbf{E}=\sum_k \mathbf{E}_k\). For a continuous distribution of volume charge density \(\rho(\mathbf{R}')\):

3-4   Gauss's Law — Strategy

Gauss's law (3-7) is most powerful when a Gaussian surface can be constructed over which \(\mathbf{E}\cdot d\mathbf{s}\) is constant. Three symmetry classes permit this:

• Spherical: sphere of radius \(R\) for a point charge, charged ball, or shell.
• Cylindrical (line): coaxial cylinder for an infinite line charge or cylindrical charge column.
• Planar: pillbox straddling an infinite sheet of surface charge.

Moral: if symmetry fits, use (3-7) directly. Otherwise find \(V\) first by integration, then \(\mathbf{E}=-\nabla V\).

3-5   Electric Potential

Because \(\nabla\times\mathbf{E}=0\), by the null identity \(\nabla\times(\nabla V)\equiv 0\) we can write \(\mathbf{E}\) as the gradient of a scalar:

The negative sign makes \(V\) increase in the direction opposite \(\mathbf{E}\), consistent with potential energy per unit charge. The potential difference between two points is path-independent:

Taking the reference at infinity, the potentials of standard charge distributions are:

Field lines and equipotentials are perpendicular — \(\nabla V\) is normal to surfaces of constant \(V\).

3-6   Conductors in Static Electric Field

A conductor has loosely bound electrons that redistribute freely under any interior field. Equilibrium (static) is reached very quickly — for copper the relaxation time is \(\sim 10^{-19}\) s. In equilibrium:

At the conductor / free-space boundary, applying (3-8) to a narrow rectangular contour and (3-7) to a shallow pillbox:

Consequences:

• A conductor's surface is an equipotential.
• All net charge resides on the surface; none in the bulk.
• Field lines meet the surface normally.
• Surface charge concentrates where curvature is highest — basis of the lightning rod.

3-7   Dielectrics — Polarization

In a dielectric, electrons are bound. An applied \(\mathbf{E}\) displaces them slightly, creating induced dipoles. The macroscopic measure is the polarization vector \(\mathbf{P}\) — electric dipole moment per unit volume (C/m²).

Polarized matter behaves as if it carried equivalent bound charges:

The total bound charge on any isolated dielectric body is zero.

3-8   Electric Flux Density \(\mathbf{D}\) and Dielectric Constant

In a material medium the divergence postulate (3-4) must account for bound charge: \(\nabla\cdot\mathbf{E}=(\rho+\rho_p)/\epsilon_0\). Defining the electric flux density

absorbs the bound charge and leaves only free charge on the right:

For a linear, isotropic medium \(\mathbf{P}=\epsilon_0\chi_e\mathbf{E}\), so

where \(\epsilon_r\) is the dimensionless dielectric constant (relative permittivity), \(\chi_e\) the electric susceptibility, and \(\epsilon\) the absolute permittivity (F/m).

Dielectric strength (§3-8.1) is the maximum \(E\) a material withstands before breakdown. Not to be confused with \(\epsilon_r\):

3-9   Boundary Conditions for Electrostatic Fields

Applying the integral forms (3-8) to a narrow contour and (3-100) to a shallow pillbox straddling the interface:

For a charge-free dielectric–dielectric interface (\(\rho_s=0\)):

For a conductor–dielectric interface (medium 2 is a conductor, \(\mathbf{D}_2=0\)): \(E_{1t}=0\) and \(D_{1n}=\rho_s\) — the special case (3-71), (3-72) generalized to \(\mathbf{D}\).

3-10   Capacitance and Capacitors

Because \(\rho_s\), \(\mathbf{E}\), and \(V\) all scale linearly with \(Q\) in a static conductor system, the ratio \(Q/V\) depends only on geometry and the medium:

Procedure to find \(C\) for a two-conductor capacitor: (1) choose coordinates; (2) assume \(\pm Q\) on the conductors; (3) find \(\mathbf{E}\) from Gauss's law or from \(\rho_s\); (4) integrate \(V_{12}=-\int_{2}^{1}\mathbf{E}\cdot d\mathbf{\ell}\); (5) take \(C=Q/V_{12}\).

Series / parallel: capacitors in parallel add (\(C_\text{par}=\sum C_k\)); capacitors in series combine as reciprocals (\(1/C_\text{ser}=\sum 1/C_k\)) — the opposite of resistors, because charge is shared in series and voltage in parallel.

Electrostatic shielding (§3-10.3): a grounded conducting shell enclosing a body decouples it from external charges — the coupling capacitance vanishes. This is why sensitive electronics live inside grounded metal enclosures.

3-11   Electrostatic Energy and Forces

In terms of charges and potentials — the work to assemble \(N\) point charges:

The factor \(\tfrac{1}{2}\) avoids double-counting pair interactions. \(V_k\) is the potential at \(Q_k\) due to all other charges.

In terms of field quantities — substituting \(\rho=\nabla\cdot\mathbf{D}\) and applying a vector identity:

Energy density:

Note: (3-165) gives the interaction (mutual) energy only — self-energies of ideal point charges diverge. (3-170) and (3-176) do include self-energy, which is why they agree with each other but can exceed (3-165) for the same system.

Stored energy in a capacitor (equivalent forms):

Force by virtual displacement. For an isolated system (constant charges), work done by the field equals the decrease in stored energy, giving:

The sign flip is not a contradiction — in the constant-\(V\) case the battery supplies additional energy equal to twice the mechanical work.

Chapter 3 at a Glance

• Two postulates fix everything: \(\nabla\cdot\mathbf{E}=\rho/\epsilon_0\), \(\nabla\times\mathbf{E}=0\).
• Integral forms: \(\oint\mathbf{E}\cdot d\mathbf{s}=Q/\epsilon_0\) (Gauss), \(\oint\mathbf{E}\cdot d\mathbf{\ell}=0\) (conservative).
• Scalar potential: \(\mathbf{E}=-\nabla V\); \(V=q/(4\pi\epsilon_0 R)\) for a point charge, referenced at infinity.
• Conductor (static): \(\mathbf{E}=0\) inside, surface equipotential, \(E_t=0\), \(E_n=\rho_s/\epsilon_0\).
• Dielectric: \(\mathbf{D}=\epsilon_0\mathbf{E}+\mathbf{P}=\epsilon\mathbf{E}\); \(\nabla\cdot\mathbf{D}=\rho_\text{free}\).
• BCs: \(E_{1t}=E_{2t}\); \(D_{1n}-D_{2n}=\rho_s\). Dielectric–dielectric: \(\tan\alpha_2/\tan\alpha_1=\epsilon_2/\epsilon_1\).
• Capacitance: \(C=Q/V\); parallel-plate \(\epsilon S/d\); coax \(2\pi\epsilon L/\ln(b/a)\); sphere \(4\pi\epsilon/(1/R_i-1/R_o)\).
• Energy: \(W_e=\tfrac{1}{2}\int\epsilon E^2\,dv=\tfrac{1}{2}CV^2\); force \(\mathbf{F}_Q=-\nabla W_e\) at constant charge.

Problems and Solutions

Review Questions (Tier 1)

Textbook Practice Problems

Textbook Practice — Approach Hints

Textbook Practice Problems

Textbook Practice — Approach Hints

Textbook Practice Problems

Textbook Practice — Approach Hints

Chapter 6 — Static Magnetic Fields

Key Theory — Chapter 6

Condensed from Cheng, Field and Wave Electromagnetics, §6-1 through §6-13. Read this before attempting the problems below.

6-1   The Magnetic Force and the Definition of B

A charge \(q\) at rest in an electric field experiences \(\mathbf{F}_e = q\mathbf{E}\). When the charge moves with velocity \(\mathbf{u}\), experiment shows a second force appears: one perpendicular to both \(\mathbf{u}\) and a fixed direction at each point, with magnitude proportional to \(q\), to \(|\mathbf{u}|\), and to the component of \(\mathbf{u}\) perpendicular to that fixed direction. This magnetic force is expressed as

and the new vector field \(\mathbf{B}\) (webers per square meter, or teslas) is the magnetic flux density. Combining electric and magnetic effects gives Lorentz's force equation:

Lorentz's equation can itself be taken as a fundamental postulate of the electromagnetic model — it defines both \(\mathbf{E}\) (via the force on a stationary charge) and \(\mathbf{B}\) (via the force on a moving one). It cannot be derived from other postulates.

6-2   Fundamental Postulates of Magnetostatics in Free Space

In free space, magnetostatics is built on exactly two postulates specifying the divergence and curl of \(\mathbf{B}\):

No magnetic monopoles. Because \(\nabla\cdot\mathbf{B} = 0\), magnetic flux lines always close on themselves — there is no magnetic analog of electric charge density. Cutting a bar magnet doesn't produce an isolated N or S pole; it produces two smaller magnets, and this continues down to atomic dimensions. Equation (6-9) is the law of conservation of magnetic flux.

Ampère's circuital law says the circulation of \(\mathbf{B}\) around any closed path equals \(\mu_0\) times the total current through the enclosed surface. Taking \(\nabla\cdot(\nabla\times\mathbf{B}) = 0\) in (6-7) immediately gives \(\nabla\cdot\mathbf{J} = 0\), consistent with steady-current continuity (5-45).

6-3   Vector Magnetic Potential A

Because \(\mathbf{B}\) is solenoidal, it can be expressed as the curl of another vector field:

\(\mathbf{A}\) is called the vector magnetic potential, measured in Wb/m. Defining a vector requires specifying both its curl (done by (6-15)) and its divergence — we are free to choose the latter. Substituting (6-15) into the curl postulate and expanding \(\nabla\times\nabla\times\mathbf{A} = \nabla(\nabla\cdot\mathbf{A}) - \nabla^2\mathbf{A}\) gives

Choosing the Coulomb gauge

reduces this to a vector Poisson equation:

Each Cartesian component is a scalar Poisson equation identical in form to \(\nabla^2 V = -\rho/\epsilon_0\) from electrostatics. The solution is therefore

Physical meaning of A: the line integral of \(\mathbf{A}\) around a closed contour equals the magnetic flux through the enclosed surface:

6-4   Biot–Savart Law

For a thin wire, \(\mathbf{J}\,dv' = I\,d\ell'\). Equation (6-23) becomes

Taking the curl and doing the \(\nabla(1/R) = -\mathbf{a}_R/R^2\) bookkeeping yields Biot–Savart law:

where \(\mathbf{a}_R\) is the unit vector from source element to field point. Biot–Savart computes \(\mathbf{B}\) directly from a current-carrying circuit; it is the magnetic analogue of Coulomb's law for \(\mathbf{E}\). Ampère's law is more convenient when symmetry (cylindrical, planar, toroidal) provides a path over which \(|\mathbf{B}|\) is constant; Biot–Savart is the tool when it doesn't.

6-5   The Magnetic Dipole

A small current loop of area \(S\) carrying current \(I\) behaves as a magnetic dipole with moment

where \(\mathbf{a}_n\) follows the right-hand rule from the current direction. At distances \(R \gg\) loop size, the vector potential is

and the magnetic flux density is

which is exactly analogous to the field of an electric dipole \((\mathbf{p}, V)\) — with \(\mathbf{A}\) playing the role of \(V\), and the same angular dependence.

6-6   Magnetization and Equivalent Current Densities

In a material, orbital and spin motions of electrons create microscopic magnetic dipoles. An applied field partially aligns them, yielding a macroscopic magnetization vector \(\mathbf{M}\) (A/m) — magnetic moment per unit volume. A magnetized body can be replaced, as far as its external field is concerned, by an equivalent volume current density and surface current density:

Uniformly magnetized bar magnets have no volume current (\(\nabla\times\mathbf{M} = 0\) for constant \(\mathbf{M}\)) but carry a surface current \(\mathbf{M}\times\mathbf{a}_n\) that effectively turns them into solenoids of lineal current density \(|\mathbf{M}|\).

6-7   Magnetic Field Intensity H and the Constitutive Relation

In matter, Ampère's law must account for both free current \(\mathbf{J}\) and bound (magnetization) current \(\mathbf{J}_m\): \((1/\mu_0)\nabla\times\mathbf{B} = \mathbf{J} + \nabla\times\mathbf{M}\). Rearranging motivates the definition of a new fundamental field:

called the magnetic field intensity. Its governing equation involves only free current:

For linear, isotropic media, \(\mathbf{M} = \chi_m\mathbf{H}\) with magnetic susceptibility \(\chi_m\) (dimensionless). Substituting into (6-75) yields the constitutive relation

6-8   Magnetic Circuits

For a toroidal ferromagnetic core with \(N\) turns carrying current \(I\), Ampère's law gives \(H = NI/\ell\) and \(B = \mu H\), so the flux is \(\Phi = BA = NI/\mathcal{R}\), where the reluctance is

and \(NI\) is the magnetomotive force (mmf, in ampere-turns). Magnetic circuits obey KVL/KCL analogues:

The analogy: \(\mathscr{V}\leftrightarrow NI\), \(I\leftrightarrow\Phi\), \(R\leftrightarrow\mathcal{R}\), \(\sigma\leftrightarrow\mu\).

6-9   Behavior of Magnetic Materials

Three classes are distinguished by the sign and magnitude of \(\chi_m\):

• Diamagnetic (\(\chi_m \sim -10^{-5}\), \(\mu_r \lesssim 1\)) — induced moments oppose the applied field (Lenz's law at the atomic scale). Bismuth, copper, silver, gold, water. No permanent magnetism; effect vanishes when field is removed.
• Paramagnetic (\(\chi_m \sim +10^{-5}\), \(\mu_r \gtrsim 1\)) — existing molecular dipoles partially align with the field but are disrupted by thermal agitation. Aluminium, magnesium, titanium, tungsten. Temperature-dependent.
• Ferromagnetic (\(\mu_r\) from 50 to over \(10^6\)) — strong coupling between atoms forms aligned domains. Iron, nickel, cobalt, and alloys. Nonlinear B–H relation and hysteresis: as H cycles, B traces a loop. Important parameters: remanence \(B_r\) (B at H=0, enables permanent magnets), coercivity \(H_c\) (H needed to drive B back to zero), and the Curie temperature above which coupling collapses and the material becomes paramagnetic. "Soft" magnetic materials have narrow loops (low loss — transformers); "hard" materials have wide loops (large \(H_c\) — permanent magnets).

6-10   Boundary Conditions for Magnetostatic Fields

From \(\nabla\cdot\mathbf{B} = 0\) applied to a pillbox at the interface, and \(\oint\mathbf{H}\cdot d\ell = I_{\text{free}}\) applied to a flat contour straddling the interface:

For two linear media, the normal condition becomes \(\mu_1 H_{1n} = \mu_2 H_{2n}\). The tangential \(\mathbf{H}\) is continuous at any interface between physical media (\(\mathbf{J}_s = 0\)); it is discontinuous only at the surface of an idealised perfect conductor or superconductor. Compare with electrostatics: \(\mathbf{D}\) normal jumps by \(\rho_s\), \(\mathbf{E}\) tangential is continuous.

6-11   Inductance

Mutual flux from loop \(C_1\) linking \(C_2\) is \(\Phi_{12} = \int_{S_2}\mathbf{B}_1\cdot d\mathbf{s}_2\). By Biot–Savart, \(\mathbf{B}_1 \propto I_1\), so

where \(\Lambda_{12} = N_2\Phi_{12}\) is the flux linkage. Carrying the algebra through for two multi-turn circuits yields Neumann's formula:

Reciprocity gives \(L_{12} = L_{21}\). Self-inductance \(L\) is defined analogously by setting both loops equal. Inductance is a purely geometrical quantity (for a linear medium), independent of the current magnitude. For a solenoid, \(L \propto N^2\).

6-12   Magnetic Energy

The work done against induced emf to build up current \(I_1\) in an inductor of self-inductance \(L_1\) is stored as magnetic energy:

For a system of N coupled loops:

Expressed in terms of field quantities:

with magnetic energy density

These forms are exact analogues of \(w_e = \tfrac{1}{2}\mathbf{D}\cdot\mathbf{E}\) in electrostatics. Computing \(W_m\) from the field integral and equating to \(\tfrac{1}{2}LI^2\) is often an easier route to inductance than counting flux linkages.

6-13   Magnetic Forces and Torques

Force on a current-carrying element:

Two parallel wires carrying currents \(I_1, I_2\) separated by distance \(d\) experience

Same direction → attraction; opposite directions → repulsion. (Opposite sense to Coulomb's law for charges — a subtle point.)

Torque on a current loop of dipole moment \(\mathbf{m}\) in a uniform \(\mathbf{B}\):

This is the operating principle of the d-c motor: a split-ring commutator reverses the loop current every half turn to keep \(\mathbf{T}\) aligned and the rotor spinning.

Hall effect: a current \(\mathbf{J}\) flowing in a magnetic field \(\mathbf{B}\) develops a transverse voltage \(V_h\) across the conductor. The sign of \(V_h\) reveals whether carriers are electrons or holes — an essential diagnostic for distinguishing n- and p-type semiconductors.

Chapter 6 at a Glance

• Defining equation: Lorentz force \(\mathbf{F} = q(\mathbf{E} + \mathbf{u}\times\mathbf{B})\).
• Postulates (free space): \(\nabla\cdot\mathbf{B} = 0\), \(\nabla\times\mathbf{B} = \mu_0\mathbf{J}\); integral form gives flux conservation and Ampère's circuital law.
• In matter: \(\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}\); Ampère's law becomes \(\nabla\times\mathbf{H} = \mathbf{J}\); constitutive relation \(\mathbf{B} = \mu\mathbf{H}\).
• Vector potential: \(\mathbf{B}=\nabla\times\mathbf{A}\), Coulomb gauge \(\nabla\cdot\mathbf{A}=0\), \(\nabla^2\mathbf{A}=-\mu_0\mathbf{J}\); \(\Phi = \oint\mathbf{A}\cdot d\ell\).
• Biot–Savart: \(\mathbf{B} = (\mu_0 I/4\pi)\oint d\ell'\times\mathbf{a}_R/R^2\).
• Magnetic dipole: \(\mathbf{m}=IS\mathbf{a}_n\); \(\mathbf{B}\) falls off as \(1/R^3\) with the same angular pattern as an electric dipole.
• Boundary conditions: \(B_n\) continuous; \(H_t\) jumps by \(J_s\) (zero for ordinary media).
• Inductance: \(L = \Lambda/I\) — geometry only; Neumann formula for mutual inductance.
• Energy: \(W_m = \tfrac{1}{2}LI^2 = \tfrac{1}{2}\int\mathbf{H}\cdot\mathbf{B}\,dv\); density \(w_m = B^2/(2\mu)\).
• Forces: \(\mathbf{F} = I\int d\ell\times\mathbf{B}\), torque \(\mathbf{T}=\mathbf{m}\times\mathbf{B}\); parallel wires attract if currents are codirectional.
• Materials: diamagnetic, paramagnetic, ferromagnetic; hysteresis with remanence \(B_r\) and coercivity \(H_c\); Curie temperature; Hall effect.

Problems and Solutions

Part A — Biot-Savart, Ampère's Law, Magnetostatics (Sections 6-1 to 6-7)

Part B — Magnetic Circuits, Inductance, Energy, Forces (Sections 6-8 to 6-13)

Textbook Practice Problems

Textbook Practice — Approach Hints

Chapter 7 — Time-Varying Fields and Maxwell's Equations

Key Theory — Chapter 7

Condensed from Cheng, Field and Wave Electromagnetics, §7-1 through §7-7. Read this before attempting the problems below.

7-1   Why Static Models Are Not Enough

Chapters 3–6 built two independent models — electrostatic and magnetostatic:

In the static case \(\{\mathbf{E}, \mathbf{D}\}\) and \(\{\mathbf{B}, \mathbf{H}\}\) are decoupled: an electrostatic field can drive a steady current, which produces a static magnetic field, but \(\mathbf{E}\) can be found from the charge distribution alone. The magnetic field is only a consequence.

Chapter 7 shows that a time-varying magnetic field produces an electric field, and a time-varying electric field produces a magnetic field. The two curl equations must therefore be modified. The resulting four equations are Maxwell's equations — the foundation of all electromagnetic theory.

7-2   Faraday's Law — Fundamental Postulate

In 1831 Faraday showed experimentally that a changing magnetic flux through a loop induces an emf. Following the deductive approach, we take the point-form relation as postulate:

Applying Stokes's theorem gives the integral form:

Equation (7-1) applies at every point of space whether or not a physical circuit exists. A time-varying magnetic field makes \(\mathbf{E}\) non-conservative — it can no longer be expressed as the gradient of a scalar potential alone.

Transformer emf (stationary circuit, changing B)

For a stationary closed circuit, (7-2) becomes

This is Faraday's law of electromagnetic induction. The induced emf equals the negative rate of change of flux linkage; the minus sign (Lenz's law) asserts that the induced current opposes the flux change that created it. This is the principle of transformers. Time-varying flux in a ferromagnetic core also induces local eddy currents in the core itself; lamination reduces this eddy-current loss.

Motional emf (moving conductor in static B)

A conductor moving with velocity \(\mathbf{u}\) through a static \(\mathbf{B}\) experiences magnetic forces on its free charges. In the conductor's frame, these are equivalent to an induced field \(\mathbf{u}\times\mathbf{B}\) acting along it. For a closed circuit partially in motion,

called the motional emf or "flux-cutting" emf. Only parts of the circuit moving across flux lines contribute.

For a circuit that both moves and lies in a time-varying field, both contributions add — but they combine cleanly into the single expression \(\mathscr{V} = -d\Phi/dt\) with the total time derivative taken along the moving loop.

7-3   Displacement Current and the Modification of Ampère's Law

The static curl equation \(\nabla\times\mathbf{H} = \mathbf{J}\) is inconsistent with charge conservation \(\nabla\cdot\mathbf{J} = -\partial\rho/\partial t\) in the time-varying case, because \(\nabla\cdot(\nabla\times\mathbf{H}) \equiv 0\) would force \(\nabla\cdot\mathbf{J} = 0\). Maxwell's brilliant repair: add a term \(\partial\mathbf{D}/\partial t\) — the displacement current density:

Now \(\nabla\cdot(\nabla\times\mathbf{H}) = \nabla\cdot\mathbf{J} + \partial\rho/\partial t = 0\) automatically, by continuity. The term \(\partial\mathbf{D}/\partial t\) has the dimension of current density (A/m²) but requires no actual charge motion — a time-varying \(\mathbf{D}\) between the plates of an air capacitor produces a magnetic field just as a conduction current would. A time-varying electric field generates a magnetic field.

Maxwell's Equations

The four self-consistent time-varying equations:

The four fundamental fields are \(\mathbf{E}, \mathbf{D}, \mathbf{B}, \mathbf{H}\) — twelve scalar unknowns total. The two curl equations supply six scalar relations; the two constitutive relations \(\mathbf{D} = \epsilon\mathbf{E}\) and \(\mathbf{H} = \mathbf{B}/\mu\) supply six more. The two divergence equations are not independent — they can be derived from the curl equations and continuity. Together with Lorentz's force equation, Maxwell's equations describe all macroscopic electromagnetic phenomena.

In the static limit (\(\partial/\partial t \to 0\)) they reduce to the two pairs of Table 7-1.

7-4   Potential Functions for Time-Varying Fields

Since \(\nabla\cdot\mathbf{B} = 0\) still holds, we retain \(\mathbf{B} = \nabla\times\mathbf{A}\) (6-15). Substituting into Faraday's law:

The quantity in parentheses is curl-free, so it can be written as \(-\nabla V\):

Now \(\mathbf{E}\) has two contributions: an "electrostatic" part \(-\nabla V\) from charge distribution, and an "induction" part \(-\partial\mathbf{A}/\partial t\) from time-varying magnetic field. In the static limit only the first survives.

Lorenz gauge and wave equations

Substituting the potentials into the Ampère–Maxwell equation gives \(\nabla(\nabla\cdot\mathbf{A}) - \nabla^2\mathbf{A} = \mu\mathbf{J} - \mu\epsilon\nabla(\partial V/\partial t) - \mu\epsilon\partial^2\mathbf{A}/\partial t^2\). Choosing the Lorenz gauge

decouples the equations into two non-homogeneous wave equations:

Their solutions travel with velocity \(u = 1/\sqrt{\mu\epsilon}\). In free space \(u = 1/\sqrt{\mu_0\epsilon_0} = c\) — the speed of light. This is the central result linking electromagnetism to optics: light is an electromagnetic wave.

7-5   Electromagnetic Boundary Conditions

The integral forms of Maxwell's equations applied to a pillbox or flat contour straddling an interface give four boundary conditions that are identical in form to those of the static case, because \(\partial\mathbf{D}/\partial t\) and \(\partial\mathbf{B}/\partial t\) have no surface integrals when the pillbox/contour is collapsed:

In words: tangential E is continuous; tangential H jumps by the surface current; normal D jumps by the surface charge; normal B is continuous.

Between two lossless media (\(\rho_s = 0\), \(\mathbf{J}_s = 0\)) all four field components have simple continuity conditions across the interface. At an interface with a perfect conductor, all fields inside vanish, so \(\mathbf{E}\) is normal to the surface (\(E_{1n} = \rho_s/\epsilon_1\)) and \(\mathbf{H}\) is tangential (\(|\mathbf{H}_{1t}| = |\mathbf{J}_s|\)) — fundamental results for waveguide and antenna analysis.

7-6   Wave Equations and Their Solutions

In a source-free region (\(\rho = 0\), \(\mathbf{J} = 0\)), \(\mathbf{E}\) and \(\mathbf{B}\) themselves satisfy homogeneous wave equations. For a spherically symmetric scalar-potential wave about a point source,

is a solution — a spherical wave diverging from the origin at velocity \(u = 1/\sqrt{\mu\epsilon}\). The argument \(t - R/u\) means the value of \(V\) at distance \(R\) and time \(t\) was "caused" by the source at the earlier time \(t - R/u\): this is retardation, a direct consequence of finite propagation speed. The corresponding retarded potentials for arbitrary time-varying sources are

These reduce to the static Poisson-equation solutions when time dependence is negligible ("quasi-static" regime). For high-frequency sources — antennas — retardation cannot be ignored.

7-7   Time-Harmonic (Phasor) Electromagnetics

Because Maxwell's equations are linear, sinusoidal sources of frequency \(\omega\) produce sinusoidal fields of the same frequency everywhere. Represent a real sinusoidal field as the real part of a complex phasor:

Under this substitution, \(\partial/\partial t \to j\omega\), and Maxwell's equations become algebraic in \(j\omega\):

Helmholtz's equation

The wave equations (7-63) and (7-65) become Helmholtz equations:

where

is the wavenumber. The Lorenz gauge in phasor form is \(\nabla\cdot\mathbf{A} + j\omega\mu\epsilon V = 0\).

Retarded potentials in phasor form

The phasor solutions contain the retardation factor \(e^{-jkR}\):

These are the foundation for antenna theory in Chapter 11: the radiation from a current distribution is computed by integrating the phasor current over the source volume with the retardation phase factor.

Chapter 7 at a Glance

• New postulate: \(\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t\) — a changing B makes E non-conservative.
• Faraday's law: \(\mathscr{V} = -d\Phi/dt\); transformer emf (stationary, changing B) + motional emf (moving, static B) both contained in this single total derivative.
• Displacement current: \(\partial\mathbf{D}/\partial t\) added to Ampère's law to save charge conservation; a changing E makes H, even with no free current.
• Maxwell's equations (Table 7-2): four coupled equations in \(\mathbf{E}, \mathbf{D}, \mathbf{B}, \mathbf{H}\); static model is a special case.
• Potentials: \(\mathbf{B} = \nabla\times\mathbf{A}\), \(\mathbf{E} = -\nabla V - \partial\mathbf{A}/\partial t\). In the Lorenz gauge, both \(V\) and \(\mathbf{A}\) obey non-homogeneous wave equations with propagation speed \(u = 1/\sqrt{\mu\epsilon}\).
• Free-space speed: \(1/\sqrt{\mu_0\epsilon_0} = c\) — the speed of light. Light is an electromagnetic wave.
• Boundary conditions: same as static cases — tangential E continuous, tangential H jumps by \(\mathbf{J}_s\), normal D jumps by \(\rho_s\), normal B continuous.
• Retardation: \(V(\mathbf{r},t) = (1/4\pi\epsilon)\int\rho(\mathbf{r}', t-R/u)/R\,dv'\) — the source's state at the earlier time reaches the field point now.
• Phasor form: \(\partial/\partial t \to j\omega\), wave equations become Helmholtz \(\nabla^2 + k^2\), wavenumber \(k = \omega\sqrt{\mu\epsilon}\), retarded potentials acquire the factor \(e^{-jkR}/R\).

Problems and Solutions

Textbook Practice Problems

Textbook Practice — Approach Hints