Electromagnetic Field Theory (EMFT) — Comprehensive Revision Notes

Part 1

Vector Analysis

Electric and magnetic fields are vector fields defined in three-dimensional space and time. Vector calculus provides the compact language of Maxwell's equations. The coordinate system is selected according to the symmetry of the problem, and differential operators encode local behaviour while integral theorems relate them to global flux and circulation.

The Four Fundamental Operators

\[ \nabla\phi\;\;(\text{gradient}),\qquad \nabla\!\cdot\!\mathbf{A}\;\;(\text{divergence}),\qquad \nabla\!\times\!\mathbf{A}\;\;(\text{curl}),\qquad \nabla^2\phi\;\;(\text{Laplacian}) \]

Side-by-side illustration of Cartesian, cylindrical, and spherical coordinate systems with unit vectors and volume elements labelled. — The three standard orthogonal coordinate systems used in EMFT: Cartesian \((x,y,z)\), cylindrical \((\rho,\phi,z)\), and spherical \((r,\theta,\phi)\). Coordinate system choice is dictated by problem symmetry.

Coordinate Systems — Choose by Symmetry

Cartesian \((x,y,z)\)

Unit vectors \(\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\) are constant everywhere.
\(dV = dx\,dy\,dz\)
\(dr^2 = dx^2+dy^2+dz^2\)

Cylindrical \((\rho,\phi,z)\)

Unit vectors \(\hat{\boldsymbol{\rho}},\hat{\boldsymbol{\phi}},\hat{\mathbf{z}}\).
\(x=\rho\cos\phi,\;y=\rho\sin\phi\)
\(dV=\rho\,d\rho\,d\phi\,dz\)
\(dr^2=d\rho^2+\rho^2 d\phi^2+dz^2\)

Spherical \((r,\theta,\phi)\)

Unit vectors \(\hat{\mathbf{r}},\hat{\boldsymbol{\theta}},\hat{\boldsymbol{\phi}}\).
\(z=r\cos\theta\)
\(dV=r^2\sin\theta\,dr\,d\theta\,d\phi\)

Gradient, Divergence, Curl, and Laplacian

Gradient (Cartesian)

\[ \nabla\phi = \frac{\partial\phi}{\partial x}\hat{\mathbf{x}}+\frac{\partial\phi}{\partial y}\hat{\mathbf{y}}+\frac{\partial\phi}{\partial z}\hat{\mathbf{z}} \]

Points in the direction of maximum rate of increase of \(\phi\). The electric field is \(\mathbf{E}=-\nabla V\).

Divergence

Cartesian: \(\nabla\!\cdot\!\mathbf{A}=\partial_x A_x+\partial_y A_y+\partial_z A_z\)

Cylindrical: \(\nabla\!\cdot\!\mathbf{A}=\tfrac{1}{\rho}\partial_\rho(\rho A_\rho)+\tfrac{1}{\rho}\partial_\phi A_\phi+\partial_z A_z\)

Spherical: \(\nabla\!\cdot\!\mathbf{A}=\tfrac{1}{r^2}\partial_r(r^2 A_r)+\tfrac{1}{r\sin\theta}\partial_\theta(\sin\theta A_\theta)+\tfrac{1}{r\sin\theta}\partial_\phi A_\phi\)

Curl (Cartesian)

\[ \nabla\!\times\!\mathbf{A}=\begin{vmatrix}\hat{\mathbf{x}}&\hat{\mathbf{y}}&\hat{\mathbf{z}}\\\partial_x&\partial_y&\partial_z\\A_x&A_y&A_z\end{vmatrix} \]

GATE Tip — Coordinate Factors

Spherical divergence carries \(1/r^2\) and \(r^2 A_r\), plus \(\sin\theta\) factors for the \(\theta\)-component. These factors are exactly the Jacobian of the volume element — a tell-tale sign to check when computing grad/div/curl in spherical coordinates.

Key Vector Identities

Must-Remember Identities

\begin{align*} \nabla\!\times\!(\nabla\phi)&=\mathbf{0} &\text{(gradient fields are curl-free)}\\ \nabla\!\cdot\!(\nabla\!\times\!\mathbf{A})&=0 &\text{(curl fields are divergence-free)}\\ \nabla\!\times\!(\nabla\!\times\!\mathbf{A})&=\nabla(\nabla\!\cdot\!\mathbf{A})-\nabla^2\mathbf{A}\\ \nabla\!\cdot\!(\phi\mathbf{A})&=\phi(\nabla\!\cdot\!\mathbf{A})+\mathbf{A}\!\cdot\!(\nabla\phi)\\ \nabla\!\cdot\!(\mathbf{A}\!\times\!\mathbf{B})&=\mathbf{B}\!\cdot\!(\nabla\!\times\!\mathbf{A})-\mathbf{A}\!\cdot\!(\nabla\!\times\!\mathbf{B}) \end{align*}

Physical Meaning

\(\nabla\!\times\!\nabla\phi=\mathbf{0}\) confirms the electrostatic field \(\mathbf{E}=-\nabla V\) is conservative.
\(\nabla\!\cdot\!\nabla\!\times\!\mathbf{A}=0\) confirms the magnetic field is solenoidal, so \(\mathbf{B}=\nabla\!\times\!\mathbf{A}\).
The BAC-CAB identity \(\nabla\!\times\!\nabla\!\times\!\mathbf{A}=\nabla\nabla\!\cdot\!\mathbf{A}-\nabla^2\mathbf{A}\) is the engine of the wave equation.
A GATE favourite: "Which field can represent an electrostatic field?" — Test: \(\nabla\!\times\!\mathbf{F}\stackrel{?}{=}\mathbf{0}\).

Integral Theorems — The Bridge

Divergence Theorem (Gauss)

\[ \oint_S \mathbf{A}\cdot d\mathbf{S} = \int_V(\nabla\!\cdot\!\mathbf{A})\,dV \]

Flux through a closed surface equals the divergence integrated over the enclosed volume.

Stokes' Theorem

\[ \oint_C \mathbf{A}\cdot d\mathbf{l} = \int_S(\nabla\!\times\!\mathbf{A})\cdot d\mathbf{S} \]

Circulation of a vector around a closed loop equals the curl integrated through the enclosed surface.

Why These Matter

These two theorems are the bridges between the differential and integral forms of Maxwell's equations. Every integral form of Maxwell's equations is obtained by applying one of these theorems to the corresponding differential form.

The Dirac Delta Function

Definition — 1-D and 3-D

\[ \delta(x-a)=0\;\;(x\ne a),\qquad\int_{-\infty}^{\infty}\delta(x-a)\,dx=1,\qquad\int f(x)\delta(x-a)\,dx=f(a) \]

\[ \delta^3(\mathbf{r}-\mathbf{r}') = \delta(x-x')\,\delta(y-y')\,\delta(z-z') \]

Application to Point Charges

A point charge \(Q\) at \(\mathbf{r}_0\) expressed as a volume charge density: \(\rho_v(\mathbf{r})=Q\,\delta^3(\mathbf{r}-\mathbf{r}_0)\). Sanity check: \(\int\rho_v\,dV=Q\). Gauss's law then yields \(E=Q/(4\pi\varepsilon_0 r^2)\).

Green's Function Identity

\[ \nabla^2\!\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right) = -4\pi\,\delta^3(\mathbf{r}-\mathbf{r}') \]

This identity is the foundation of the integral solution of Poisson's equation.

Part 2

Electrostatics

Electrostatics deals with charges at rest and the fields they produce. The fundamental law is Coulomb's (1785), from which Gauss's law, potential theory, and energy relations all follow.

Coulomb's Law and Electric Field

Coulomb's Law

\[ \mathbf{F}_{12} = \frac{Q_1 Q_2}{4\pi\varepsilon_0 R^2}\,\hat{\mathbf{R}}_{12} \]

\(\varepsilon_0 = 8.854\times 10^{-12}\) F/m \(\approx\tfrac{1}{36\pi}\times 10^{-9}\). Force is inverse-square, central, and satisfies Newton's third law.

Electric Field Intensity

\[ \mathbf{E} = \frac{\mathbf{F}}{Q_{\text{test}}} = \frac{Q}{4\pi\varepsilon_0 R^2}\,\hat{\mathbf{R}}\quad(\text{V/m}) \]

Superposition: \(\mathbf{E}=\sum_i \mathbf{E}_i\). Continuous: \(\mathbf{E}=\int\dfrac{dQ}{4\pi\varepsilon_0 R^2}\hat{\mathbf{R}}\) where \(dQ=\rho_v dV,\;\rho_s dS,\;\rho_l dl\).

Electric field lines radiating from a positive charge and terminating on a negative charge, illustrating Coulomb force and field superposition. — Electric field lines originate on positive charges and terminate on negative charges. The force on a test charge is \(\mathbf{F}=Q\mathbf{E}\).

Gauss's Law — The Power of Symmetry

Integral and Differential Forms

\[ \oint_S \mathbf{D}\cdot d\mathbf{S} = Q_{\text{enc}}\qquad\Leftrightarrow\qquad \nabla\!\cdot\!\mathbf{D}=\rho_v \]

\(\mathbf{D}=\varepsilon\mathbf{E}\) is the electric flux density (C/m²). Gauss's law is always true, but only useful when symmetry allows \(|\mathbf{D}|\) to be constant on a Gaussian surface.

Spherical symmetry → Gaussian sphere
Cylindrical symmetry (infinite line/coax) → Gaussian cylinder
Planar symmetry (infinite sheet) → Gaussian pillbox

Standard Electric Field Results

Configuration	\(\mathbf{E}\)	Falloff
Point charge \(Q\)	\(\dfrac{Q}{4\pi\varepsilon_0 r^2}\hat{\mathbf{r}}\)	\(\sim 1/r^2\)
Infinite line \(\rho_l\)	\(\dfrac{\rho_l}{2\pi\varepsilon_0\rho}\hat{\boldsymbol{\rho}}\)	\(\sim 1/\rho\)
Infinite sheet \(\rho_s\)	\(\dfrac{\rho_s}{2\varepsilon_0}\hat{\mathbf{n}}\)	Uniform
Inside uniform sphere (\(r<a\))	\(\dfrac{\rho_v r}{3\varepsilon_0}\hat{\mathbf{r}}\)	\(\propto r\)
Outside uniform sphere (\(r>a\))	\(\dfrac{Q}{4\pi\varepsilon_0 r^2}\hat{\mathbf{r}}\)	\(\sim 1/r^2\)
Between parallel-plate capacitor	\(\dfrac{\rho_s}{\varepsilon_0}\hat{\mathbf{n}}\)	Uniform
Inside coaxial cable (between \(a,b\))	\(\dfrac{\rho_l}{2\pi\varepsilon_0\rho}\hat{\boldsymbol{\rho}}\)	\(\sim 1/\rho\)

GATE Trap

The field inside a uniformly charged sphere grows linearly with \(r\), not as \(1/r^2\). A common GATE favourite: "Field at the centre of a charged sphere?" — Answer: zero.

Electric Potential and Dipole

Potential

\[ V = -\int_{\infty}^{P}\mathbf{E}\cdot d\mathbf{l},\qquad \mathbf{E}=-\nabla V \]

Point charge: \(V=\dfrac{Q}{4\pi\varepsilon_0 R}\). Work to move charge \(q\): \(W=q(V_B-V_A)\). Conservative: \(\oint\mathbf{E}\cdot d\mathbf{l}=0\).

Electric Dipole \(\mathbf{p}=Q\mathbf{d}\)

\[ V = \frac{p\cos\theta}{4\pi\varepsilon_0 r^2}, \qquad \mathbf{E} = \frac{p}{4\pi\varepsilon_0 r^3}\bigl(2\cos\theta\,\hat{\mathbf{r}}+\sin\theta\,\hat{\boldsymbol{\theta}}\bigr) \]

The dipole field falls as \(E\sim 1/r^3\) — faster than a point charge's \(1/r^2\).

Energy in the Electrostatic Field

Stored Energy

\[ W_E = \tfrac{1}{2}\sum_i Q_i V_i = \tfrac{1}{2}\int_V\rho_v V\,dV = \tfrac{1}{2}\int_V\mathbf{D}\cdot\mathbf{E}\,dV \]

The third form shows that energy resides in the field itself, not on the charges. Energy density: \(w_E=\tfrac{1}{2}\varepsilon E^2\) (J/m³).

Dielectric Insertion into a Capacitor

Battery disconnected: \(Q\) fixed, \(V\) drops, \(W\) decreases (work done by the dielectric).
Battery connected: \(V\) fixed, \(Q\) rises, \(W\) increases (battery supplies extra energy).

Method of Images

Concept

Replace a conductor (or dielectric interface) with a fictitious image charge that satisfies the boundary condition on the conductor surface. The uniqueness theorem guarantees correctness.

Point Charge over Grounded Plane

Charge \(+Q\) at height \(h\) above an infinite grounded plane. Image: \(-Q\) at \(-h\).

\[ F = -\frac{Q^2}{4\pi\varepsilon_0(2h)^2}\;\;(\text{attractive}) \]

\[ \rho_s(\rho) = -\frac{Qh}{2\pi(\rho^2+h^2)^{3/2}} \]

Dielectric Interface Image Strength

\[ Q'=Q\,\frac{\varepsilon_1-\varepsilon_2}{\varepsilon_1+\varepsilon_2} \]

Point charge +Q at height h above a grounded conducting plane, with image charge -Q shown at the mirror position below the plane and representative field lines. — Method of images: a real charge \(+Q\) above a grounded conducting plane is replaced by an image charge \(-Q\) at its mirror position. The field in the upper half-space is identical to that of the charge pair.

Poisson's and Laplace's Equations

The Two Key Equations

\[ \nabla^2 V = -\frac{\rho_v}{\varepsilon}\quad(\text{Poisson})\qquad\qquad \nabla^2 V = 0\quad(\text{Laplace, charge-free}) \]

Derived from \(\mathbf{E}=-\nabla V\) and \(\nabla\!\cdot\!\mathbf{D}=\rho_v\).

Uniqueness Theorem

A solution of Laplace's (or Poisson's) equation satisfying all prescribed boundary conditions is unique. Therefore, any method — separation of variables, method of images, or numerical — that yields such a solution is correct.

Solution recipe: (1) Identify symmetry and choose coordinate system. (2) Write \(\nabla^2 V=0\) (or Poisson) in that system. (3) Integrate and apply boundary conditions to fix integration constants. (4) Obtain \(\mathbf{E}=-\nabla V\), then \(\rho_s\), \(Q\), and \(C\).

Part 3

Conductors, Dielectrics, and Capacitance

Current Density and the Continuity Equation

Current Density

Convection: \(\mathbf{J}=\rho_v\mathbf{u}\). Conduction (point-form of Ohm's law): \(\mathbf{J}=\sigma\mathbf{E}\). Total current: \(I=\int_S\mathbf{J}\cdot d\mathbf{S}\).

Continuity Equation

\[ \nabla\!\cdot\!\mathbf{J}+\frac{\partial\rho_v}{\partial t}=0 \]

Charge is conserved everywhere in space.

Charge Relaxation: In a conducting medium, \(\rho_v(t)=\rho_v(0)\,e^{-t/T_r}\) where \(T_r=\varepsilon/\sigma\). For copper \(T_r\sim 10^{-19}\) s (essentially instantaneous); for mica \(T_r\) is several hours.

Resistance and the RC Duality

Resistance and RC Duality

\[ R = \frac{\ell}{\sigma A}\quad(\text{uniform conductor})\qquad\qquad \boxed{RC = \frac{\varepsilon}{\sigma}} \]

For any geometry, replace \(\varepsilon\to\sigma\) in the capacitance formula to get \(R\) of its leaky version. Example: coaxial capacitor \(C=2\pi\varepsilon L/\ln(b/a)\) → leakage resistance \(R=\ln(b/a)/(2\pi\sigma L)\).

Conductors and Dielectric Polarisation

Inside a Perfect Conductor (Static)

\(\mathbf{E}=0\) and \(\rho_v=0\) inside. All free charge resides on the surface. \(\mathbf{E}\) just outside is purely normal; the conductor is an equipotential.

Dielectric Polarisation

\[ \mathbf{P}=\varepsilon_0\chi_e\mathbf{E},\qquad \mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}=\varepsilon\mathbf{E} \]

\(\varepsilon_r=1+\chi_e\). Bound charges: \(\rho_{pb}=-\nabla\!\cdot\!\mathbf{P}\), \(\rho_{sb}=\mathbf{P}\!\cdot\!\hat{\mathbf{n}}\).

Boundary Conditions — Electrostatics

At a Dielectric–Dielectric Interface

\[ E_{1t}=E_{2t}\qquad D_{1n}-D_{2n}=\rho_s \]

If no free surface charge: \(D_{1n}=D_{2n}\). Field refraction law: \(\tan\theta_1/\tan\theta_2=\varepsilon_1/\varepsilon_2\).

At a conductor–dielectric surface: \(E_t=0\) and \(D_n=\rho_s\). These conditions follow from applying Maxwell's equations in integral form to a pillbox or loop straddling the interface.

Capacitance — Standard Results

Geometry	Capacitance
Parallel plate (area \(A\), gap \(d\))	\(C=\varepsilon A/d\)
Coaxial (inner \(a\), outer \(b\), length \(L\))	\(C=2\pi\varepsilon L/\ln(b/a)\)
Concentric spheres (inner \(a\), outer \(b\))	\(C=4\pi\varepsilon/\bigl(1/a-1/b\bigr)\)
Isolated sphere (radius \(a\))	\(C=4\pi\varepsilon a\)
Two dielectrics in series (plate cap.)	\(1/C=d_1/(\varepsilon_1 A)+d_2/(\varepsilon_2 A)\)

Series and Parallel Combinations

Series: \(1/C_{\text{eq}}=\sum 1/C_i\). Parallel: \(C_{\text{eq}}=\sum C_i\). Stacked dielectrics between plates are capacitors in series.

Part 4

Magnetostatics

Magnetostatics deals with the fields produced by steady (DC) currents. The fundamental law is the Biot–Savart law, and Ampère's circuital law provides a powerful tool when symmetry permits.

Biot–Savart Law

Magnetic Field of a Current Element

\[ d\mathbf{H} = \frac{I\,d\mathbf{l}\times\hat{\mathbf{R}}}{4\pi R^2}\quad(\text{A/m})\qquad\qquad \mathbf{B}=\mu_0\mathbf{H},\;\;\mu_0=4\pi\times 10^{-7}\;\text{H/m} \]

Standard Field Results

Infinite wire: \(H=I/(2\pi\rho)\)
Circular loop centre (radius \(a\)): \(H=I/(2a)\)
On-axis at distance \(z\) from loop: \(H=Ia^2/[2(a^2+z^2)^{3/2}]\)
Infinite solenoid (\(n\) turns/m): \(H=nI\)
Toroid: \(H=NI/(2\pi\rho)\)

Ampère's Circuital Law and Magnetic Flux

Ampère's Law (Magnetostatics)

\[ \oint_C\mathbf{H}\cdot d\mathbf{l} = I_{\text{enc}}\qquad\Leftrightarrow\qquad \nabla\!\times\!\mathbf{H}=\mathbf{J} \]

Gauss's Law for Magnetism

\[ \oint_S\mathbf{B}\cdot d\mathbf{S}=0\qquad\Leftrightarrow\qquad \nabla\!\cdot\!\mathbf{B}=0 \]

No magnetic monopoles — all field lines form closed loops.

When Does Ampère's Law Work?

Only when symmetry allows \(H\) to be constant along an Amperian loop: infinite wire, infinite solenoid, toroid, coaxial cable.

Magnetic Potentials

Magnetic Vector Potential \(\mathbf{A}\)

Since \(\nabla\!\cdot\!\mathbf{B}=0\): \(\mathbf{B}=\nabla\!\times\!\mathbf{A}\).

In Coulomb gauge (\(\nabla\!\cdot\!\mathbf{A}=0\)): \(\nabla^2\mathbf{A}=-\mu\mathbf{J}\).

\[ \mathbf{A}(\mathbf{r})=\frac{\mu}{4\pi}\int_V\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}dV' \]

Magnetic flux: \(\Psi=\oint_C\mathbf{A}\cdot d\mathbf{l}\).

Magnetic Scalar Potential \(V_m\)

Valid only in current-free regions (\(\mathbf{J}=0\)):

\[ \mathbf{H}=-\nabla V_m,\qquad \nabla^2 V_m=0 \]

Useful for magnetic circuits and permanent-magnet analysis.

Forces, Torques, and the Hall Effect

Lorentz Force

\[ \mathbf{F} = Q(\mathbf{E}+\mathbf{u}\times\mathbf{B}) \]

Force on Current Conductors

Wire: \(\mathbf{F}=\int I\,d\mathbf{l}\times\mathbf{B}\)

Parallel wires: \(F/L=\mu_0 I_1 I_2/(2\pi d)\)

Magnetic moment: \(\mathbf{m}=I\mathbf{S}\). Torque: \(\mathbf{T}=\mathbf{m}\times\mathbf{B}\). Potential energy: \(U=-\mathbf{m}\cdot\mathbf{B}\).

Hall Effect

\[ V_H = \frac{IB}{nqt} \]

\(n\) = carrier density, \(t\) = thickness. The sign of \(V_H\) distinguishes \(n\)-type from \(p\)-type. Parallel currents attract; anti-parallel repel.

Magnetic Materials

Magnetisation Relations

\[ \mathbf{M}=\chi_m\mathbf{H},\qquad \mu_r=1+\chi_m,\qquad \mathbf{B}=\mu\mathbf{H}=\mu_0(\mathbf{H}+\mathbf{M}) \]

Type	\(\chi_m\)	Examples
Diamagnetic	\(<0\), small (\(\sim10^{-5}\))	Cu, Bi, Ag, water
Paramagnetic	\(>0\), small (\(\sim10^{-3}\))	Al, Pt, Mn
Ferromagnetic	\(\gg 1\), nonlinear, hysteresis	Fe, Ni, Co

Magnetic Boundary Conditions

\[ B_{1n}=B_{2n}\qquad H_{1t}-H_{2t}=K\;\text{(surface current density)} \]

Field refraction: \(\tan\theta_1/\tan\theta_2=\mu_1/\mu_2\). At a perfect conductor surface: \(B_n=0,\;H_t=K\).

Magnetic Circuits and the Electrical Analogy

Reluctance and MMF

\[ \mathscr{F}=NI\;\text{(ampere-turns)}\qquad \mathscr{R}=\frac{\ell}{\mu A}\qquad \Psi=\frac{\mathscr{F}}{\mathscr{R}} \]

Electric Quantity	Magnetic Analogue
EMF \(V\)	MMF \(NI\)
Current \(I\)	Flux \(\Psi\)
Resistance \(R=\ell/\sigma A\)	Reluctance \(\mathscr{R}=\ell/\mu A\)
Conductivity \(\sigma\)	Permeability \(\mu\)
KVL	Hopkinson's law (loop)
KCL	Flux continuity (node)

Inductance and Magnetic Energy

Definitions

Self-inductance: \(L=N\Psi/I=\lambda/I\). Mutual inductance: \(M_{12}=N_2\Psi_{12}/I_1\), and \(M_{12}=M_{21}\). Coupling coefficient: \(k=M/\sqrt{L_1 L_2}\), with \(0\le k\le 1\).

Geometry	Inductance
Solenoid (\(N\) turns, length \(\ell\), area \(A\))	\(L=\mu N^2 A/\ell\)
Toroid (\(N\) turns, mean radius \(\rho_0\))	\(L\approx\mu N^2 A/(2\pi\rho_0)\)
Coaxial (per unit length)	\(L'=\dfrac{\mu}{2\pi}\ln(b/a)\)
Two parallel wires (per unit length)	\(L'=\dfrac{\mu}{\pi}\cosh^{-1}(d/2a)\)

Neumann Formula and Magnetic Energy

\[ M_{12}=\frac{\mu}{4\pi}\oint\!\!\oint\frac{d\mathbf{l}_1\cdot d\mathbf{l}_2}{R},\qquad W_m=\tfrac{1}{2}LI^2=\int_V\frac{B^2}{2\mu}\,dV \]

Energy density: \(w_m=B^2/(2\mu)=\tfrac{1}{2}\mu H^2\).

Part 5

Time-Varying Fields and Maxwell's Equations

Faraday's Law of Induction

Faraday's Law

\[ \mathcal{E} = -\frac{d\Psi}{dt} = -\frac{d}{dt}\int_S\mathbf{B}\cdot d\mathbf{S} \]

\[ \oint_C\mathbf{E}\cdot d\mathbf{l} = -\int_S\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{S}\qquad\Leftrightarrow\qquad \nabla\!\times\!\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t} \]

Two Flavours of EMF

Transformer EMF: loop fixed, \(\mathbf{B}\) time-varying.

Motional EMF: loop moves, \(\mathbf{B}\) static: \(\mathcal{E}_m=\oint(\mathbf{u}\times\mathbf{B})\cdot d\mathbf{l}\).

In the general case, both contribute.

Lenz's Law

The induced EMF opposes the change in flux that caused it. This is energy conservation in disguise — the negative sign in Faraday's law is its mathematical statement.

Displacement Current — Maxwell's Insight

The Inconsistency

Static Ampère: \(\nabla\!\times\!\mathbf{H}=\mathbf{J}\) implies \(\nabla\!\cdot\!\mathbf{J}=0\). But the continuity equation requires \(\nabla\!\cdot\!\mathbf{J}=-\partial_t\rho_v\ne 0\) for time-varying charge distributions.

Maxwell's Fix (1865)

Adding the displacement current \(\mathbf{J}_d=\partial_t\mathbf{D}\) resolves the inconsistency:

\[ \nabla\!\times\!\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t} \]

This single insight predicted electromagnetic waves and unified optics with electromagnetism. In a capacitor gap, \(\mathbf{J}=0\) but \(\partial_t\mathbf{D}\ne 0\), so the Amperian loop closes through the gap: \(I_d=\varepsilon A\,\partial_t E=I_{\text{wire}}\).

Maxwell's Equations — The Grand Summary

Law	Differential Form	Integral Form
Gauss (Electric)	\(\nabla\!\cdot\!\mathbf{D}=\rho_v\)	\(\oint\mathbf{D}\cdot d\mathbf{S}=Q_{\text{enc}}\)
Gauss (Magnetic)	\(\nabla\!\cdot\!\mathbf{B}=0\)	\(\oint\mathbf{B}\cdot d\mathbf{S}=0\)
Faraday	\(\nabla\!\times\!\mathbf{E}=-\dfrac{\partial\mathbf{B}}{\partial t}\)	\(\oint\mathbf{E}\cdot d\mathbf{l}=-\dfrac{d}{dt}\int\mathbf{B}\cdot d\mathbf{S}\)
Ampère–Maxwell	\(\nabla\!\times\!\mathbf{H}=\mathbf{J}+\dfrac{\partial\mathbf{D}}{\partial t}\)	\(\oint\mathbf{H}\cdot d\mathbf{l}=I_{\text{enc}}+\dfrac{d}{dt}\int\mathbf{D}\cdot d\mathbf{S}\)

Constitutive Relations

\[ \mathbf{D}=\varepsilon\mathbf{E}\qquad \mathbf{B}=\mu\mathbf{H}\qquad \mathbf{J}=\sigma\mathbf{E}\qquad \mathbf{F}=Q(\mathbf{E}+\mathbf{u}\times\mathbf{B}) \]

Key Observation

For static fields \(\partial/\partial t=0\), so \(\mathbf{E}\) and \(\mathbf{B}\) decouple. Maxwell's equations plus the Lorentz force law constitute complete classical electrodynamics.

Retarded Potentials

Scalar and Vector Potentials (Time-Varying)

\[ \mathbf{E}=-\nabla V - \frac{\partial\mathbf{A}}{\partial t},\qquad \mathbf{B}=\nabla\!\times\!\mathbf{A} \]

Lorenz Gauge and Wave Equations

Gauge condition: \(\nabla\!\cdot\!\mathbf{A}+\mu\varepsilon\,\partial_t V=0\). Under this gauge, \(V\) and \(\mathbf{A}\) each satisfy uncoupled wave equations:

\[ \nabla^2 V - \mu\varepsilon\,\frac{\partial^2 V}{\partial t^2}=-\frac{\rho_v}{\varepsilon},\qquad \nabla^2\mathbf{A}-\mu\varepsilon\,\frac{\partial^2\mathbf{A}}{\partial t^2}=-\mu\mathbf{J} \]

Solutions are the retarded potentials: \(t_r=t-|\mathbf{r}-\mathbf{r}'|/c\) is the retarded time, confirming that information propagates at \(c\).

General Boundary Conditions

At Any Interface (\(\hat{\mathbf{n}}\) from medium 2 into medium 1)

\[ \hat{\mathbf{n}}\cdot(\mathbf{D}_1-\mathbf{D}_2)=\rho_s,\qquad \hat{\mathbf{n}}\cdot(\mathbf{B}_1-\mathbf{B}_2)=0 \]

\[ \hat{\mathbf{n}}\times(\mathbf{E}_1-\mathbf{E}_2)=\mathbf{0},\qquad \hat{\mathbf{n}}\times(\mathbf{H}_1-\mathbf{H}_2)=\mathbf{K} \]

At a perfect conductor (\(\sigma\to\infty\)): \(\mathbf{E}=\mathbf{B}=0\) inside; just outside \(E_t=0\), \(D_n=\rho_s\), \(B_n=0\), \(H_t=K\).

Part 6

Electromagnetic Wave Propagation

The Wave Equation

Derivation from Maxwell's Equations

In a source-free, linear, homogeneous, isotropic medium (\(\rho_v=0\), \(\mathbf{J}=0\)), taking the curl of Faraday's law and substituting Ampère–Maxwell:

\[ \boxed{\nabla^2\mathbf{E}-\mu\varepsilon\,\frac{\partial^2\mathbf{E}}{\partial t^2}=\mathbf{0}} \]

The same equation holds for \(\mathbf{H}\). This is the 3-D vector wave equation.

Wave Speed and Refractive Index

\[ v=\frac{1}{\sqrt{\mu\varepsilon}},\qquad c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}\approx 3\times 10^8\;\text{m/s},\qquad n=c/v=\sqrt{\mu_r\varepsilon_r} \]

The fact that \(c\) falls out from purely electrical and magnetic constants convinced Maxwell that light is an electromagnetic wave.

Plane Waves in Lossless Media

Plane Wave Parameters

\[ \mathbf{E}=E_0\,e^{j(\omega t-\beta z)}\,\hat{\mathbf{x}},\quad \beta=\omega\sqrt{\mu\varepsilon}\;\text{(rad/m)},\quad \lambda=2\pi/\beta=v/f,\quad v_p=\omega/\beta \]

\[ \eta=\frac{E}{H}=\sqrt{\frac{\mu}{\varepsilon}},\quad \eta_0=120\pi\approx 377\;\Omega \]

\(\mathbf{E}\perp\mathbf{H}\perp\) direction of propagation ⟹ TEM wave.

Three-dimensional diagram of a TEM plane wave travelling in the z-direction, showing orthogonal sinusoidal E and H field oscillations and the Poynting vector. — Transverse electromagnetic (TEM) plane wave: the electric field \(\mathbf{E}\), magnetic field \(\mathbf{H}\), and direction of propagation \(\mathbf{k}\) form a right-handed orthogonal triad. The Poynting vector \(\mathbf{S}=\mathbf{E}\times\mathbf{H}\) points in the direction of power flow.

Plane Waves in Lossy Media

Complex Propagation Constant

\[ \gamma=\alpha+j\beta=j\omega\sqrt{\mu\varepsilon}\sqrt{1+\frac{\sigma}{j\omega\varepsilon}} \]

Loss tangent: \(\tan\delta=\sigma/(\omega\varepsilon)\). \(\ll 1\): good dielectric; \(\gg 1\): good conductor.

Good Dielectric (\(\sigma/\omega\varepsilon\ll 1\))

\[ \alpha\approx\frac{\sigma}{2}\sqrt{\frac{\mu}{\varepsilon}},\qquad \beta\approx\omega\sqrt{\mu\varepsilon} \]

Behaves nearly like a lossless medium.

Good Conductor (\(\sigma/\omega\varepsilon\gg 1\))

\[ \alpha=\beta\approx\sqrt{\pi f\mu\sigma},\qquad \eta\approx\sqrt{\frac{\pi f\mu}{\sigma}}\angle 45^\circ \]

Strong attenuation; \(\mathbf{H}\) lags \(\mathbf{E}\) by \(45^\circ\).

Skin Depth and Surface Resistance

\[ \delta=\frac{1}{\alpha}=\frac{1}{\sqrt{\pi f\mu\sigma}}\quad\text{(depth where amplitude }\to 1/e),\qquad R_s=\frac{1}{\sigma\delta}=\sqrt{\frac{\pi f\mu}{\sigma}}\;\Omega \]

Copper: \(\delta\approx 8.5\) mm at 60 Hz; \(\approx 2\;\mu\)m at 1 GHz. Hence thin silver plating suffices at RF.

Poynting Vector and Power Flow

Poynting's Theorem

\[ \mathbf{S}=\mathbf{E}\times\mathbf{H}\;\text{(W/m}^2\text{)} \]

\[ -\oint_S(\mathbf{E}\times\mathbf{H})\cdot d\mathbf{S} = \frac{\partial}{\partial t}\int_V\!\left(\tfrac{1}{2}\varepsilon E^2+\tfrac{1}{2}\mu H^2\right)dV+\int_V\!\sigma E^2\,dV \]

Energy flowing in = rate of increase of stored energy + Ohmic dissipation.

Time-average: \(\mathbf{S}_{\text{avg}}=\tfrac{1}{2}\,\text{Re}\{\mathbf{E}\times\mathbf{H}^*\}\). For a plane wave in lossless medium: \(S_{\text{avg}}=|E_0|^2/(2\eta)\).

Polarisation of Plane Waves

Types of Polarisation

Linear: \(x\)- and \(y\)-components in phase (or \(180^\circ\) out). Axial ratio AR→∞.
Circular: equal amplitudes, phase difference \(\pm90^\circ\). AR=1. RHCP or LHCP by sense of rotation.
Elliptical: general case (unequal amplitudes or non-quadrature phase). AR between 1 and ∞.

Reflection and Transmission

Normal Incidence — Fresnel Coefficients

\[ \Gamma = \frac{\eta_2-\eta_1}{\eta_2+\eta_1},\qquad \tau = \frac{2\eta_2}{\eta_2+\eta_1}=1+\Gamma \]

Power conservation: \(|\Gamma|^2+(\eta_1/\eta_2)|\tau|^2=1\). At a perfect conductor (\(\eta_2=0\)): \(\Gamma=-1\), total reflection, SWR=\(\infty\).

Oblique Incidence — Snell's Laws and Special Angles

\[ \theta_i=\theta_r,\qquad n_1\sin\theta_i=n_2\sin\theta_t \]

Brewster's angle (no reflection for parallel/TM polarisation): \(\tan\theta_B=\sqrt{\varepsilon_2/\varepsilon_1}=n_2/n_1\).

Critical angle (total internal reflection when \(n_1>n_2\)): \(\sin\theta_c=n_2/n_1\).

Applications

Polarised sunglasses use Brewster's angle to eliminate glare. Optical fibres use total internal reflection beyond \(\theta_c\).

Part 7

Transmission Lines

Distributed Parameters and Telegrapher's Equations

Per-Unit-Length Model and Telegrapher's Equations

\[ -\frac{\partial V}{\partial z}=RI+L\frac{\partial I}{\partial t},\qquad -\frac{\partial I}{\partial z}=GV+C\frac{\partial V}{\partial t} \]

Phasor Form — Propagation Constant and Characteristic Impedance

\[ \gamma=\sqrt{(R+j\omega L)(G+j\omega C)}=\alpha+j\beta \]

\[ Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}} \]

Lossless line (\(R=G=0\)): \(Z_0=\sqrt{L/C}\in\mathbb{R}\), \(\beta=\omega\sqrt{LC}\), \(v_p=1/\sqrt{LC}\).

Distributed RLGC lumped-element model of a transmission line showing series R and L elements on the top conductor and shunt G and C elements between conductors, repeated for each infinitesimal section dz. — Distributed RLGC equivalent circuit of a transmission line. Each infinitesimal section \(dz\) contains series resistance \(R\,dz\) and inductance \(L\,dz\) on the top rail, with shunt conductance \(G\,dz\) and capacitance \(C\,dz\) between the rails.

Distortionless and Lossless Lines

Lossless Line (\(R=G=0\))

\(\gamma=j\beta\), \(\alpha=0\). No attenuation and no distortion — all frequencies travel at \(v_p=1/\sqrt{LC}\).

Distortionless Condition (Heaviside)

\[ \boxed{\frac{R}{L}=\frac{G}{C}} \]

Then \(\alpha\) is frequency-independent and \(\beta\propto\omega\) — pulses retain their shape as they attenuate uniformly. Loading coils (Pupin coils) were historically added to telephone lines to approach this condition.

Reflection and Input Impedance

Reflection Coefficient and Input Impedance

\[ \Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0},\qquad \Gamma(l)=\Gamma_L\,e^{-j2\beta l} \]

\[ \boxed{Z_{\text{in}}(l)=Z_0\,\frac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}} \]

Special Case	\(Z_{\text{in}}\)
Short-circuit (\(Z_L=0\))	\(jZ_0\tan\beta l\)
Open-circuit (\(Z_L=\infty\))	\(-jZ_0\cot\beta l\)
Matched (\(Z_L=Z_0\))	\(Z_0\)
Quarter-wave (\(l=\lambda/4\))	\(Z_0^2/Z_L\)
Half-wave (\(l=\lambda/2\))	\(Z_L\)

VSWR, Smith Chart, and Matching

Standing Wave Parameters

\[ \text{VSWR}=\frac{V_{\max}}{V_{\min}}=\frac{1+|\Gamma_L|}{1-|\Gamma_L|},\qquad \text{Return loss: }RL=-20\log_{10}|\Gamma|\;\text{dB} \]

Matched line: \(|\Gamma|=0\), VSWR=1, RL→∞.

Matching Techniques

Quarter-wave transformer: \(Z_{0T}=\sqrt{Z_0 Z_L}\) — works for real loads.
Single stub: short-circuited stub at distance \(d\) cancels susceptance.
Double stub: fixed spacing (often \(\lambda/8\)).
Lumped L-section: series and shunt reactance.

Smith chart showing constant-resistance circles (r-circles) and constant-reactance arcs (x-arcs) on the complex reflection coefficient plane, with the unit circle boundary. — The Smith chart: constant-resistance circles (orange) all pass through the open-circuit point \(\Gamma=+1\), and constant-reactance arcs (blue) all pass through the same point. One full revolution around the chart corresponds to a \(\lambda/2\) line length. Moving toward the source is clockwise.

Smith Chart Rules

One full turn = \(\lambda/2\). Moving toward the source is clockwise; toward the load is counter-clockwise. The centre (\(\Gamma=0\)) corresponds to the matched condition \(Z_L=Z_0\).

Special Transmission Line Geometries

Type	\(L'\) (H/m)	\(C'\) (F/m)
Coaxial (\(a,b\))	\(\dfrac{\mu}{2\pi}\ln(b/a)\)	\(\dfrac{2\pi\varepsilon}{\ln(b/a)}\)
Two-wire (\(a,d\))	\(\dfrac{\mu}{\pi}\cosh^{-1}(d/2a)\)	\(\dfrac{\pi\varepsilon}{\cosh^{-1}(d/2a)}\)
Parallel-plate (\(w,d\))	\(\dfrac{\mu d}{w}\)	\(\dfrac{\varepsilon w}{d}\)

Universal TEM Line Identities

\[ L'C'=\mu\varepsilon,\qquad G'/C'=\sigma/\varepsilon \]

Coaxial characteristic impedance: \(Z_0=\dfrac{60}{\sqrt{\varepsilon_r}}\ln(b/a)\;\Omega\).

Part 8

Waveguides and Cavity Resonators

Rectangular Waveguide — TE and TM Modes

Cutoff Frequency

\[ \boxed{f_c^{(mn)}=\frac{1}{2\sqrt{\mu\varepsilon}}\sqrt{\left(\frac{m}{a}\right)^{\!2}+\left(\frac{n}{b}\right)^{\!2}}} \]

Propagation Constants and Mode Rules

\[ k_c=\sqrt{(m\pi/a)^2+(n\pi/b)^2},\qquad \beta=\sqrt{k^2-k_c^2}\;\text{(real if }f>f_c\text{)} \]

Dominant mode: \(TE_{10}\), \(f_c=c/2a\) (assuming \(a>b\)).
\(TM_{m0}\) and \(TM_{0n}\) modes are forbidden (\(E_z=0\) trivially).
\(TE_{00}\) mode is forbidden (\(H_z=0\)).

Cross-section of a rectangular waveguide showing the TE10 dominant mode electric field distribution as half-sine variation in the x-direction and uniform in the y-direction, with field arrows. — Electric field distribution of the dominant \(TE_{10}\) mode in a rectangular waveguide. The transverse electric field follows a half-sine variation in the \(x\)-direction and is uniform in the \(y\)-direction.

Waveguide Parameters

Guide Wavelength, Phase and Group Velocities, Impedance

\[ \lambda_g=\frac{\lambda}{\sqrt{1-(f_c/f)^2}},\qquad v_p=\frac{c}{\sqrt{1-(f_c/f)^2}},\qquad v_g=c\sqrt{1-(f_c/f)^2} \]

\[ \boxed{v_p\cdot v_g=c^2},\qquad Z_{TE}=\frac{\eta}{\sqrt{1-(f_c/f)^2}},\qquad Z_{TM}=\eta\sqrt{1-(f_c/f)^2} \]

Phase Velocity Does Not Violate Relativity

\(v_p>c\) is the speed of phase, not energy or information. The group velocity \(v_g<c\) is the actual signal speed. The product \(v_p v_g=c^2\) links the two.

Below cutoff (\(f<f_c\)): \(\beta\) becomes imaginary → evanescent wave with exponential amplitude decay — no power is transported.

Circular Waveguide and Cavity Resonators

Circular Waveguide (radius \(a\))

Modes are \(TE_{np}\) and \(TM_{np}\), involving Bessel functions \(J_n\).

Dominant mode: \(TE_{11}\), \(k_c a=1.8412\) (first root of \(J'_1\)).

\[ f_c^{TE_{11}} = \frac{1.8412}{2\pi a\sqrt{\mu\varepsilon}} \]

\(TM_{01}\) has \(k_c a=2.4048\) (first root of \(J_0\)).

Rectangular Cavity Resonator (\(a\times b\times d\))

\[ f_{mnp}=\frac{1}{2\sqrt{\mu\varepsilon}}\!\sqrt{\!\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2+\left(\frac{p}{d}\right)^2} \]

For \(a>d>b\), dominant mode is \(TE_{101}\).

Quality Factor

\[ Q = 2\pi\,\frac{\text{energy stored}}{\text{energy lost per cycle}}=\frac{\omega W}{P_L} \]

Typical metallic cavity: \(Q\sim 10^4\) — far higher than any lumped LC circuit.

Practical Note

Microwave ovens use the \(TE_{10}\) mode at 2.45 GHz in standard waveguide WR-340. For single-mode operation the band is chosen so that only the dominant mode propagates; higher-order modes are deliberately suppressed.

Part 9

Antennas and Radiation

Hertzian Dipole Radiation

Far-Field of a Short Dipole (\(d\ell\ll\lambda\))

\[ E_\theta = j\,\frac{\eta_0 I_0\,d\ell}{2\lambda r}\sin\theta\,e^{-j\beta r},\qquad H_\phi=E_\theta/\eta_0 \]

Pattern: \(\sin\theta\) — nothing radiated along the axis (\(\theta=0,\pi\)); maximum at \(\theta=\pi/2\) (broadside).

Radiated Power and Radiation Resistance

\[ P_{\text{rad}}=40\pi^2\!\left(\frac{d\ell}{\lambda}\right)^{\!2} I_0^2,\qquad R_{\text{rad}}=80\pi^2\!\left(\frac{d\ell}{\lambda}\right)^{\!2}\;\Omega \]

Half-wave dipole: \(R_{\text{rad}}\approx 73\;\Omega\), directivity \(D\approx 1.64\) (2.15 dBi).

Figure-of-eight radiation pattern of a Hertzian dipole showing maximum radiation broadside to the dipole axis and null radiation along the dipole axis, with the dipole oriented vertically. — Radiation pattern of the Hertzian (short) dipole: a toroidal (figure-of-eight in cross-section) pattern with nulls along the dipole axis and maximum radiation broadside (\(\theta=90^\circ\)).

Antenna Parameters and Friis Equation

Key Antenna Parameters

\[ U(\theta,\phi)=r^2 S_{\text{avg}},\quad D=\frac{4\pi U_{\max}}{P_{\text{rad}}},\quad G=e_r D,\quad A_e=\frac{\lambda^2}{4\pi}G \]

\(U\): radiation intensity (W/sr); \(D\): directivity; \(G\): gain; \(A_e\): effective aperture; \(e_r\): radiation efficiency.

Friis Transmission Equation

\[ \boxed{\frac{P_r}{P_t}=G_t G_r\!\left(\frac{\lambda}{4\pi R}\right)^{\!2}} \]

Free-space spreading gives \(1/R^2\) dependence; gains multiply. This is the foundation of every link budget.

Radar Equation

\[ P_r = \frac{P_t G_t G_r \lambda^2 \sigma_{\text{rcs}}}{(4\pi)^3 R^4} \]

Round-trip propagation gives \(1/R^4\); \(\sigma_{\text{rcs}}\) is the radar cross-section of the target.

Antenna Arrays

Uniform Linear Array (ULA) of \(N\) Elements

\[ \text{AF}(\theta)=\frac{\sin(N\psi/2)}{\sin(\psi/2)},\qquad \psi=\beta d\cos\theta+\alpha \]

Pattern multiplication: \(E_{\text{total}}=E_{\text{element}}\times\text{AF}\).

Broadside: \(\alpha=0\), maximum at \(\theta=90^\circ\). End-fire: \(\alpha=-\beta d\), maximum along the array axis.

The Electromagnetic Spectrum

Band	Frequency Range	Typical Applications
Radio	\(10^3\)–\(10^9\) Hz	AM/FM, TV, cellular
Microwave	\(10^9\)–\(10^{12}\) Hz	Radar, satellite, WiFi, 5G
Infrared	\(10^{12}\)–\(10^{14}\) Hz	Thermal imaging, IR remote control
Visible	\(\sim 4\)–\(7\times10^{14}\) Hz (400–700 nm)	Human vision
Ultraviolet	\(10^{15}\)–\(10^{16}\) Hz	Sterilisation, fluorescence
X-ray	\(10^{16}\)–\(10^{19}\) Hz	Medical imaging
Gamma ray	>\(10^{19}\) Hz	Nuclear decay

Common Microwave Bands

L (1–2 GHz), S (2–4), C (4–8), X (8–12), Ku (12–18), K, Ka. 2.4 GHz: WiFi / microwave ovens; 5 GHz: WiFi; 28 & 39 GHz: 5G mmWave.

Part 10

Worked Examples (GATE Style)

Example 1 — Electrostatics Superposition

Problem

Charges \(Q_1=+4\) nC at \((0,0,0)\) m and \(Q_2=-2\) nC at \((3,0,0)\) m. Find \(\mathbf{E}\) at \(P=(0,4,0)\) m.

Solution

From \(Q_1\): \(R_1=4\) m, \(\hat{\mathbf{R}}_1=\hat{\mathbf{y}}\).

\[ E_1=\frac{4\times 10^{-9}}{4\pi(8.854\times 10^{-12})(16)}\approx 2.25\;\text{V/m in }\hat{\mathbf{y}} \]

From \(Q_2\): \(\mathbf{R}_2=(-3,4,0)\), \(R_2=5\) m, \(\hat{\mathbf{R}}_2=(-0.6,0.8,0)\). Since \(Q_2\) is negative, field points toward \(Q_2\):

\[ E_2\approx 0.72\;\text{V/m}\;\;\Rightarrow\;\; \mathbf{E}_2=(0.432,-0.576,0)\;\text{V/m} \]

Total: \(\mathbf{E}_P=(0.432,\;1.674,\;0)\) V/m, \(|\mathbf{E}_P|\approx 1.73\) V/m.

Example 2 — Reflection Coefficient and VSWR

Problem

Lossless line: \(Z_0=50\;\Omega\), \(Z_L=100+j50\;\Omega\). Find \(\Gamma_L\) and VSWR.

Solution

\[ \Gamma_L=\frac{(50+j50)}{(150+j50)}=\frac{10000+j5000}{25000}=0.4+j0.2 \]

\[ |\Gamma_L|=\sqrt{0.20}\approx 0.447,\qquad \text{VSWR}=\frac{1.447}{0.553}\approx 2.62 \]

Return loss: \(RL=-20\log_{10}(0.447)\approx 7.0\) dB.

Example 3 — Plane Wave Parameters

Problem

A 1 GHz plane wave in lossless medium with \(\varepsilon_r=4\), \(\mu_r=1\). Find \(\beta\), \(v_p\), \(\lambda\), \(\eta\).

Solution

\(v_p=c/\sqrt{\mu_r\varepsilon_r}=3\times 10^8/2=1.5\times 10^8\) m/s.
\(\beta=\omega/v_p=2\pi\times 10^9/(1.5\times 10^8)\approx 41.89\) rad/m.
\(\lambda=v_p/f=0.15\) m = 15 cm.
\(\eta=\eta_0/\sqrt{\varepsilon_r}=120\pi/2\approx 188.5\;\Omega\).

Sanity check: \(\lambda f=0.15\times 10^9=1.5\times 10^8=v_p\) ✓

Example 4 — Waveguide Analysis

Problem

Air-filled rectangular waveguide, \(a=2.5\) cm, \(b=1.0\) cm. Find \(f_c\) of the dominant mode and \(\lambda_g\) at 10 GHz.

Solution

Dominant mode \(TE_{10}\): \(f_c=c/(2a)=3\times 10^8/(2\times 0.025)=6\) GHz.

At \(f=10\) GHz: \((f_c/f)^2=0.36\), \(\sqrt{1-0.36}=0.8\).

\[ \lambda_0=c/f=3\;\text{cm}\;\Rightarrow\;\lambda_g=\lambda_0/0.8=3.75\;\text{cm} \]

\[ v_p=c/0.8=3.75\times 10^8\;\text{m/s},\quad v_g=0.8c=2.4\times 10^8\;\text{m/s} \]

Verify: \(v_p v_g=(3.75)(2.4)\times 10^{16}=9\times 10^{16}=c^2\) ✓

Example 5 — Skin Depth in Copper

Problem

Skin depth in copper (\(\sigma=5.8\times 10^7\) S/m, \(\mu=\mu_0\)) at 1 MHz and at 1 GHz.

Solution

\[ \delta=\frac{1}{\sqrt{\pi f\mu_0\sigma}} \]

At \(f=10^6\) Hz: \(\delta\approx 66\;\mu\)m.

Since \(\delta\propto 1/\sqrt{f}\), scaling by \(\sqrt{1000}\approx 31.6\): \(\delta_{1\,\text{GHz}}\approx 2.1\;\mu\)m.

Interpretation: at GHz frequencies only the outermost few micrometres of copper conduct. Thin silver plating on RF cables is therefore sufficient.

Part 11

Summary, Formula Recap, and GATE Aids

Constants and Free-Space Values

Constant	Value
\(\varepsilon_0\) (vacuum permittivity)	\(8.854\times 10^{-12}\) F/m \(=\tfrac{1}{36\pi}\times 10^{-9}\)
\(\mu_0\) (vacuum permeability)	\(4\pi\times 10^{-7}\) H/m
\(c\) (speed of light)	\(3\times 10^8\) m/s
\(\eta_0\) (free-space impedance)	\(120\pi\approx 377\;\Omega\)
\(e\) (elementary charge)	\(1.602\times 10^{-19}\) C

Useful Identities

\[ c=\frac{1}{\sqrt{\mu_0\varepsilon_0}},\quad \eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}},\quad \frac{\eta_0}{c}=\mu_0,\quad c\,\eta_0=\frac{1}{\varepsilon_0} \]

Quick Formula Recap — Statics

Quantity	Formula
Coulomb force	\(F=Q_1 Q_2/(4\pi\varepsilon R^2)\)
\(E\)-field of point charge	\(E=Q/(4\pi\varepsilon r^2)\)
\(E\)-field of line charge	\(E=\rho_l/(2\pi\varepsilon\rho)\)
\(E\)-field of sheet	\(E=\rho_s/(2\varepsilon)\)
Potential of point charge	\(V=Q/(4\pi\varepsilon r)\)
Dipole potential	\(V=p\cos\theta/(4\pi\varepsilon r^2)\)
Energy density (E)	\(w_E=\tfrac{1}{2}\varepsilon E^2\)
Parallel-plate capacitor	\(C=\varepsilon A/d\)
Coaxial capacitor	\(C=2\pi\varepsilon L/\ln(b/a)\)
Resistance	\(R=\ell/(\sigma A)\)
RC duality	\(RC=\varepsilon/\sigma\)
Ohm's law (point form)	\(\mathbf{J}=\sigma\mathbf{E}\)
Biot–Savart	\(d\mathbf{H}=I\,d\mathbf{l}\times\hat{\mathbf{R}}/(4\pi R^2)\)
Field of long wire	\(H=I/(2\pi\rho)\)
Field at loop centre	\(H=I/(2a)\)
Field, infinite solenoid	\(H=nI\)
Force on current-carrying wire	\(\mathbf{F}=\int I\,d\mathbf{l}\times\mathbf{B}\)
Energy density (B)	\(w_m=B^2/(2\mu)\)
Solenoid inductance	\(L=\mu N^2 A/\ell\)
Reluctance	\(\mathscr{R}=\ell/(\mu A)\)

Quick Formula Recap — Waves and Lines

Quantity	Formula
Wave speed	\(v=1/\sqrt{\mu\varepsilon}\)
Intrinsic impedance	\(\eta=\sqrt{\mu/\varepsilon}\)
Poynting vector	\(\mathbf{S}=\mathbf{E}\times\mathbf{H}\)
Average power (plane wave)	\(\|E_0\|^2/(2\eta)\)
Skin depth	\(\delta=1/\sqrt{\pi f\mu\sigma}\)
Loss tangent	\(\tan\delta=\sigma/(\omega\varepsilon)\)
Reflection coefficient	\(\Gamma=(\eta_2-\eta_1)/(\eta_2+\eta_1)\)
Brewster's angle	\(\tan\theta_B=\sqrt{\varepsilon_2/\varepsilon_1}\)
Critical angle	\(\sin\theta_c=n_2/n_1\)
Char. impedance (lossless)	\(Z_0=\sqrt{L/C}\)
Input impedance	\(Z_0(Z_L+jZ_0\tan\beta l)/(Z_0+jZ_L\tan\beta l)\)
Quarter-wave transformer	\(Z_{0T}=\sqrt{Z_0 Z_L}\)
VSWR	\((1+\|\Gamma\|)/(1-\|\Gamma\|)\)
Distortionless condition	\(R/L=G/C\)
Waveguide cutoff	\(f_c=\tfrac{1}{2\sqrt{\mu\varepsilon}}\sqrt{(m/a)^2+(n/b)^2}\)
Guide wavelength	\(\lambda_g=\lambda/\sqrt{1-(f_c/f)^2}\)
\(v_p v_g\) (air waveguide)	\(=c^2\)
Friis equation	\(P_r/P_t=G_t G_r(\lambda/4\pi R)^2\)
Half-wave dipole \(R_{\text{rad}}\)	\(\approx 73\;\Omega\)

GATE Pitfalls and Tips

Units — Always Verify!

\(\mathbf{E}\): V/m \(\mathbf{D}\): C/m² \(\mathbf{H}\): A/m \(\mathbf{B}\): T = Wb/m² \(\mathbf{P}\): C/m² \(\mathbf{M}\): A/m \(\mathbf{J}\): A/m² \(\mathbf{S}\): W/m²

Field Falloff with Distance

Point: \(\sim 1/r^2\). Line: \(\sim 1/\rho\). Sheet: constant (distance-independent). Inside a uniformly charged sphere: \(\mathbf{E}\propto r\) (linear, not zero, not \(1/r^2\)).

Boundary Conditions Summary

Tangential \(\mathbf{E}\) always continuous. Normal \(\mathbf{B}\) always continuous. \(D_n\) jumps by \(\rho_s\). \(H_t\) jumps by \(K\).

Skin Depth and Losses

\(\delta=1/\sqrt{\pi f\mu\sigma}\). In good conductors \(\alpha=\beta=1/\delta\). \(\delta\propto 1/\sqrt{f}\).

Waveguides

Dominant mode in rectangular: \(TE_{10}\), \(f_c=c/2a\). Modes \(TM_{m0}\), \(TM_{0n}\), \(TE_{00}\) are all forbidden. \(v_p v_g=c^2\) (air-filled).

Transmission Lines

Quarter-wave transformer for real loads: \(Z_{0T}=\sqrt{Z_0 Z_L}\). Smith chart: one full turn = \(\lambda/2\); moving toward source is clockwise. Distortionless: \(R/L=G/C\). Lossless: \(R=G=0\).

Polarisation and Antennas

Circular polarisation: equal amplitudes, \(\pm 90^\circ\) phase difference. Brewster angle: only for parallel (TM) polarisation. Friis: received power falls as \(1/R^2\); radar as \(1/R^4\).

Study Strategy for GATE

Master vector calculus first — every EMFT problem reduces to grad, div, or curl.
Memorise standard field formulas for point, line, sheet, sphere, and coaxial geometries.
Understand boundary conditions thoroughly — they are a consistent high-weightage topic.
Practice Maxwell's equations in both integral and differential forms.
Drill transmission-line problems on the Smith chart.
Know waveguide cutoff and mode-existence rules cold.
Derive, don't memorise blindly. A derivation protects you when question wording shifts.
Solve previous-year papers — patterns recur: skin depth, \(\Gamma\), \(f_c\), Friis, polarisation.

Suggested Time Allocation

Statics: 30% | Time-varying + Maxwell: 15% | EM Waves: 20% | Lines & Guides: 25% | Antennas: 10%.

Conceptual Map — How Topics Connect

Conceptual dependency map linking Vector Analysis to Electrostatics and Magnetostatics, which flow into Time-Varying Fields, then Maxwell's Equations, then EM Waves, and finally branching into Transmission Lines, Waveguides and Cavities, and Antennas and Radiation. — Topic dependency map for EMFT: every downstream subject builds on Maxwell's equations, which in turn unify the static laws. The path Vector Analysis → Statics → Maxwell → Waves → Guided Structures & Antennas represents the logical progression of the course.

The Unifying Thread

Every topic downstream depends on Maxwell's equations. Master them first — they unify everything learned in statics into a single framework supporting wave propagation, guidance, and radiation.

Recommended References

M. N. O. Sadiku, Elements of Electromagnetics — Oxford University Press. (Primary text)
W. H. Hayt & J. A. Buck, Engineering Electromagnetics — McGraw-Hill.
D. K. Cheng, Field and Wave Electromagnetics — Pearson.
J. D. Kraus, Electromagnetics — McGraw-Hill.
D. J. Griffiths, Introduction to Electrodynamics — Pearson.
D. M. Pozar, Microwave Engineering — for transmission lines, waveguides, and the Smith chart.
C. A. Balanis, Antenna Theory: Analysis and Design — for antennas and arrays.
GATE previous-year papers (EE and EC streams).

Online Resources

NPTEL lectures on Electromagnetic Theory — Prof. Pradeep Kumar K, IIT Madras.
MIT OpenCourseWare 8.03 — Physics III: Vibrations and Waves.
NPTEL courses on Transmission Lines and Waveguides — excellent for Smith chart practice.