GATE EC Electromagnetics Formula Sheet

Organised by the official GATE 2027 EC syllabus, Section 8: Electromagnetics. Each worked example was checked in code. Practise these: GATE EC Electromagnetics previous year questions.

Maxwell's equations

Syllabus: Differential and integral forms and their interpretation, boundary conditions, wave equation, Poynting vector.

Maxwell's equations and displacement current

\nabla \cdot \mathbf{D} = \rho_v,\quad \nabla \cdot \mathbf{B} = 0,\quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},\quad \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}

\begin{aligned} \nabla \cdot \mathbf{D} &= \rho_v, \\ \nabla \cdot \mathbf{B} &= 0, \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \times \mathbf{H} &= \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \end{aligned}

Symbols

$\mathbf{D} = \varepsilon\mathbf{E}$ : electric flux density (C/m²)
$\mathbf{B} = \mu\mathbf{H}$ : magnetic flux density (T)
$\partial\mathbf{D}/\partial t$ : displacement current density (A/m²)

Applies when

Linear, isotropic media. In phasor form ∂/∂t becomes jω, so the displacement current density is jωεE.

Common mistake

Forgetting the displacement current term, which is what lets current 'flow' through a capacitor.

Worked example

E = 10 sin(2π × 10⁹ t) V/m in free space: peak displacement current density: 0.556 A/m²

Boundary conditions

E_{t1} = E_{t2}, \qquad D_{n1} - D_{n2} = \rho_s, \qquad H_{t1} - H_{t2} = K, \qquad B_{n1} = B_{n2}

\begin{aligned} E_{t1} &= E_{t2}, \\ D_{n1} - D_{n2} &= \rho_s, \\ H_{t1} - H_{t2} &= K, \\ B_{n1} &= B_{n2} \end{aligned}

Symbols

$E_t, H_t$ : tangential components
$D_n, B_n$ : normal components
$\rho_s, K$ : surface charge (C/m²) and surface current (A/m)

Applies when

At an interface between two media. For a charge-free dielectric interface, tan θ₁ / tan θ₂ = ε₁ / ε₂ for the field lines.

Common mistake

Making the normal E continuous: it is the normal D (= εE) that is continuous without surface charge.

Worked example

E meets a boundary from εr = 1 into εr = 4 at 45° to the normal: the angle in the second medium: 75.96 ° from the normal

Poynting vector

\mathbf{S} = \mathbf{E} \times \mathbf{H}, \qquad S_{avg} = \frac{|E_0|^2}{2\eta}

\begin{aligned} \mathbf{S} &= \mathbf{E} \times \mathbf{H}, \\ S_{avg} &= \frac{|E_0|^2}{2\eta} \end{aligned}

Symbols

$E_0$ : peak electric field amplitude (V/m)
$\eta$ : intrinsic impedance, 377 Ω in free space

Applies when

The time average of a uniform plane wave in a lossless medium. With RMS values there is no factor of 2.

Common mistake

Using E₀²/η with a peak amplitude: the average needs the ½.

Worked example

E_{0}

= 100 V/m in free space: 13.27 W/m²

Plane waves and properties

Syllabus: Reflection and refraction, polarization, phase and group velocity, propagation through various media, skin depth.

Wave parameters

\beta = \omega\sqrt{\mu\varepsilon}, \qquad v_p = \frac{1}{\sqrt{\mu\varepsilon}} = \frac{c}{\sqrt{\mu_r\varepsilon_r}}, \qquad \eta = \sqrt{\frac{\mu}{\varepsilon}} = \frac{377}{\sqrt{\varepsilon_r}}\ \Omega\ (\mu_r = 1)

\begin{aligned} \beta &= \omega\sqrt{\mu\varepsilon}, \\ v_p &= \frac{1}{\sqrt{\mu\varepsilon}} \\ &= \frac{c}{\sqrt{\mu_r\varepsilon_r}}, \\ \eta &= \sqrt{\frac{\mu}{\varepsilon}} \\ &= \frac{377}{\sqrt{\varepsilon_r}}\ \Omega\ (\mu_r = 1) \end{aligned}

Symbols

$\beta$ : phase constant (rad/m)
$\eta$ : intrinsic impedance (Ω)

Applies when

Lossless, non-magnetic medium for the last form. A lossless medium with constant

\varepsilon_r

is non-dispersive, so

v_{g}

v_{p}

Common mistake

Dividing 377 by

\varepsilon_r

instead of its square root.

Worked example

εr = 4, µr = 1: intrinsic impedance: 188.4 Ω

Normal-incidence reflection

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

\begin{aligned} \Gamma &= \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}, \\ \tau &= \frac{2\eta_2}{\eta_2 + \eta_1} \\ &= 1 + \Gamma \end{aligned}

Symbols

$\Gamma, \tau$ : reflection and transmission coefficients for E
$\eta_1, \eta_2$ : intrinsic impedances, incident side first

Applies when

A plane wave hitting a plane boundary head-on. The reflected power fraction is |Γ|².

Common mistake

Taking τ = 1 − Γ: the tangential E is continuous, so 1 + Γ = τ.

Worked example

From air into εr = 4: -0.333 (11.1% of the power reflects)

Snell's law, Brewster and critical angles

n_1\sin\theta_1 = n_2\sin\theta_2, \qquad \tan\theta_B = \frac{n_2}{n_1}, \qquad \sin\theta_c = \frac{n_2}{n_1}\ (n_1 > n_2)

\begin{aligned} n_1\sin\theta_1 &= n_2\sin\theta_2, \\ \tan\theta_B &= \frac{n_2}{n_1}, \\ \sin\theta_c &= \frac{n_2}{n_1}\ (n_1 > n_2) \end{aligned}

Symbols

$n = \sqrt{\varepsilon_r}$ : refractive index
$\theta_B$ : Brewster angle: no reflection for parallel polarisation
$\theta_c$ : critical angle for total internal reflection

Applies when

Non-magnetic dielectrics. The Brewster angle exists only for parallel (TM) polarisation.

Common mistake

Using the ratio of permittivities directly in

\tan\theta_B

: it is the ratio of the refractive indices, the square root.

Worked example

Air to εr = 4: the Brewster angle: 63.43 °

Skin depth

\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}, \qquad \text{fields fall as } e^{-z/\delta}

\begin{aligned} \delta &= \frac{1}{\sqrt{\pi f \mu \sigma}}, \\ \text{fields fall as } e^{-z/\delta} \end{aligned}

Symbols

$\sigma$ : conductivity (S/m)
$f$ : frequency (Hz)

Applies when

Good conductors, σ ≫ ωε. The field falls to 1/e (37%) in one skin depth.

Common mistake

Using ω in place of πf, which is off by √2.

Worked example

Copper, σ = 5.8 × 10⁷ S/m, at 1 MHz: 66.1 µm

Polarisation sense

\mathbf{E} = \hat{x}E_x\cos(\omega t - \beta z) + \hat{y}E_y\cos(\omega t - \beta z + \phi):\quad 0 < \phi < \pi \Rightarrow \text{left-handed},\ -\pi < \phi < 0 \Rightarrow \text{right-handed}

\begin{gathered} \mathbf{E} = \hat{x}E_x\cos(\omega t - \beta z) \\ +\, \hat{y}E_y\cos(\omega t - \beta z + \phi) \\ 0 < \phi < \pi: \text{left-handed} \\ -\pi < \phi < 0: \text{right-handed} \end{gathered}

Symbols

$\phi$ : phase of $E_{y}$ relative to $E_{x}$

Applies when

Wave travelling along +z, IEEE convention (judged looking along the direction of travel). φ = 0 or π is linear; |

E_{x}

| = |

E_{y}

| with φ = ±π/2 is circular.

Common mistake

Forgetting to flip the rule for a wave travelling along −z.

Worked example

E_{x}

= 2,

E_{y}

= 1, φ = π/2: the axial ratio: 2 (left-handed elliptical)

Transmission lines

Syllabus: Equations, characteristic impedance, impedance matching, impedance transformation, S-parameters, Smith chart.

Characteristic impedance

Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \;\to\; \sqrt{\frac{L}{C}}\ (\text{lossless}), \qquad v_p = \frac{1}{\sqrt{LC}}

\begin{aligned} Z_0 &= \sqrt{\frac{R + j\omega L}{G + j\omega C}} \; \\ &\to\; \sqrt{\frac{L}{C}}\ (\text{lossless}), \\ v_p &= \frac{1}{\sqrt{LC}} \end{aligned}

Symbols

$R, L, G, C$ : per-unit-length line constants

Applies when

Lossless or low-loss lines for the simple form.

Common mistake

Using total inductance and capacitance of a length of line: Z₀ uses per-unit-length values, and it doesn't depend on length.

Worked example

L = 250 nH/m, C = 100 pF/m: 50 Ω (

v_{p}

= 2 × 10⁸ m/s)

Input impedance of a line

Z_{in} = Z_0\,\frac{Z_L + jZ_0\tan\beta l}{Z_0 + jZ_L\tan\beta l}, \qquad \frac{\lambda}{4}:\ Z_{in} = \frac{Z_0^2}{Z_L}, \qquad \frac{\lambda}{2}:\ Z_{in} = Z_L

\begin{aligned} Z_{in} &= Z_0\,\frac{Z_L + jZ_0\tan\beta l}{Z_0 + jZ_L\tan\beta l}, \\ \frac{\lambda}{4}:\ Z_{in} &= \frac{Z_0^2}{Z_L}, \\ \frac{\lambda}{2}:\ Z_{in} &= Z_L \end{aligned}

Symbols

$\beta l$ : electrical length (rad)
$Z_L$ : load impedance (Ω)

Applies when

Lossless line. A shorted line gives jZ₀ tan βl, an open one −jZ₀ cot βl.

Common mistake

Measuring l from the generator: it is the distance from the load.

Worked example

A λ/4 line of 50 Ω ending in 100 Ω: 25 Ω

Reflection coefficient and VSWR

\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}, \qquad \text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}, \qquad |\Gamma| = \frac{\text{VSWR} - 1}{\text{VSWR} + 1}

\begin{aligned} \Gamma_L &= \frac{Z_L - Z_0}{Z_L + Z_0}, \\ \text{VSWR} &= \frac{1 + |\Gamma|}{1 - |\Gamma|}, \\ |\Gamma| &= \frac{\text{VSWR} - 1}{\text{VSWR} + 1} \end{aligned}

Symbols

$\Gamma_L$ : load reflection coefficient

Applies when

Lossless line, so |Γ| is the same everywhere on it. Voltage minima are λ/2 apart, with a maximum λ/4 from each.

Common mistake

Reading a VSWR below 1: it is always 1 or more.

Worked example

100 Ω on a 50 Ω line: 2 (Γ = 1/3)

Quarter-wave transformer

Z_q = \sqrt{Z_0 Z_L}

Symbols

$Z_q$ : characteristic impedance of the λ/4 section (Ω)

Applies when

Real load, at the design frequency only. A complex load first needs a length of line to make it real.

Common mistake

Using the arithmetic mean of the two impedances.

Worked example

Matching 200 Ω to 50 Ω: 100 Ω

Waveguides, optical fibres and antennas

Syllabus: Rectangular and circular waveguides, light propagation in optical fibers, dipole and monopole antennas, linear antenna arrays.

Rectangular waveguide

f_c = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}, \qquad v_p = \frac{c}{\sqrt{1 - (f_c/f)^2}}, \qquad v_p v_g = c^2

\begin{aligned} f_c &= \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}, \\ v_p &= \frac{c}{\sqrt{1 - (f_c/f)^2}}, \\ v_p v_g &= c^2 \end{aligned}

Symbols

$a, b$ : inner width and height, a > b (m)
$m, n$ : mode indices; TE₁₀ is the dominant mode

Applies when

Air-filled guide. Only modes with

f_{c}

below the operating frequency propagate.

Common mistake

Expecting

v_{p}

≤ c inside a guide: the phase velocity exceeds c, and the group velocity is below it.

Worked example

WR-90 guide, a = 2.286 cm: TE₁₀ cut-off: 6.56 GHz

Optical fibre numerical aperture

NA = \sin\theta_a = \sqrt{n_1^2 - n_2^2}, \qquad V = \frac{2\pi a}{\lambda} NA \ (\text{single mode if } V < 2.405)

\begin{gathered} NA = \sin\theta_a = \sqrt{n_1^2 - n_2^2} \\ V = \frac{2\pi a}{\lambda} NA \\ \text{(single mode if } V < 2.405) \end{gathered}

Symbols

$n_1, n_2$ : core and cladding indices
$\theta_a$ : acceptance half-angle in air
$a$ : core radius

Applies when

Step-index fibre, light entering from air.

Common mistake

Using n₁ − n₂ without the squares.

Worked example

n_{1}

= 1.48,

n_{2}

= 1.46: 0.2425

Dipole radiation resistance

R_r = 80\pi^2\left(\frac{dl}{\lambda}\right)^2\ (\text{short dipole}), \qquad R_r \approx 73\ \Omega\ (\lambda/2), \qquad R_r \approx 36.5\ \Omega\ (\lambda/4 \text{ monopole})

\begin{gathered} \text{Short dipole:} \\ R_r = 80\pi^2\left(\frac{dl}{\lambda}\right)^2 \\ \lambda/2 \text{ dipole: } R_r \approx 73\ \Omega \\ \lambda/4 \text{ monopole: } R_r \approx 36.5\ \Omega \end{gathered}

Symbols

$dl$ : length of a short (Hertzian) dipole, dl ≪ λ

Applies when

Free space, uniform current for the Hertzian dipole. A monopole over a ground plane has half the dipole's radiation resistance.

Common mistake

Using the Hertzian formula for a half-wave dipole, whose current isn't uniform.

Worked example

dl = λ/50: 0.316 Ω

Uniform linear array

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

\begin{aligned} AF(\psi) &= \frac{\sin(N\psi/2)}{N\sin(\psi/2)}, \\ \psi &= \beta d\cos\theta + \alpha \end{aligned}

Symbols

$N$ : number of elements
$d$ : element spacing
$\alpha$ : progressive phase shift
$\theta$ : angle from the array axis

Applies when

Identical, equally spaced, equal-amplitude elements. α = 0 gives a broadside beam; α = −βd gives end-fire.

Common mistake

Measuring θ from broadside when the formula measures it from the array axis.

Worked example

N = 4, d = λ/2, broadside: the first null from the axis: 60 ° (cos θ = 1/2)

Practise these: GATE EC Electromagnetics previous year questions

Source for the section headings: the official GATE 2027 EC syllabus, IIT Madras. The formulas, examples and notes are Studyloaf's own.

GATE EC Electromagnetics Formula Sheet

Maxwell's equations

Maxwell's equations and displacement current

Boundary conditions

Poynting vector

Plane waves and properties

Wave parameters

Normal-incidence reflection

Snell's law, Brewster and critical angles

Skin depth

Polarisation sense

Transmission lines

Characteristic impedance

Input impedance of a line

Reflection coefficient and VSWR

Quarter-wave transformer

Waveguides, optical fibres and antennas

Rectangular waveguide

Optical fibre numerical aperture

Dipole radiation resistance

Uniform linear array

Other GATE EC formula sheets