Electromagnetic Fields
Electromagnetic Fields in GATE EE covers electrostatics, magnetostatics, Maxwell's equations, plane waves, and transmission-line fundamentals. The exam favours application of the integral laws to simple symmetric geometries and boundary-condition reasoning.
Electrostatics
- Gauss's law: ∮D·dS = Q_enc with D = εE.
- Potential V = −∫E·dl and E = −∇V; in charge-free regions ∇²V = 0.
- At an interface with no free surface charge, D_n is continuous; E_t is always continuous.
Magnetostatics
- Ampere's law: ∮H·dl = I_enc with B = μH.
- Inductance L = NΦ/I and stored energy ½LI².
Maxwell and plane waves
- ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t, ∇·D = ρ, ∇·B = 0.
- Lossless medium: η = √(μ/ε), v = 1/√(με), β = ω√(με).
- Skin depth δ = 1/√(πfμσ).
Worked example: a coaxial cable with inner radius a = 1 mm, outer b = 5 mm, dielectric ε_r = 2.5. Capacitance per metre C = 2πε/ln(b/a) = 2π(2.5)(8.854×10⁻¹²)/ln 5 = 1.391×10⁻¹⁰/1.609 = 8.64×10⁻¹¹ F/m ≈ 86.4 pF/m. The geometry enters only through ln(b/a), and the answer scales linearly with ε_r.
Traps that cost marks
- Using ε₀ alone when a relative permittivity is quoted.
- Swapping the boundary conditions: normal D and tangential E are the continuous pair (absent free charge).
- Poynting vector direction: verify the sign of E × H rather than assuming propagation along +z.