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3.5 Electromagnetics

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FE Electrical and Computer Exam Exam

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1 FE Electrical & Computer: Exam Map, Handbook & Strategy

1.1 Current Exam Facts 1.2 CBT Format & Scaled Scoring 1.3 The FE Reference Handbook 1.4 Approved Calculators & Test-Day Logistics 1.5 Study Plan & Time Strategy

2 Mathematics, Economics & Ethics

2.1 Algebra, Calculus & Differential Equations 2.2 Complex Numbers & Laplace Transforms 2.3 Probability & Statistics 2.4 Engineering Economics 2.5 Ethics & Professional Practice

3 Circuits, Electronics & Electromagnetics

3.1 DC Circuit Analysis 3.2 AC Steady-State & Phasors 3.3 Transient Response (RL, RC, RLC) 3.4 Electronics & Devices 3.5 Electromagnetics

4 Signals, Systems, Controls & Communications

4.1 Linear Systems & Transfer Functions 4.2 Signal Processing & Sampling 4.3 Control Systems 4.4 Communications

5 Digital Systems, Computers, Networks & Software

5.1 Number Systems & Boolean Algebra 5.2 Combinational & Sequential Logic 5.3 Computer Architecture & Microprocessors 5.4 Computer Networks 5.5 Software Engineering & Algorithms

6 Power, Materials & Final Readiness

6.1 Power Fundamentals & Three-Phase 6.2 Transformers & Rotating Machines 6.3 Power Electronics & Transmission 6.4 Properties of Electrical Materials & Engineering Sciences 6.5 Integrated Practice & Final Readiness

3.5 Electromagnetics

Key Takeaways Explain This Topic AI

Maxwell's equations (Gauss's laws, Faraday's law, Ampere's law) describe how charges and currents create electric and magnetic fields.

The intrinsic impedance of free space is eta_0 = sqrt(mu_0/epsilon_0), approximately 377 ohms (120*pi).

Capacitance of a parallel-plate capacitor is C = epsilon A/d; inductance of a solenoid is L = mu N^2*A/length.

Electromagnetic waves travel at c = 1/sqrt(mu_0*epsilon_0) = 3x10^8 m/s in free space, with wavelength lambda = c/f.

A transmission line is matched and reflection-free when the load equals the characteristic impedance Z_L = Z_0 (reflection coefficient = 0).

Last updated: June 2026

Maxwell's equations at a conceptual level

The four Maxwell's equations unify electricity and magnetism, and the FE tests them conceptually rather than by surface/line-integral derivation:

Gauss's law (electric): the electric flux out of a closed surface equals the enclosed charge divided by epsilon. Charges are the sources of E fields.

Gauss's law (magnetic): the net magnetic flux through any closed surface is zero (no magnetic monopoles); B field lines always close on themselves.

Faraday's law: a time-changing magnetic field induces an electromotive force (EMF): EMF = -d(flux)/dt. This underlies generators, transformers, and inductors.

Ampere's law (with Maxwell's displacement-current correction): a current and/or a changing electric field produces a magnetic field.

Together they predict self-propagating electromagnetic waves , in which changing E and B fields continuously regenerate each other and travel outward at the speed of light. In an electrostatics-only problem (no time variation), only the two Gauss-type relations matter; in magnetostatics, Ampere's law I = integral of H around a loop and the Biot-Savart law govern the B field of steady currents.

The key conceptual takeaways the FE tests: only the electric Gauss law has a non-zero right-hand side (charge), because there are no magnetic monopoles; a changing magnetic flux is what induces voltage (Faraday), so a static field induces nothing; and Maxwell's displacement-current term is what lets fields propagate through a vacuum where no conduction current exists.

Fields, forces, capacitance, and inductance

The electric field from a point charge is E = Q/(4 pi epsilon r^2); the force on a charge in a field is F = qE (Coulomb's law gives F = Q_1 Q_2/(4 pi epsilon r^2) between two charges). The magnetic force on a moving charge is the Lorentz force F = q*(v x B), perpendicular to both velocity and field.

Energy-storage elements derive from field geometry:

Parallel-plate capacitor: C = epsilon A/d, where epsilon = epsilon_r epsilon_0 and epsilon_0 = 8.854x10^-12 F/m. Stored energy = (1/2) C V^2.

Solenoid/coil inductance: L = mu N^2 A/length, where mu = mu_r mu_0 and mu_0 = 4 pi*10^-7 H/m. Stored energy = (1/2) L I^2.

Increasing plate area A or relative permittivity epsilon_r raises capacitance; decreasing plate spacing d also raises it. Because N appears squared , doubling a coil's turns quadruples its inductance — a frequently tested sensitivity. Inserting a high-permeability magnetic core (large mu_r) likewise multiplies inductance, just as a high-permittivity dielectric multiplies capacitance.

For electric and magnetic energy density in a field region, the FE Handbook gives w_E = (1/2) epsilon E^2 and w_B = (1/2)*B^2/mu (joules per cubic meter); integrating these over the field volume recovers the (1/2)CV^2 and (1/2)LI^2 lumped-element results, tying the field picture to the circuit picture.

Plane waves and the 377-ohm free-space impedance

In free space, electromagnetic waves propagate at the speed of light:

c = 1/sqrt(mu_0*epsilon_0) = 3x10^8 m/s, with wavelength lambda = c/f.

The ratio of the electric- to magnetic-field magnitude of a plane wave is the intrinsic (characteristic) impedance of the medium: eta = sqrt(mu/epsilon). For free space this evaluates to eta_0 = sqrt(mu_0/epsilon_0) = 120*pi, approximately 377 ohms — worth memorizing exactly because it appears throughout antenna, radiation, and wave-impedance problems.

  • Constant Symbol Value
  • Permittivity of free space epsilon_0 8.854x10^-12 F/m
  • Permeability of free space mu_0 4 pi 10^-7 H/m
  • Speed of light c 3x10^8 m/s
  • Free-space intrinsic impedance eta_0 approx 377 ohms (120*pi)

Worked example: a 1 GHz wave in free space has lambda = c/f = (3x10^8)/(1x10^9) = 0.3 m. A plane wave with E = 7.54 V/m has H = E/eta_0 = 7.54/377 = 0.02 A/m.

Transmission-line basics

At high frequency, a line's distributed inductance and capacitance set its characteristic impedance Z_0 = sqrt(L/C) for a lossless line — a real number in ohms (e.g., 50 ohm coax). When a line of impedance Z_0 is terminated in a load Z_L, the reflection coefficient at the load is:

Gamma = (Z_L - Z_0)/(Z_L + Z_0).

Key cases:

Matched (Z_L = Z_0): Gamma = 0, no reflection, all power delivered — the design goal.

Open circuit (Z_L = infinity): Gamma = +1, full reflection.

Short circuit (Z_L = 0): Gamma = -1, full reflection with phase inversion.

The standing wave ratio is SWR = (1 + |Gamma|)/(1 - |Gamma|); SWR = 1 means a perfectly matched line. A signal advances one wavelength when the line length equals lambda, so quarter-wave (lambda/4) sections act as impedance transformers, with input impedance Z_in = Z_0^2 / Z_L. Worked example: a 75 ohm line feeding a 300 ohm load has Gamma = (300-75)/(300+75) = 225/375 = 0.6, so SWR = (1+0.6)/(1-0.6) = 1.6/0.4 = 4. The magnitude of Gamma ranges from 0 (matched) to 1 (total reflection); SWR ranges from 1 (matched) to infinity (open/short).

A quarter-wave transformer matches a real load R_L to a line Z_0 by choosing an intermediate section of impedance Z_0' = sqrt(Z_0 * R_L). For the 75-to-300 ohm case, a lambda/4 section of sqrt(75*300) = sqrt(22,500) = 150 ohm would give a perfect match. Transmission-line effects matter when the line length is a non-trivial fraction of a wavelength (roughly > lambda/10); at low frequency or short lengths a wire is just a node and reflections are negligible. The signal's propagation velocity on the line is v = 1/sqrt(L'C') where L' and C' are per-unit-length values, always at or below c.