Coulomb and Gauss for E-fields, potential and capacitance, then Biot-Savart and Ampere for B-fields, inductance, and the energy stored in each field.
Static fields are the foundation the entire Electromagnetics topic is built on: get the constants, the geometry, and the energy bookkeeping right here and the dynamics and transmission-line questions become bookkeeping on top of them. The handbook gives you a compact toolbox under "Electrostatic Fields" and "Magnetic Fields" (FE Reference Handbook, Electrical and Computer Engineering), but the symbols hide a lot of structure — what is a point versus a line versus a sheet source, when a field falls off as 1/r2 versus 1/r
, and which permittivity or permeability to use. This concept makes you fluent in all of it. Expect 1 to 2 of your 4 to 6 Electromagnetics questions to live right here, and they are usually fast points if you keep the geometry and units straight.
Coulomb's law and the electric field
Two point charges exert a force along the line joining them: like charges repel, opposite charges attract. The electric field E is that force per unit positive test charge, so F=QE. For a single point charge Q the field points radially outward (for Q>0) and falls off as the inverse square of distance. In free space or air use ε0=8.85×10−12F/m; in a dielectric use ε=εrε0. Work in vectors when directions matter — superpose the contributions of each charge as vectors, not magnitudes.
E=4πεr2Qar,F=QE
Line and sheet sources: the geometry sets the falloff
Symmetry changes how the field decays. A point charge gives E∝1/r2; an infinite line charge of density ρL (C/m) gives a radial field that decays only as 1/r; an infinite sheet of charge ρs (C/m2) gives a uniform field that does not decay with distance at all. Recognizing the source geometry is half the battle on these problems — a question that mentions "per meter of length" is a line source, and "per unit area" is a sheet.
Eline=2πεrρL,Esheet=2ερs
Gauss's law
Gauss's law is the fast path whenever the geometry is symmetric. It says the flux of the electric flux density D=εE through any closed surface equals the charge enclosed — fields outside the surface contribute nothing. Pick a Gaussian surface that matches the symmetry (sphere for a point charge, cylinder for a line, pillbox for a sheet) so that D is constant and either parallel or perpendicular to the surface, then D comes straight out of the integral.
∮SD⋅dS=Qenc,D=εE
Potential, voltage, and stored energy
Potential difference is work per unit charge moved against the field, V=−∫p1p2E⋅dr. The most-tested special case is the parallel-plate field: a uniform E=V/d pointing from the + plate to the − plate. Energy can be tracked either as a field integral 21εE2 over the volume or, for a capacitor, with the lumped forms below — they agree.
Eplates=dV,WE=∫V21εE2dV
Capacitance and stored charge
Capacitance is the charge stored per volt, C=q/v, set purely by geometry and the dielectric. For parallel plates of area A separated by d with a dielectric of permittivity ε=εrε0, capacitance scales with area and inversely with spacing. The energy stored has three equivalent forms; pick whichever matches the quantities you already have.
C=dεA,W=21Cv2=2Cq2=21qv
Magnetostatics: Biot-Savart and Ampere
Steady currents make magnetic fields. The handbook gives the result for a long straight wire on the z-axis: the magnetic field strength H circles the wire (right-hand rule) and decays as 1/r, with flux density B=μH. This is exactly what Ampere's law ∮H⋅dl=Ienc delivers when you take a circular Amperian loop of radius r. Use μ0=4π×10−7H/m in air, μ=μrμ0 in magnetic material. The force on a current-carrying conductor in a field is F=IL×B.
H=2πrI,B=μH,∮H⋅dl=Ienc
Inductance and magnetic energy
Inductance relates flux linkage to current, Nϕ=Li. For a coil of N turns on a core of cross-section A, mean magnetic path length ℓ, and permeability μ, inductance grows as N2 and falls with path length. The magnetic-circuit view writes L=N2/R with reluctance R=ℓ/(μA) — the magnetic analog of resistance. Magnetic energy parallels electric energy: a field integral 21μH2 over the volume, or the lumped 21Li2.
L=ℓN2μA=RN2,WH=21Li2
Exam strategy
First classify the source — point, line, sheet, or current — because that fixes the falloff and the right formula before you touch numbers. Use Gauss's law (electric) or Ampere's law (magnetic) whenever the geometry is symmetric; they are far faster than integrating Coulomb or Biot-Savart contributions. Watch the constants: ε0=8.85×10−12F/m and μ0=4π×10−7H/m are on the formula sheet, but 1/(4πε0)≈8.99×109 is not — derive it. Keep E (V/m), D (C/m2), H (A/m), and B (T) distinct, and remember the energy forms come in matched electric/magnetic pairs: 21Cv2 mirrors 21Li2.
Key equations
Coulomb forceF2=4πεr2Q1Q2ar12
Force on charge 2 from charge 1; r in m, Q in C, ε=εrε0 (F/m). Positive product = repulsive (along ar12
Point-charge E fieldE=4πεr2Qar,F=QE
Line and sheet fieldsEline=2πεrρL,Esheet=2ερs
Gauss's law∮SD⋅dS=Qenc,D=εE
Parallel-plate E fieldE=dV
Uniform field between plates, directed from + to −; V
Parallel-plate capacitanceC=dεA=dεrε0A
Capacitor stored energyW=21Cv2=2Cq2=21qv
Electric / magnetic field energyWE=∫V21εE2dV,WH=∫V21μH2dV
Long-wire magnetic fieldH=2πrI,B=μH
H
Ampere's law∮H⋅dl=Ienc
Circulation of H around a closed loop equals enclosed current; the fast path for symmetric geometries.
Force on a conductorF=IL×B
Force on length L of conductor carrying current I in flux density B; magnitude BILsinθ
InductanceL=ℓN2μA=RN2,R=μAℓ
Inductor stored energyW=21Li2
Magnetic energy in joules; the dual of 21Cv2
Worked examples
Coulomb force and field between two charges
Problem. Charge Q1=+20nC and Q2=−15nC sit 5.0cm apart in air. Find the magnitude and nature of the force between them, and the magnitude of the field Q1 alone produces at Q2's location.
Solution. Use F=4πε0r2∣Q1Q2∣
F=4πε0r2∣Q1Q2∣≈1.08mN(attractive)
Parallel-plate capacitor: C, charge, energy, field
Problem. A parallel-plate capacitor has plates of area A=0.020m2 separated by d=0.50mm of dielectric with εr=4.0
Solenoid: inductance, B-field, and stored energy
Problem. A 200-turn air-core solenoid has length ℓ=0.30m and cross-sectional area A=4.0cm2, carrying I=3.0A
Common pitfalls
•Forgetting the 1/(4πε0)≈8.99×109 prefactor: the handbook lists ε0, not the combined Coulomb constant. Compute 1/(4πε0) yourself or you will be off by ∼1010.
•Using the point-charge 1/r2 falloff for a line or sheet source. A line charge field goes as 1/r and an infinite sheet field is uniform — classify the source first.
•Omitting εr (or μr): air uses ε0
•Adding field magnitudes instead of vectors when multiple charges are present. Superpose E as vectors (components), then take the magnitude.
•Mixing up H (A/m) and B (T): the wire formula gives H=I/(2πr); multiply by μ to get B
•Treating inductance as linear in turns. L∝N2 — doubling turns quadruples L, not doubles it.
•Unit slips on area and length: cm2=10−4m2 and mm=10−3m
References
NCEES FE Reference Handbook — Electrical and Computer Engineering: Electrostatics / Electrostatic Fields — Coulomb's law, point/line/sheet fields, Gauss's law, voltage, parallel-plate E.
NCEES FE Reference Handbook — Electrical and Computer Engineering: Magnetic Fields and Capacitors/Inductors — Long-wire H/B, force on a conductor, capacitance, inductance, and stored-energy forms.
Hayt & Buck, Engineering Electromagnetics — Standard derivations of Gauss's law and Ampere's law with symmetric surfaces/loops.
B. Electrodynamics
Electrodynamics and Maxwell's Equations
Faraday's law and displacement current close Maxwell's four equations, which then predict plane waves, the Poynting power flow, and every wave parameter you will compute.
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C. Transmission lines (high frequency)
High-Frequency Transmission Lines
Characteristic impedance and propagation, the load reflection coefficient and VSWR, the input-impedance formula, and quarter-wave matching for high-frequency lines.
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).
E in V/m, radial, ∝1/r2. Multiply by a test charge to get force.
Infinite line charge ρL (C/m) falls as 1/r; infinite sheet ρs (C/m2) is uniform.
Flux of D (C/m2) through a closed surface equals enclosed charge (C). Use with symmetric Gaussian surfaces.
in volts,
d
in m,
E
in V/m.
A = plate area (m2), d = spacing (m), εr = dielectric constant; C in farads.
Three equivalent forms in joules with q=Cv.
Energy as field integrals over a volume; energy densities 21εE2 (J/m3) and 21μH2.
in A/m circles the wire (right-hand rule);
B=μH
in tesla;
r
from the wire axis (m).
in N.
N turns, core area A (m2), mean path ℓ (m), permeability μ=μrμ0; reluctance R in H−1.