Gauss's law is used as a practical shortcut to find E or D only when

the charge distribution has enough symmetry to pull D out of the surface integral as a constant

the total enclosed charge is zero

the charge distribution has enough symmetry to pull D out of the surface integral as a constant

the medium is a perfect conductor

the charge is moving at constant velocity

A parallel-plate capacitor's two dielectric layers each fill the full plate area and are stacked one above the other between the plates. The two layers behave as capacitors connected in

neither — they add as resistors

Which of the following are correctly true?

E = -grad(V); field lines are always perpendicular to equipotential surfaces.

No work is done moving a charge along an equipotential surface.

E = -grad(V); field lines are always perpendicular to equipotential surfaces.

No work is done moving a charge along an equipotential surface.

A dielectric between capacitor plates decreases capacitance for the same geometry.

Relative permittivity epsilon_r of any real dielectric is always less than 1.

The Lorentz force law F = q(E + v x B) says a moving charge in a magnetic field alone (E = 0) experiences a force that is

always perpendicular to both velocity and the field

zero unless the charge is stationary

In a magnetic circuit, an air gap of small length compared to the total core path length usually dominates the total reluctance because

air has a much lower permeability than iron, so its reluctance per unit length is far higher

air has a much higher permeability than iron

air has a much lower permeability than iron, so its reluctance per unit length is far higher

the air gap carries no flux at all

reluctances in series average rather than add

Electrostatics, Magnetostatics and Magnetic Circuits

Section 3 of the GATE Electrical Engineering (EE) paper, and the field-theory foundation underneath every transformer, machine and transmission line the rest of this paper studies as a lumped circuit. Start with Coulomb's law and how it generalises into the electric field and Gauss's law; then the potential a charge distribution produces, and what a dielectric does to a capacitor; then the magnetic analogue built from Biot-Savart's law and Ampere's law, and the vector operators — divergence and curl — that make Gauss's and Faraday's laws exact rather than approximate; and finally magnetic circuits, the practical machinery of MMF, reluctance and inductance that every transformer and rotating machine in Section 5 is built from.

1. Coulomb's law, electric field, flux density and Gauss's law

Coulomb's law gives the force between two point charges Q1 and Q2 separated by distance R as F = Q1Q2 / (4piepsilonR^2), directed along the line joining them. The electric field intensity E at a point is the force per unit positive test charge placed there, E = F/q, and electric flux density D = epsilon*E is the same field scaled by the medium's permittivity — introduced precisely so that Gauss's law can be stated without epsilon appearing inside the integral.

Gauss's law states that the net electric flux through any closed surface equals the total free charge enclosed: the closed-surface integral of D.dS = Q_enclosed. Its point (differential) form, div(D) = rho_v (the divergence of D equals the volume charge density), says the same thing at a single point rather than over a whole surface — divergence measures how much a vector field 'spreads out' from a point, and it is exactly zero wherever there is no charge to source the field.

🧠 When Gauss's law actually saves work

Gauss's law is only a shortcut when the charge distribution has enough symmetry (spherical, cylindrical or planar) that D can be pulled out of the surface integral as a constant over a cleverly chosen Gaussian surface. For a point or spherical charge, a sphere centred on it; for an infinite line charge, a coaxial cylinder; for an infinite sheet, a pillbox straddling it. Applying it to an asymmetric distribution gives a true but useless equation, because D cannot be factored out of the integral.

2. Potential, dielectrics and capacitance

Electric potential from common charge distributions
Distribution	Potential (relative to infinity, where defined)
Point charge Q	V = Q / (4piepsilon*R)
Infinite line charge, density rho_L	V = -(rho_L / 2piepsilon) * ln(R) + constant
Infinite plane sheet, density rho_S	field uniform (rho_S / 2*epsilon), V falls linearly with distance
Uniformly charged sphere (outside, r > R0)	identical to a point charge at the centre, V = Q / (4piepsilon*r)

The electric field is minus the gradient of potential, E = -grad(V), so equipotential surfaces are always perpendicular to field lines, and no work is done moving a charge along an equipotential. A dielectric placed between the plates of a capacitor polarises — its bound charges rotate to partially oppose the applied field — which reduces the net field for a given free charge and so increases capacitance by the dielectric's relative permittivity epsilon_r; every practical capacitance formula (parallel plate, coaxial, spherical) carries epsilon = epsilon_0 * epsilon_r rather than epsilon_0 alone.

⚠️ Two capacitors in series with different dielectrics

A parallel-plate capacitor with two dielectric layers stacked between the plates (each filling the full plate area) behaves as two capacitors in SERIES, not parallel, because the same D (flux density) must pass through both layers while E differs in each; a common mistake is to add the two layer capacitances directly as if they were in parallel. If instead the two dielectrics fill the area side by side (each covering part of the plate), it genuinely is two capacitors in parallel.

3. Biot-Savart's law, Ampere's law, curl, Faraday's law and Lorentz force

Biot-Savart's law gives the magnetic field dH produced by a differential current element Idl at a point, dH = (Idl x a_R) / (4piR^2) — the magnetic analogue of Coulomb's law, with a cross product replacing the scalar force because the magnetic field is inherently directional in a way the electrostatic field is not. Ampere's circuital law is its large-scale, symmetric counterpart: the closed-line integral of H.dl around any closed path equals the total current enclosed by that path, and — exactly like Gauss's law — it is only a computational shortcut when symmetry lets H be pulled out of the integral (an infinite straight wire, a solenoid, a toroid).

Curl measures a vector field's local rotation, and Ampere's law in point form is curl(H) = J (the curl of H equals current density); Faraday's law of electromagnetic induction, curl(E) = -dB/dt, says a time-varying magnetic field induces a circulating electric field, and its integral form (EMF induced = -d(flux)/dt) is what makes a generator generate. The Lorentz force law, F = q(E + v x B), is the single expression that unifies electric and magnetic force on a moving charge and is the starting point for deriving the force on a current-carrying conductor in a field.

🎯 Why div(B) = 0 has no magnetic-charge equivalent

Gauss's law for the electric field has a magnetic twin, div(B) = 0, which says magnetic field lines never begin or end at a point — there is no magnetic monopole to source them the way a charge sources D. This is why magnetic field lines always form closed loops, and why cutting a bar magnet in half produces two smaller magnets rather than an isolated north and an isolated south pole.

4. Magnetomotive force, reluctance, magnetic circuits and inductance

A magnetic circuit is the flux equivalent of an electric circuit: magnetomotive force MMF = NI (ampere-turns) plays the role of EMF, flux phi plays the role of current, and reluctance S = l / (muA) — length over permeability times area — plays the role of resistance, giving MMF = phi * S, the magnetic analogue of V = I*R. Reluctances in series add, and in parallel combine as reciprocals, exactly as resistances do, which is why an air gap in a core (very low mu, so very high reluctance for its length) dominates a magnetic circuit's total reluctance even though it may be a tiny fraction of the total path length.

Self-inductance L relates a coil's own flux linkage to its own current, L = Nphi/I, and for a magnetic circuit reduces to L = N^2/S (turns squared over reluctance) — which is why doubling the turns quadruples the inductance for the same core. Mutual inductance M relates one coil's flux linkage to the OTHER coil's current, M = N2phi12/I1, and the coupling coefficient k = M/sqrt(L1*L2) (between 0 and 1) says how much of one coil's flux actually links the other — k = 1 for perfect, leakage-free coupling.

⚠️ Reluctance in parallel — do not just add

Two parallel flux paths (say, two limbs of a core sharing the same MMF source) combine their reluctances the way parallel resistors combine: 1/S_total = 1/S1 + 1/S2, never S1 + S2 directly. The same care applies to an air gap in series with an iron path — those DO simply add, S_total = S_iron + S_gap, since the flux is forced through both in sequence with no alternative path. Confusing which topology is series and which is parallel is the standard slip in magnetic-circuit problems.

Key takeaways

Gauss's law and Ampere's law are computational shortcuts only when symmetry lets D or H be factored out of the integral — spherical, cylindrical or planar for the electric case; an infinite wire, solenoid or toroid for the magnetic.
A dielectric between capacitor plates increases capacitance by epsilon_r; two full-area dielectric layers stacked between the same plates form a SERIES combination, not a parallel one.
div(B) = 0 has no magnetic-charge source term, unlike div(D) = rho_v — magnetic field lines always close on themselves, which is why no isolated magnetic monopole has ever been found.
MMF = phi * S is the magnetic circuit's V = IR; reluctances add in series and combine as reciprocals in parallel, and a small air gap dominates total reluctance because its permeability is so much lower than iron's.
Self-inductance L = N^2/S grows with the square of turns; mutual inductance M and the coupling coefficient k = M/sqrt(L1*L2) describe how much of one coil's flux actually reaches the other.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Coulomb's law gives the force between two point charges as varying with distance R as
1. 1/R
2. 1/R^2
3. 1/R^3
4. R^2
Show answer
Answer: B — 1/R^2
Coulomb's law is an inverse-square law, F proportional to 1/R^2, exactly parallel in form to Newton's law of gravitation — the force falls off with the square of separation, not linearly or as a cube.
Gauss's law is used as a practical shortcut to find E or D only when
1. the total enclosed charge is zero
2. the charge distribution has enough symmetry to pull D out of the surface integral as a constant
3. the medium is a perfect conductor
4. the charge is moving at constant velocity
Show answer
Answer: B — the charge distribution has enough symmetry to pull D out of the surface integral as a constant
Gauss's law is always true, but it only SOLVES for the field directly when a Gaussian surface can be chosen (sphere, cylinder, pillbox) over which D is constant in magnitude and always parallel or perpendicular to the surface, letting it be factored out of the integral; without that symmetry the equation is correct but not solvable for D by itself.
A parallel-plate capacitor's two dielectric layers each fill the full plate area and are stacked one above the other between the plates. The two layers behave as capacitors connected in
1. series
2. parallel
3. neither — they add as resistors
4. short circuit
Show answer
Answer: A — series
The same electric flux density D must pass through both stacked layers in sequence (no alternative path), which is the series condition; E differs between the layers because permittivity differs, but D does not. Side-by-side dielectrics covering different parts of the same plate area would instead be a parallel combination.
Which of the following are correctly true?
1. E = -grad(V); field lines are always perpendicular to equipotential surfaces.
2. No work is done moving a charge along an equipotential surface.
3. A dielectric between capacitor plates decreases capacitance for the same geometry.
4. Relative permittivity epsilon_r of any real dielectric is always less than 1.
Show answer
Answer: A — E = -grad(V); field lines are always perpendicular to equipotential surfaces.; B — No work is done moving a charge along an equipotential surface.
E is the negative gradient of V, which makes field lines perpendicular to equipotentials by definition, and moving along an equipotential does zero work since V does not change. A dielectric INCREASES capacitance (epsilon_r > 1 for every real dielectric, not less than 1) by reducing the net field for a given free charge.
Biot-Savart's law is the magnetic analogue of
1. Gauss's law
2. Coulomb's law
3. Faraday's law
4. Ampere's law
Show answer
Answer: B — Coulomb's law
Biot-Savart's law gives the field of a differential current element the way Coulomb's law gives the force of a point charge — both are the elementary, point-source statement in their respective domain. Ampere's law is instead the large-scale, symmetric counterpart of Biot-Savart, playing the role Gauss's law plays for the electric field.
curl(E) = -dB/dt is the point form of
1. Gauss's law for the electric field
2. Ampere's circuital law
3. Faraday's law of electromagnetic induction
4. Coulomb's law
Show answer
Answer: C — Faraday's law of electromagnetic induction
This is Faraday's law in point (differential) form, saying a time-varying magnetic field induces a circulating (curling) electric field — its integral form gives the EMF a generator or a transformer secondary actually produces.
The Lorentz force law F = q(E + v x B) says a moving charge in a magnetic field alone (E = 0) experiences a force that is
1. along the direction of motion
2. always perpendicular to both velocity and the field
3. along the field direction
4. zero unless the charge is stationary
Show answer
Answer: B — always perpendicular to both velocity and the field
The v x B term is a cross product, which is by definition perpendicular to both v and B — this magnetic force does no work on the charge (it changes direction, not speed), unlike an electric force which can act along the direction of motion.
In a magnetic circuit, an air gap of small length compared to the total core path length usually dominates the total reluctance because
1. air has a much higher permeability than iron
2. air has a much lower permeability than iron, so its reluctance per unit length is far higher
3. the air gap carries no flux at all
4. reluctances in series average rather than add
Show answer
Answer: B — air has a much lower permeability than iron, so its reluctance per unit length is far higher
Reluctance S = l/(mu*A), and air's permeability mu_0 is thousands of times smaller than a typical ferromagnetic core's mu, so even a physically short air gap contributes a reluctance comparable to or larger than the entire rest of the (much longer) iron path — reluctances in series simply add, S_total = S_iron + S_gap.
If the number of turns of a coil on a fixed magnetic core is doubled, its self-inductance L = N^2/S becomes
1. doubled
2. quadrupled
3. unchanged
4. halved
Show answer
Answer: B — quadrupled
L is proportional to N-squared with reluctance S held fixed by the core geometry, so doubling N multiplies L by 2^2 = 4 — quadrupling, not merely doubling, which is the standard trap in this formula.
Two coils have self-inductances L1 = 4 H and L2 = 9 H and are perfectly (loss-free) coupled, so the coupling coefficient k equals 1. Their mutual inductance M, in henry, is ___
Numerical answer — type the value.
Show answer
Answer: 6
M = k * sqrt(L1*L2) = 1 * sqrt(4*9) = sqrt(36) = 6 H — the coupling coefficient formula rearranged, with k = 1 for the perfectly coupled case stated here.