Electromagnetic Waves • Maxwell Correction • Exam Notes

Displacement Current and Ampere-Maxwell Law

Understand charging capacitor paradox, displacement current, Ampere-Maxwell law, Maxwell equations, derivations and numerical problems.

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Section 1: Introduction

Maxwell introduced displacement current to remove a serious contradiction in Ampere's circuital law for a charging capacitor. In the connecting wire, real charges move, so conduction current exists. Between capacitor plates, no charge crosses the insulating gap, yet the electric field changes with time and the magnetic field remains continuous.

Charging capacitor paradox
A capacitor charging through wires seems to have current in the wire but no current in the gap between plates.

Continuity of current
Maxwell showed that a time-varying electric field between the plates behaves like a current and keeps the current continuous.

Ampere's law corrected
Ordinary Ampere law used only conduction current. Maxwell added displacement current.

Correct charging capacitor picture: real charge current flows only in the wires; changing electric field between plates is represented by displacement current.

Section 2: Charging Capacitor Paradox

In a charging capacitor, conduction current flows through the wire, but no conduction current flows through the gap between capacitor plates. Still, a magnetic field exists around the capacitor region. Ordinary Ampere's law gives inconsistent results for two different surfaces bounded by the same loop.

Surface S1 cuts the wireI = conduction current

Ampere's law gives a non-zero magnetic field.

Surface S2 passes between platesI = 0

Ordinary Ampere's law would give zero magnetic field, which contradicts experiment and symmetry.

Ordinary Ampere law gives different answers for S1 and S2. Maxwell's displacement current term makes both surfaces consistent.

Section 3: Displacement Current

Displacement current is the current equivalent produced by a time-varying electric field. It is not a real flow of charges across the capacitor gap, but it produces magnetic effects like current.

VacuumI_d = ε₀ dΦ_E/dt

MediumI_d = ε dΦ_E/dt

Electric fluxΦ_E = EA

ContinuityI_d = I

Proof for Charging Parallel Plate Capacitor

E = q / (ε₀A) Φ_E = EA = q / ε₀ dΦ_E/dt = (1/ε₀) dq/dt ε₀ dΦ_E/dt = dq/dt = I

Therefore, during charging of a capacitor, displacement current between the plates equals conduction current in the wire.

Section 4: Parallel Plate Capacitor Derivation

Using Electric Field

Φ_E = EA I_d = ε₀ d(EA)/dt I_d = ε₀A dE/dt

Using Voltage

E = V/d I_d = ε₀A d(V/d)/dt I_d = (ε₀A/d) dV/dt = C dV/dt

I_d = ε₀dΦ_E/dt

I_d = ε₀A dE/dt

I_d = C dV/dt

C = ε₀A/d

Section 5: Ampere-Maxwell Law

Ordinary Ampere Law∮ B · dl = μ₀I

This fails for a charging capacitor because it ignores changing electric flux between plates.

Maxwell Correction∮ B · dl = μ₀(I + I_d)∮ B · dl = μ₀I + μ₀ε₀dΦ_E/dt

Between circular capacitor plates, changing electric field acts like current and produces circular magnetic field lines around the axis.

Section 6: Magnetic Field Between Capacitor Plates

Consider circular capacitor plates of radius R. If the total displacement current between plates is I_d, the enclosed displacement current depends on the Amperian loop radius r.

For r < R

I_d,enclosed = I_d r²/R² B(2πr) = μ₀I_dr²/R² B = μ₀I_dr / (2πR²)

For r > R

I_d,enclosed = I_d B(2πr) = μ₀I_d B = μ₀I_d / (2πr)

Section 7: Maxwell Equations

Law	Integral Form	Differential Form	Physical Meaning
Gauss Law of Electrostatics	∮ E · dS = q_enc/ε₀	∇ · E = ρ/ε₀	Electric charges are sources or sinks of electric field.
Gauss Law of Magnetism	∮ B · dS = 0	∇ · B = 0	No isolated magnetic monopoles exist; magnetic field lines are closed.
Faraday Law	∮ E · dl = -dΦ_B/dt	∇ × E = -∂B/∂t	Changing magnetic flux produces non-conservative electric field.
Ampere-Maxwell Law	∮ B · dl = μ₀I + μ₀ε₀dΦ_E/dt	∇ × B = μ₀J + μ₀ε₀∂E/∂t	Conduction current and changing electric field produce magnetic field.

Section 8: Dimensional Analysis of Maxwell Equations

A. Gauss Law

[E] = M L T^-3 A^-1, [dS] = L²[∮E·dS] = M L³ T^-3 A^-1[q/ε₀] = (AT)/(M^-1L^-3T⁴A²) = M L³ T^-3A^-1

LHS = RHS.

B. Gauss Law of Magnetism

∮B · dS = 0

[B][A] is magnetic flux, measured in weber. Zero net magnetic flux through a closed surface means no magnetic monopoles.

C. Faraday Law

[E][l] = M L² T^-3A^-1[Φ_B/t] = M L² T^-3A^-1

LHS = RHS.

D. Ampere-Maxwell Law

[B][l] = M L T^-2A^-1[μ₀I] = M L T^-2A^-1

ε₀dΦ_E/dt has dimension of current, so μ₀ε₀dΦ_E/dt also has the same dimension as μ₀I.

Section 9: Solved Numericals from Displacement Current and Maxwell Theory

Section 10: PYQ and Exam-Style Practice

Each exam block contains MCQs, numerical problems, conceptual questions and case-study questions with answers and solutions.

Section 11: Important Derivations

1. I_d = ε₀dΦ_E/dt

A time-varying electric flux produces a current equivalent called displacement current.

2. I_d = ε₀A dE/dt

For uniform electric field, Φ_E = EA, so I_d = ε₀d(EA)/dt.

3. I_d = C dV/dt

Use E = V/d and C = ε₀A/d.

4. I_d = I

For charging capacitor, Φ_E = q/ε₀, so I_d = dq/dt = I.

5. Ampere-Maxwell Law

Add displacement current to conduction current: ∮B·dl = μ₀(I + I_d).

6. Magnetic Field Between Plates

Use circular Amperian loops and enclosed displacement current to get r < R and r > R formulas.

7. Dimensional Verification

Each Maxwell equation has identical LHS and RHS dimensions, as shown above.

8. E₀/B₀ = c

In a plane EM wave, electric and magnetic field amplitudes are related through wave speed.

9. Equal Energy Density

For EM waves, u_E = 1/2 ε₀E² and u_B = B²/(2μ₀), and using E = cB gives u_E = u_B.

Section 12: Common Mistakes

Displacement current is not charge flow
It is produced by changing electric field, not electrons crossing the capacitor gap.

Conduction vs displacement current
Conduction current is due to moving charges; displacement current is due to changing electric flux.

Forgetting time variation
Displacement current exists only when electric flux changes with time.

Using ordinary Ampere law
For charging capacitor, use Ampere-Maxwell law, not only ∮B·dl = μ₀I.

Missing ε₀
The correct vacuum formula is I_d = ε₀dΦ_E/dt.

Wrong area selection
For r < R, use enclosed area πr². For r > R, the full plate area is already enclosed.

Confusing r cases
B ∝ r inside plates and B ∝ 1/r outside the plate radius.

Wrong units
dE/dt is V m^-1s^-1; dV/dt is V s^-1.

Maxwell sign errors
Faraday law has a negative sign: ∮E·dl = -dΦ_B/dt.

Section 13: Final Formula Sheet

I_d = ε₀dΦ_E/dt

I_d = ε₀A dE/dt

I_d = C dV/dt

Φ_E = EA

E = q/(ε₀A)

∮B · dl = μ₀I

∮B · dl = μ₀(I + I_d)

∮B · dl = μ₀I + μ₀ε₀dΦ_E/dt

B = μ₀I_dr/(2πR²), r < R

B = μ₀I_d/(2πr), r > R

c = E₀/B₀

u_E = 1/2 ε₀E²

u_B = B²/(2μ₀)

u_E = u_B

Pressure = I/c absorbing

Pressure = 2I/c reflecting

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