Faraday's law and modified Maxwell equations

Faraday's law (1 mark)

The emf induced in a closed loop equals the negative rate of change of magnetic flux linking the loop:

emf = −dΨ/dt

The negative sign (Lenz's law) shows that the induced emf opposes the change producing it.

Correcting ∇×E = 0 (3 marks)

For static fields ∮ E·dl = 0, so ∇×E = 0. For a time-varying flux, the emf around a fixed loop is

emf = ∮ E·dl = −dΨ/dt = −d/dt ∫ B·dS

For a stationary loop only B changes with time, so the derivative can go inside the integral:

∮ E·dl = −∫ (∂B/∂t)·dS

Apply Stokes' theorem to the left side, ∮ E·dl = ∫ (∇×E)·dS:

∫ (∇×E)·dS = −∫ (∂B/∂t)·dS

This holds for any surface, so the integrands are equal:

∇×E = −∂B/∂t

So the electric field is no longer conservative when B varies with time.

Modifying ∇×H = J (4 marks)

Ampere's law for steady currents is ∇×H = J. Take the divergence of both sides. Since ∇·(∇×H) = 0 always:

∇·J = 0

But the continuity equation of charge says ∇·J = −∂ρv/∂t, which is non-zero for time-varying fields. So the equation is inconsistent. Add an unknown term J_x:

∇×H = J + J_x   ⇒   ∇·J + ∇·J_x = 0   ⇒   ∇·J_x = ∂ρv/∂t

Using Gauss's law ρv = ∇·D:

∇·J_x = ∂(∇·D)/∂t = ∇·(∂D/∂t)   ⇒   J_x = ∂D/∂t

Maxwell named this the displacement current density J_d = ∂D/∂t. The corrected equation is

∇×H = J + ∂D/∂t

and in integral form ∮ H·dl = ∫ (J + ∂D/∂t)·dS. This is consistent with charge conservation and predicts electromagnetic waves.