Time Varying Fields and Maxwells Equations Questions and Answers

Answer: A

Taking the curl of the Faraday equation produces a curl of a curl on the left and a time derivative of the curl of B on the right; expanding the left and dropping the divergence term leaves a Laplacian, and only then is the Ampere equation substituted on the right.

Answer: B

Correct answer: 1 and 2 only are correct

Statements 1 and 2 are correct: two of the four are divergence equations, and the magnetic one has zero on the right in every circumstance. Statement 3 is wrong - Maxwell’s equations relate the four field quantities but do not connect \( \mathbf{D} \) to \( \mathbf{E} \) or \( \mathbf{B} \) to \( \mathbf{H} \). The constitutive relations of the medium are needed as well before anything can be solved.

Answer: C

All three are correct. It is the rate of change of electric flux rather than any current of charge, it enters Ampere's law on the same footing as conduction current, and in copper the ratio \( \omega\epsilon/\sigma \) makes it vanishingly small at fifty hertz.

Answer: D

Correct answer: the continuity equation for charge

The four are Gauss’s law for \( \mathbf{D} \), Gauss’s law for \( \mathbf{B} \), Faraday’s law and Ampere’s law with the displacement current. Continuity is not among them - though it is not independent either, since it follows from taking the divergence of Ampere’s law and substituting the electric divergence equation.

Answer: A

Correct answer: the normal component of the electric field intensity

At a charge-free interface the normal component of \( \mathbf{D} \) is continuous, and since \( D_n = \epsilon E_n \) with different permittivities on the two sides, \( E_n \) must jump. The other three are continuous: tangential \( \mathbf{E} \) from Faraday’s law, normal \( \mathbf{B} \) from its zero divergence, and tangential \( \mathbf{H} \) in the absence of a surface current.

Answer: B

In the conductors the ratio of displacement to conduction current is of order ten to the minus seventeen, so ignoring it costs nothing. Inside a capacitor's dielectric the situation is reversed and the displacement current is all there is, which circuit theory handles as the capacitor's current.

Answer: C

Because the electric field is not conservative, the line integral between two points differs from path to path, and the voltmeter leads form part of a loop that may itself link some of the flux. This is a genuine and much-discussed pitfall in measuring around inductive circuits.

Answer: D

Setting the two time derivatives to zero decouples the electric and magnetic equations entirely, which is why electrostatics and magnetostatics can be studied separately. Restoring them couples the fields and makes waves possible.

Answer: A

The changing flux induces an electric field within the metal, and its conductivity turns that field into circulating currents that dissipate energy. Laminating the material restricts the loops available and so reduces the loss, which is why cores are built up from thin sheets.

Answer: B

The changing flux within the conductor induces currents that add to the current at the surface and subtract from it at the centre, so the current crowds into a layer whose depth falls as the inverse square root of frequency. The effective resistance rises accordingly.