Time Varying Fields and Maxwells Equations Questions and Answers

Answer: B

The matching is: a curl of H with current density (ii), b curl of E with Faraday's law of induction (iii), c divergence of D with Gauss's law (iv), d divergence of J with the continuity equation (i), the code ii iii iv i (option B).

These are Maxwell's equations and the continuity equation \(\nabla\cdot\overline{J} = -\partial\rho/\partial t\). The curl of H equals the current density plus the displacement term, and the curl of E equals the negative rate of change of B. Options A, C and D swap Gauss's law with the continuity equation or the two curl equations.

Answer: A

Correct answer: the negative rate of change of the flux linking it

\[ \mathrm{emf} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t}. \] Two features matter. It is the *rate of change* that induces, so a large steady flux induces nothing; and the minus sign is Lenz’s law, which makes the induced current oppose the change that produced it - without it, energy would not be conserved.

Answer: B

Any other sense would let the induced current reinforce the change and the arrangement would run away, creating energy from nothing. The law is therefore a statement of energy conservation as much as of electromagnetism.

Answer: C

A changing magnetic field gives the electric field a curl, so the electric field is no longer conservative and no single-valued scalar potential exists. Stokes's theorem converts this into the familiar integral statement about an electromotive force round a loop.

Answer: D

With a non-zero curl the electric field cannot be written as minus the gradient of a scalar alone; the vector potential's time derivative must be included as well. That is why a voltage round a loop threaded by changing flux depends on the path taken.

Answer: A

The charges in the moving conductor experience the Lorentz force and are driven along it, giving an emf equal to the line integral of \( \mathbf{v} \times \mathbf{B} \). It is the principle of every generator, and the transformer emf is the complementary case.

Answer: B

Nothing moves, and the changing flux gives the electric field a curl which drives the current. The two mechanisms, motional and transformer, together account for the whole of the emf in any general case, and Faraday's law covers both.

Answer: C

The expression follows from integrating \( \mathbf{v} \times \mathbf{B} \) along the conductor when all three vectors are mutually perpendicular. If the motion is not perpendicular to the field the component of velocity along the field contributes nothing.

Answer: D

Taking the divergence of the unmodified Ampere law forces the divergence of the current density to vanish, which contradicts the continuity equation whenever charge accumulates. The extra term repairs that at a stroke and predicts electromagnetic waves as a bonus.

Answer: A

It has the units of a current density but involves no motion of charge whatever, being the rate of change of the electric flux density. In a capacitor it carries the current across the dielectric and makes the circuit continuous.