Time Varying Fields and Maxwells Equations Questions and Answers

Answer: A

The correct matching is a-ii, b-i, c-iv, d-iii, which is code (A).

Faraday's law is the curl of E equal to minus the rate of change of B (ii). Gauss's law is the divergence of D equal to the charge density (i). Ampere's law, with the displacement current, is the curl of H (iv), and the continuity equation is the divergence of J equal to minus the rate of change of charge density (iii).

Codes (B), (C) and (D) pair the laws with the wrong equations.

Answer: B

Statements (i) and (ii) are correct, option (B). Statement (i) is Faraday's law, \(\nabla \times \overline{E} = -\partial \overline{B}/\partial t\), and statement (ii) gives the boundary conditions for the tangential E and the normal D at an interface with no surface charge.

Statement (iii) is wrong, because the identity has a minus sign: \(\nabla \times \nabla \times C = \nabla(\nabla \cdot C) - \nabla^{2}C\). Statement (iv) is wrong too, because the curl of a gradient is always zero. Options A, C and D contain (iii) or (iv).

Answer: C

The correct match is \(\nabla \times \overline{E}\) with \(-\partial \overline{B}/\partial t\) (iv), \(\nabla \times \overline{H}\) with \(\overline{J} + \partial \overline{D}/\partial t\) (iii), \(\nabla \cdot (\nabla \times \overline{E})\) with zero (i) and \(\nabla \cdot \overline{D}\) with \(\rho\) (ii), so the code is iv iii i ii, option (C).

These are Faraday's law, Ampere's law with displacement current, the identity that the divergence of a curl is zero, and Gauss's law. Options A, B and D exchange some of these, for example giving the charge density to the curl of H.

Answer: B

The correct matching is a-iii, b-i, c-ii, d-iv. These are Maxwell's equations. The curl of H equals the conduction current density plus the displacement current, \(\bar{J} + \partial\bar{D}/\partial t\) (iii). The divergence of D equals the volume charge density \(\rho_{v}\) (i). The curl of E equals \(-\partial B/\partial t\) (ii). The divergence of the curl of any vector is identically zero (iv).

The other options each exchange at least two of these, for example pairing the curl of E with the charge density, which would violate Faraday's law.

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.