Time Varying Fields and Maxwells Equations Questions and Answers

Answer: C

The conduction and displacement terms together form the total current density, whose divergence vanishes identically as the curl of anything must. In a good conductor at power frequency the second term is utterly negligible; in free space the first is absent altogether.

Answer: D

Correct answer: \( \nabla \cdot \mathbf{J} = -\partial\rho_v/\partial t \)

The equation says that current diverging from a point must be draining charge stored there: charge is conserved and cannot appear or vanish. The first option holds only in the steady state, and it was precisely its failure in a charging capacitor that led Maxwell to add the displacement current term to Ampere’s law.

Answer: A

This dimensionless ratio, sometimes called the loss tangent's reciprocal, decides whether a medium behaves as a conductor or a dielectric at a given frequency. Copper is a conductor to well beyond optical frequencies; dry earth is a dielectric above a few megahertz.

Answer: B

Gauss's law for electricity and for magnetism are the divergences, and Faraday's and Ampere's laws as modified are the curls. Together with the constitutive relations of the medium they determine the fields completely given the sources.

Answer: C

Correct answer: \( \nabla \cdot \mathbf{B} = 0 \)

The magnetic flux density has zero divergence everywhere and always, which is the statement that no magnetic monopole has been found - magnetic field lines close on themselves rather than beginning or ending on sources. The electric divergence equation has the charge density on its right, and the two curl equations have source terms of their own.

Answer: D

The field equations relate the fields to one another and to the sources but say nothing about the medium; the three constitutive relations supply that. For a nonlinear or anisotropic medium they become more complicated but the field equations themselves are unchanged.

Answer: A

Substituting the measured values gives very nearly three times ten to the eighth metres per second, and the agreement with the measured speed of light was Maxwell's decisive argument that light is an electromagnetic wave. The other combination gives the impedance of free space.

Answer: B

It is the ratio of the electric to the magnetic field magnitudes in a plane wave in free space, and its value of \( 120\pi \) ohms is worth carrying. The other combination of the same two constants gives the speed of the wave.

Answer: C

Faraday's law applied to a shrinking rectangle straddling the boundary gives equal tangential components whatever the medium and whatever the time variation, since the flux through a vanishing area goes to zero. It is the most robust of the four boundary conditions.

Answer: D

A finite tangential field would drive an infinite current in a perfect conductor, so it must vanish, and continuity then makes it vanish just outside as well. This condition is what fixes the modes of a waveguide and the currents on an antenna.