Gauss Law and Divergence Questions and Answers

Answer: C

Charge redistributes on the conducting shell so that the interior field cancels, and Gauss's law guarantees the redistribution is always possible. Earthing helps with the external field but the interior cancellation does not depend on it.

Answer: D

Correct answer: 4 nC, unchanged

The flux depends only on the enclosed charge, and doubling the cube changes neither the charge nor its enclosure. The flux density on the faces falls, since the same flux is spread over four times the area, but the total is fixed at 4 nC.

Answer: A

Correct answer: 360 V/m

For an infinite line charge, \[ E = \frac{\lambda}{2\pi\varepsilon_0 \rho} = \frac{10^{-9}}{2\pi \times 8.854\times10^{-12} \times 0.05} = 360\ \mathrm{V/m}. \] Note the inverse first power of the distance rather than the inverse square, which follows from the cylindrical rather than spherical geometry.

Answer: B

Correct answer: 4 C/m3

In cylindrical coordinates the radial divergence is \[ \frac{1}{\rho}\frac{d}{d\rho}(\rho D_\rho) = \frac{1}{\rho}\frac{d}{d\rho}(2\rho^2) = \frac{4\rho}{\rho} = 4. \] The metric factor is \( \rho \) here rather than \( r^2 \) as in spherical coordinates, and omitting it would give 2 - the offered trap.

Answer: C

Correct answer: 4 nC/m2

\[ \frac{12}{3} = 4\ \mathrm{nC/m^2}. \] The word average matters: the flux density need not be uniform over the surface unless the geometry is symmetric, and this figure is the total flux divided by the area rather than the value at any particular point.

Answer: D

The plane is an equipotential, and the field is always normal to an equipotential surface. That boundary condition is what the method of images is constructed to satisfy, by placing an equal and opposite charge below the plane.

Answer: A

Outside, only the total enclosed charge matters, so the fields agree. Inside, the conductor is field free while the insulator has a field rising linearly from the centre, because charge is distributed through its volume.

Answer: B

Correct answer: Gauss's law requires a closed surface

The law relates the flux out of a closed surface to the charge it encloses, and an open hemisphere encloses nothing - there is no inside and outside to speak of. The flux through it can be computed by integration, but the law says nothing about it and cannot be applied.

Answer: C

Only the enclosed charges count, and their sum is ten nanocoulombs. The external charge distorts the field at every point of the surface and so changes the local flux density, but its net contribution to the integral is exactly zero.

Answer: D

Correct answer: zero

A uniform vector field has zero divergence, since nothing is varying with position, so the charge density is zero throughout. The field is being produced by sources outside the region - an infinite sheet of charge being the standard example of one that produces a uniform field with no charge in it.