Your Answer : (Not Answered)

Correct Answer : C

The imaginary part of the permittivity accounts for the lag of the polarisation behind the field, which dissipates energy quite apart from any free carriers. It is what heats food in a microwave oven, since water's relaxation falls in that band.

Your Answer : (Not Answered)

Correct Answer : B

The three are points on one continuum: as the phase difference grows from zero to ninety degrees the line opens into an ellipse and then closes into a circle. Unequal amplitudes give an ellipse at every phase difference except zero.

Your Answer : (Not Answered)

Correct Answer : D

The free space wavelength is three metres and the square root of 2.25 is 1.5, so the wavelength in the medium is two metres. A relative permittivity of 2.25 corresponds to a refractive index of exactly 1.5, which is that of ordinary glass.

Your Answer : (Not Answered)

Correct Answer : A

Correct answer: \( \beta = 2\pi/\lambda \)

The phase constant is the phase advance per unit distance, and a full wavelength corresponds to \( 2\pi \) radians, so \( \beta = 2\pi/\lambda \) with units of radians per metre. It is the spatial counterpart of the angular frequency \( \omega = 2\pi/T \), and the phase velocity follows as \( v = \omega/\beta \).

Your Answer : (Not Answered)

Correct 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.

Your Answer : (Not Answered)

Correct 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.

Your Answer : (Not Answered)

Correct 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.

Your Answer : (Not Answered)

Correct Answer : A

The expression gives \( 2 \times 10^{-7} \times 10^{4}/0.1 = 2 \times 10^{-2} \) N/m, and opposing currents repel. On a short circuit of ten kiloamperes the same conductors would push apart with two hundred newtons per metre, which is why busbar bracing matters.

Your Answer : (Not Answered)

Correct Answer : C

Each plane produces an image and each of those must be imaged in the other plane, but the fourth coincides with the original position and the series terminates. For a wedge angle that divides exactly into a full turn the series always terminates in this way.

Your Answer : (Not Answered)

Correct Answer : B

Poisson's equation with a non-zero right hand side has a non-zero solution even with zero boundary values, the charge itself raising or lowering the interior potential. Only the homogeneous Laplace case would give zero throughout.

Your Answer : (Not Answered)

Correct Answer : A

Laplace's equation is linear and homogeneous, so scaling the boundary data scales the whole solution, and with it the field, the surface charges and the stored energy by the square. The capacitance, being a ratio, stays the same.

Your Answer : (Not Answered)

Correct Answer : A

The product is \( 10^{-6} \times 10^{8} = 100 \) seconds, so the voltage falls to about a third in a minute and a half. This is why a large capacitor is fitted with a discharge resistor and why one should never assume a disconnected capacitor is safe to handle.

Your Answer : (Not Answered)

Correct Answer : A

The molecules cannot turn fast enough to follow a rapidly reversing field, so their contribution drops out and the permittivity falls from about eighty at low frequency toward the value set by electronic polarisation alone. The energy absorbed during that relaxation is what heats food in a microwave oven.

Your Answer : (Not Answered)

Correct Answer : A

The first two are the defining conditions from which both combination rules follow. The energy goes as the square of the voltage, which is why doubling the working voltage of a capacitor bank quadruples the energy it can deliver.

Your Answer : (Not Answered)

Correct Answer : D

The loss is exactly half whatever the resistance, because a smaller resistance simply makes a larger current flow for a shorter time. The reason given is therefore false, though the assertion it accompanies is a standard and correct result.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: bound charge on its surfaces but no net charge overall

The field aligns or induces molecular dipoles, and within the material the head of one dipole cancels the tail of its neighbour, so charge appears only where that cancellation fails - at the two surfaces, positive on one and negative on the other. The magnitudes are equal, so the material remains neutral overall, and the charge is bound because it cannot leave the molecules that carry it.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: 1 and 2 only are correct

Statements 1 and 2 are correct: the density goes as \( E^2 \), and assembling like charges requires work against their repulsion. Statement 3 has the sign wrong - a bound pair of opposite charges has negative energy, since work would have to be done to separate them, and that is precisely what being bound means.

Your Answer : (Not Answered)

Correct Answer : A

A uniform field has zero divergence, so the charge density is zero throughout. The charge producing it lies elsewhere, on the plates that bound the region, which is exactly the situation inside a parallel plate capacitor.

Your Answer : (Not Answered)

Correct Answer : D

Correct answer: where the surface curves most sharply

Charge accumulates where the radius of curvature is smallest, since the mutual repulsion pushes it as far apart as the shape allows and a sharp region offers the greatest escape from its neighbours. The field just outside is \( \sigma/\epsilon_0 \), so the field is largest there too - which is why discharge begins at points and why a lightning conductor is pointed.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: \( 9 \times 10^{9} \)

With \( \epsilon_0 = 8.85\times10^{-12} \) F/m, \[ \frac{1}{4\pi\epsilon_0} = 8.99\times10^{9} \approx 9\times10^{9}\ \mathrm{N\,m^2\,C^{-2}}. \] The 8.85 offered is the permittivity itself rather than the constant built from it, \( 3\times10^{8} \) is the speed of light, and \( 4\pi\times10^{-7} \) is the permeability of free space.