Electric Potential and Energy Questions and Answers

Answer: A

It is a purely geometrical property, depending on the size and shape of the body and on the medium surrounding it but not on how much charge it happens to carry. For a sphere it comes out as \( 4\pi\epsilon_0 a \), which is the standard result and the reason capacitance is measured in farads per metre of characteristic dimension.

Answer: B

The energy gained is the charge times the potential difference, and the two particles carry equal charge magnitudes, so the energies are equal. The proton being about eighteen hundred times the more massive, its speed is smaller by the square root of that factor, some forty-three times slower than the electron.

Answer: C

The whole conductor and its cavity sit at one potential, so the field within the cavity is zero and no force acts. The potential being a hundred volts rather than zero makes no difference whatever, since only the gradient of the potential produces a force and a constant has no gradient.

Answer: D

The energy is \( -\mathbf{p} \cdot \mathbf{E} \), which is most negative when the moment lies along the field. That is the position of stable equilibrium, and turning the dipole to lie against the field raises the energy to \( +pE \), giving a total swing of twice the product between the two extreme orientations.

Answer: A

Each corner is at a distance of half the diagonal, that is \( a/\sqrt{2} \), and being scalars the four contributions simply add. The field at the same point, by contrast, is exactly zero, since the four vector contributions cancel in opposite pairs, which is a good illustration of how differently the two quantities behave.

Answer: B

Each pair of diagonally opposite charges contributes equal and opposite vectors at the centre, so the whole cancels. The potential there is nevertheless four times a single contribution, because potentials add as scalars, and the contrast between the two results is the point of the arrangement.

Answer: C

Like charges are being pushed closer together against their mutual repulsion, so work must be done and the field energy rises. Taken to the limit of a point charge the energy diverges, which is one of the well known difficulties of the classical theory of the electron and a sign that the point charge is an idealisation.

Answer: D

The field falls as the inverse square of the radius between the spheres, so it is largest where the radius is smallest. Breakdown therefore always begins at the inner electrode, which is why the inner conductor of a high voltage cable is made as large as the design will allow rather than as small as possible.

Answer: A

The three partial derivatives of \( xyz \) are \( yz \), \( xz \) and \( xy \), each of which is one at the given point, and reversing the sign gives the field. The magnitude of the field there is therefore the square root of three, directed along the negative of the diagonal through the origin.

Answer: B

Taking the divergence of the gradient relation and using the vanishing divergence of D in a charge-free region gives \( \nabla^2 V = 0 \). Where charge is present the right hand side becomes minus the density over the permittivity, and the equation is then Poisson's, which is the subject of the next topic in this series.