A zero field in a region implies that the potential there is
Answer: C
The field is the negative gradient, so a vanishing field means a vanishing gradient and hence a constant potential. Confusing a constant potential with a zero one accounts for a good many mistakes in conductor problems.
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The potential of a dipole varies with distance as
Answer: D
Correct answer: the inverse square
The two charges of a dipole nearly cancel, so their potentials nearly cancel too, and what survives falls off one power faster than a single charge - as \( 1/r^2 \) rather than \( 1/r \). The field, being the gradient, falls faster still, as \( 1/r^3 \).
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The potential of a dipole on the plane bisecting the line joining its charges is
Answer: A
Every point of that plane is equidistant from the two equal and opposite charges, so their scalar contributions cancel exactly. The field there is not zero, however, being directed from the positive charge toward the negative one.
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The potential of a continuous charge distribution is found by
Answer: B
Each element contributes its own charge over \( 4\pi\epsilon_0 \) times its distance, and being scalars these add without regard to direction. That scalar addition is why the potential is usually computed first and the field obtained from it afterwards.
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The energy required to assemble a system of point charges is
Answer: C
The first charge costs nothing, the second is brought in against the field of the first, and so on, which is why the total is a sum over pairs. Written compactly it is half the sum of each charge times the potential of the others at its position.
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The energy stored in an electric field, per unit volume, is
Answer: D
It may also be written as \( \tfrac{1}{2}\mathbf{D} \cdot \mathbf{E} \), and integrating it over all space gives the same total as the pairwise sum over charges. The factor of one half arises in the same way as in the energy of a charged capacitor.
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The energy of a system of two like charges brought closer together
Answer: A
Pushing like charges together requires work, and that work is stored as field energy. Two opposite charges behave the other way: bringing them together releases energy and the stored energy is negative relative to infinite separation.
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Bringing two opposite charges together from infinity
Answer: B
They attract, so the field does the work and energy must be taken out of the system to keep them from accelerating. A bound system such as an atom therefore has a negative electrostatic energy relative to its separated parts.
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The potential 0.5 m from a point charge of 2 nC is about
Answer: C
Correct answer: 36 V
\[ V = \frac{Q}{4\pi\varepsilon_0 r} = \frac{2\times10^{-9}}{4\pi \times 8.854\times10^{-12} \times 0.5} = 36\ \mathrm{V}. \] Note the units: potential is in volts, and an answer quoted in volts per metre has computed the field instead.
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A charge of 3 nC is moved from a point at 100 V to one at 40 V. The work done by the field is
Answer: D
Correct answer: 180 nJ
\[ W = q(V_1 - V_2) = 3\times10^{-9} \times 60 = 180\ \mathrm{nJ}. \] The charge moves from high potential to low, so the field does the work and the sign is positive. Moving it the other way would require that work to be supplied from outside.
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