The field on either side of an infinite sheet of charge of surface density \( \rho_s \) is
Answer: C
Gauss's law applied to a box straddling the sheet gives equal flux out of two faces whose area does not change with distance, so the field is uniform. It is directed away from the sheet on both sides for a positive density.
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Between two parallel sheets carrying equal and opposite surface charge densities, the field is
Answer: D
Each sheet contributes \( \rho_s/2\epsilon_0 \) and between them the two contributions point the same way and add. Outside the pair they point oppositely and cancel, which is why a parallel plate capacitor confines its field.
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On the axis of a uniformly charged ring, the electric field at the centre of the ring is
Answer: A
Every element of the ring is matched by a diametrically opposite element whose contribution is equal and opposite, so the whole cancels. The axial field rises from zero, reaches a maximum at a distance of the ring radius over root two, and then falls away.
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Far from a uniformly charged ring along its axis, the field approaches that of
Answer: B
At distances much greater than the ring radius the structure of the source ceases to matter and only its total charge does, so the field tends to the inverse square form. Every bounded charge distribution with a net charge behaves this way at a distance.
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Close to the surface of a large uniformly charged disc, on its axis, the field approaches that of
Answer: C
At a distance small compared with the radius the disc looks unbounded from the point in question, so the field tends to \( \rho_s/2\epsilon_0 \). At the other extreme, far away, it tends to the point charge form instead.
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A charge distributed along a thin wire is described by a
Answer: D
Idealising the wire as having no thickness makes the charge per unit length the natural description, and integrating it along the wire gives the total charge. The field close to the wire diverges, which is a consequence of the idealisation rather than of physics.
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The total charge in a volume with charge density \( \rho_v \) is obtained by
Answer: A
Charge density is charge per unit volume, so the total follows from \( Q = \int_v \rho_v \, dv \) with the appropriate volume element for the coordinate system in use. For a uniform density the integral reduces to the density times the volume.
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An electric field line at any point is drawn so that
Answer: B
Field lines cannot cross, since the field has one direction at each point, and their density conveys the magnitude. They begin on positive charge and end on negative charge, or run to infinity, which is the picture Gauss's law makes quantitative.
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The arrangement to which the name electric dipole is given is
Answer: C
The dipole moment is the product of the charge and the separation, directed from the negative charge to the positive one. Because the net charge is zero the field falls faster than that of a point charge, which is what makes the dipole a distinct case worth studying.
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The field of an electric dipole falls off with distance as
Answer: D
The two opposite charges nearly cancel at a distance, and what survives is the difference of two inverse square terms, which falls as the inverse cube. The potential correspondingly falls as the inverse square rather than the inverse first power.
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