The capacitance of two concentric spheres of radii \( a \) and \( b \) becomes that of an isolated sphere when
Answer: D
The concentric sphere formula reduces to \( 4\pi\epsilon a \) as the outer conductor recedes, which is exactly the isolated sphere result. Physically the return conductor has moved so far away that only the inner sphere matters.
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Two capacitors, one charged and one uncharged, are connected in parallel. The total stored energy afterwards is
Answer: A
Charge is conserved but energy is not: the redistribution current flows through the resistance of the connecting wires and radiates or heats them. The loss is independent of that resistance, exactly as in charging a capacitor from a source.
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A guard ring is used in a precision capacitance measurement to
Answer: B
A ring surrounding the measuring electrode, held at the same potential but connected separately, carries away the fringing flux and leaves the field between the main electrodes truly uniform. The capacitance then agrees with the simple formula to a high accuracy, which is what a standard capacitor requires.
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The force of attraction between the plates of a charged capacitor, at constant charge, is
Answer: C
The force is the energy density times the area, and at constant charge the field between the plates does not change as they are moved, so neither does the force. At constant voltage instead the field rises as the plates close and the force grows as the inverse square of the separation.
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A dielectric slab partly inserted between the plates of a charged capacitor is
Answer: D
Inserting the slab raises the capacitance, and at constant charge that lowers the stored energy, so the slab is pulled into the region of stronger field. The same conclusion holds at constant voltage once the work done by the source is properly accounted for, which is a classic examination trap.
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The capacitance of a cable rises with age in service chiefly because
Answer: A
Water has a relative permittivity of about eighty, so even a small ingress raises the effective permittivity of the insulation appreciably. A rising capacitance measured on a routine test is therefore one of the indications of a cable taking in moisture, alongside a falling insulation resistance.
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A stack of \( n \) identical plates interleaved to form a capacitor gives a capacitance of
Answer: B
Each adjacent pair of plates forms one gap, and n plates give n minus one gaps, all effectively in parallel. This is how a variable air capacitor obtains a useful value from a small volume, and it is why such capacitors are built with many thin vanes rather than two large ones.
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Two capacitors of very different values are connected in series across an alternating supply. Most of the voltage appears across
Answer: C
The same current, and therefore the same charge, passes through both, so the voltages divide inversely with the capacitances and the smaller one takes almost all of it. On a direct supply the long-term division would instead be settled by the ratio of the leakage resistances, which is a quite different calculation.
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The relative permittivity of air at ordinary pressure is taken as unity because
Answer: D
The measured value is about 1.00059, so treating air as a vacuum introduces an error far smaller than that of any ordinary measurement. It does rise with pressure, which is why compressed gas capacitors and gas-insulated switchgear behave measurably differently from air-filled ones.
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A capacitor of 1 \( \mu \)F with a leakage resistance of 100 M\( \Omega \) loses its charge with a time constant of
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.
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