A parallel plate capacitor with plates of 100 cm² and a gap of 0.5 mm filled with mica of relative permittivity 6 has a capacitance of about
Answer: B
The area is \( 10^{-2} \) m², so \( 6 \times 8.854 \times 10^{-12} \times 10^{-2}/(5 \times 10^{-4}) = 1.06 \times 10^{-9} \) F. Mica capacitors of this construction were the standard precision component before plastic films displaced them.
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Doubling both the plate area and the separation of a capacitor leaves its capacitance
Answer: C
The area appears in the numerator and the separation in the denominator, so doubling both cancels exactly. Recognising such a cancellation avoids recomputing the whole expression, and the same reasoning shows what happens when a capacitor is scaled up in every linear dimension at once.
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Scaling every linear dimension of a parallel plate capacitor by a factor \( k \) changes its capacitance by
Answer: D
The area grows as \( k^2 \) and the separation as \( k \), so the ratio grows as the first power of the scale. Capacitance therefore has the dimensions of a permittivity times a length, which is the same conclusion the isolated sphere formula reaches.
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Comparing a capacitor charged at constant voltage with one charged at constant charge as a dielectric is inserted, the stored energy
Answer: A
At constant voltage the energy is proportional to capacitance and so rises; at constant charge it is inversely proportional and so falls. In both cases the slab is drawn in, because the correct criterion at constant voltage includes the work done by the source and not merely the stored energy.
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The energy stored per unit volume in a dielectric of relative permittivity 4 at a field of 1 MV/m is about
Answer: B
Substituting in \( \tfrac{1}{2}\epsilon_0\epsilon_r E^2 \) gives \( 0.5 \times 8.854 \times 10^{-12} \times 4 \times 10^{12} = 17.7 \) J/m³. Even at a megavolt per metre the density is modest, which is why electrostatic energy storage cannot compete with chemical storage on a volume basis.
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A composite capacitor has two layers whose product of thickness and reciprocal permittivity is the same for each. The two layers then
Answer: C
The capacitance per unit area of a layer is the permittivity over the thickness, so equal values of thickness over permittivity means equal capacitances, and two equal capacitances in series divide the voltage equally. The fields in them differ, being inversely proportional to their permittivities.
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The reason a variable capacitor for a radio tuner uses air rather than a solid dielectric is that air
Answer: D
Interleaved vanes must slide past one another repeatedly, which no solid dielectric would survive, and air introduces almost no loss at radio frequency. The penalty is a small capacitance for the volume, which is why such capacitors are physically large for a few hundred picofarads.
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