Chapter 24 — Capacitance & Dielectrics

14 points · 3 extra credit

Chapter 24 — Capacitance & Dielectrics

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given. Use $\varepsilon_0=8.85\times10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$ and $k=8.99\times10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$.

1. 1. (2 pts ••) A parallel-plate capacitor has plate area $A=0.012\,\text{m}^2$ and separation $d=1.5\,\text{mm}$, connected to a $12\,\text{V}$ battery.

   (a) Find the capacitance.

   (b) Find the charge on each plate.

   (c) Find the energy stored.

   (d) Find the electric field between the plates.

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2. 2. (2 pts ••) Three capacitors have capacitances $6.0\,\mu\text{F}$, $3.0\,\mu\text{F}$, and $2.0\,\mu\text{F}$.

   (a) Find the equivalent capacitance if all three are in series.

   (b) Find the equivalent capacitance if all three are in parallel.

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3. 3. (3 pts •••) A $6.0\,\mu\text{F}$ capacitor ($C_1$) is connected in series with the parallel combination of two $3.0\,\mu\text{F}$ capacitors ($C_2$ and $C_3$). This combination is connected across a $24\,\text{V}$ battery.

   (a) Find the equivalent capacitance of the network.

   (b) Find the total charge delivered by the battery.

   (c) Find the voltage across, and the charge on, each of the three capacitors.

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4. 4. (2 pts ••) A $4.0\,\mu\text{F}$ capacitor is charged to $50\,\text{V}$ and then disconnected from the battery. Its plate separation is then doubled.

   (a) Find the new capacitance.

   (b) Find the new charge on the plates.

   (c) Find the new voltage.

   (d) Find the change in stored energy, and explain where that energy came from.

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5. 5. (2 pts ••) An air-filled capacitor with $C_0=100\,\text{pF}$ is connected to a $12\,\text{V}$ battery. While it remains connected, a dielectric of constant $K=3.0$ is inserted, completely filling the gap.

   (a) Find the new capacitance.

   (b) Find the new charge on the plates.

   (c) Does the stored energy increase or decrease, and where does the difference come from?

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6. 6. (1 pt •) A capacitor stores $4.5\times10^{-4}\,\text{J}$ of energy when charged to $30\,\text{V}$. Find its capacitance.

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7. 7. (2 pts ••) A spherical capacitor has an inner sphere of radius $a=1.0\,\text{cm}$ and a concentric outer shell of radius $b=1.2\,\text{cm}$ (its capacitance is $C=4\pi\varepsilon_0\dfrac{ab}{b-a}$).

   (a) Find the capacitance.

   (b) Find the energy stored when it is charged to $500\,\text{V}$.

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8. Extra Credit (3 pts •••)

   A cylindrical capacitor consists of an inner conductor of radius $a$ and a coaxial outer conductor of radius $b$, both of length $L\gg b$, carrying charges $\pm Q$.

   (a) Use Gauss's law to find the electric field between the conductors.

   (b) Integrate the field to find the potential difference between them, and hence show that $C=\dfrac{2\pi\varepsilon_0 L}{\ln(b/a)}$.

   (c) Show that when the gap $d=b-a$ is very small compared with $a$, this reduces to the parallel-plate result $C=\varepsilon_0 A/d$.

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