Q: A capacitor is connected to a 12 V battery and stores a charge of 960 pC. What is its capacitance?

$$C = \dfrac{Q}{V} = \dfrac{960 \times 10$$^{-12}$$ \ C}{12 \ V} = 80 \ pF$$

Q: What is the energy stored in a capacitor of $$80 \ pF$$ charged to $$12 \ V$$?

$$U = \dfrac{1}{2}CV^2$ = \dfrac{1}{2} \times 80 \times 10$$^{-12}$$ \ F \times (12 \ V)^2$ = 5.76 \times 10$$^{-9}$$ \ J$$

Q: A capacitor is rated for a maximum working voltage of $$50 \ V$$ and has a dielectric with breakdown field $$E_{breakdown} = 500 \ MV/m. If $$the plate separation is $$0.1 \ mm$$, is it safe to operate the capacitor at its rated voltage?

Yes, because $$E = \dfrac{V}{d} = \dfrac{50}{1 \times 10$$^{-4}$$} = 0.5 \times 10$$^{6}$$ \ V/m = 0.5 \ MV/m$$, which is much less than $$500 \ MV/m$$

Question 1

A parallel-plate capacitor is constructed using two circular plates of radius 12 cm separated by 5.0 mm. What is the capacitance of this capacitor in picofarads ($$pF$$)? Use $$\$$varepsilon_0$$ = 8.85 	imes 10$$^{-12}$$ \ F/m.$$

Accepted Answer

$$C = \dfrac{\$$varepsilon_0$$ \pi R^2$}{d} = 80 \ pF$$

Question 2

Which of the following changes will increase the capacitance of a parallel-plate capacitor the most?

Accepted Answer

Increase the plate area and decrease the separation between plates

Question 3

A capacitor is connected to a 12 V battery and stores a charge of 960 pC. What is its capacitance?

Accepted Answer

$$C = \dfrac{Q}{V} = \dfrac{960 	imes 10$$^{-12}$$ \ C}{12 \ V} = 80 \ pF$$

Question 4

What is the energy stored in a capacitor of $$80 \ pF$$ charged to $$12 \ V$$?

Accepted Answer

$$U = \dfrac{1}{2}CV^2$ = \dfrac{1}{2} 	imes 80 	imes 10$$^{-12}$$ \ F 	imes (12 \ V)^2$ = 5.76 	imes 10$$^{-9}$$ \ J$$

Question 5

If a dielectric material with dielectric constant $$\kappa = 4 is $$inserted between the plates of a capacitor, how does the capacitance change?

Accepted Answer

The capacitance increases by a factor of 4

Question 6

Which equation correctly relates the electric field $$E$$ inside a parallel-plate capacitor to the potential difference $$V$$ and plate separation $$d$$?

Accepted Answer

$$E = \dfrac{V}{d}$$

Question 7

A capacitor is made from two plates separated by an insulator. What is the primary role of the insulator (dielectric) between the plates?

Accepted Answer

To increase the capacitance by reducing the electric field and allowing more charge to be stored for the same voltage

Question 8

Three capacitors with capacitances $$C_1$$, $$C_2$$, and $$C_3$$ are connected in series. What is the equivalent capacitance $$C_{series}$$?

Accepted Answer

$$\dfrac{1}{C_{series}} = \dfrac{1}{C_1$} + \dfrac{1}{C_2$} + \dfrac{1}{C_3$}$$

Question 9

Which combination of capacitors will result in an equivalent capacitance greater than any individual capacitance?

Accepted Answer

Capacitors connected in parallel

Question 10

A capacitor is rated for a maximum working voltage of $$50 \ V$$ and has a dielectric with breakdown field $$E_{breakdown} = 500 \ MV/m. If $$the plate separation is $$0.1 \ mm$$, is it safe to operate the capacitor at its rated voltage?

Accepted Answer

Yes, because $$E = \dfrac{V}{d} = \dfrac{50}{1 	imes 10$$^{-4}$$} = 0.5 	imes 10$$^{6}$$ \ V/m = 0.5 \ MV/m$$, which is much less than $$500 \ MV/m$$

Question 11

Which of the following best describes the energy stored in a capacitor?

Accepted Answer

$$U = \dfrac{1}{2}CV^2$

Question 12

A parallel-plate capacitor is filled with a dielectric material of dielectric constant $$\kappa. $$Which formula gives its capacitance?

Accepted Answer

$$C = \kappa \dfrac{\$$varepsilon_0$$ A}{d}$$

Question 13

If you want to store a large amount of energy in a capacitor, which combination should you choose?

Accepted Answer

Large capacitance and large potential difference

Question 14

A capacitor is made from two plates of area $$A$$ separated by distance $$d. If $$the area is doubled and the separation is halved, how does the capacitance change?

Accepted Answer

The capacitance increases by a factor of 4

Question 15

Which of the following materials has the highest dielectric constant according to the provided table?

Accepted Answer

Water

Question 16

In a real-world scenario, why is it important to consider the breakdown field of a dielectric when designing a capacitor?

Accepted Answer

To prevent the dielectric from failing and causing a short circuit at high voltages

Question 17

Which of the following statements about capacitors in series is correct?

Accepted Answer

The equivalent capacitance is less than any individual capacitance

Question 18

A capacitor stores a charge $$Q at a $$potential difference $$V. $$What is the relationship between $$Q$$, $$C$$, and $$V$$?

Accepted Answer

$$C = \dfrac{Q}{V}$$

Question 19

In a circuit, why might you use a combination of capacitors rather than a single capacitor?

Accepted Answer

To achieve a desired capacitance value that may not be available in a single capacitor

Capacitors: Principles, Practice, and Circuits

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