Q: A parallel plate capacitor with area $$A = 2.0\ \mathrm{m}^2$$ and separation $$d = 2.0\ \mathrm{mm} is $$filled with a dielectric of $$\kappa = 5. $$What is its capacitance?

$$C = \dfrac{\kappa \$$varepsilon_0$$ A}{d} = 4.43 \times 10$$^{-8}$$\ \mathrm{F}$$

Question 1

A solid insulating sphere of radius $$a$$ carries a total charge $$Q$$ uniformly distributed throughout its volume. What is the electric potential $$V(r) at a $$point outside the sphere ($$r \u003e a$$), taking $$V(\infty) = 0$$?

Accepted Answer

$$V(r) = \dfrac{Q}{4\pi\$$varepsilon_0$$ r}$$

Question 2

For the same sphere, what is the electric potential $$V(r) at a $$point inside the sphere ($$r \u003c a$$)?

Accepted Answer

$$V(r) = \dfrac{Q}{8\pi\$$varepsilon_0$$ a} \left(3 - \dfrac{r^2$}{a^2$}\right)$$

Question 3

Given the potential $$V(x, y, z) = 3xy^2$ - 2yz$$, what is the $$x-$$component of the electric field $$E_x at $$any point?

Accepted Answer

$$E_x = -\dfrac{\partial V}{\partial x} = -3y^2$

Question 4

A parallel plate capacitor has plates of area $$A$$ separated by distance $$d. $$What is the capacitance $$C in $$vacuum?

Accepted Answer

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

Question 5

A cylindrical capacitor consists of an inner cylinder of radius $$a$$ and an outer cylinder of radius $$b$$ ($$b \u003e a$$), both of length $$l. $$What is the capacitance $$C$$?

Accepted Answer

$$C = \dfrac{2\pi\$$varepsilon_0$$ l}{\ln(b/a)}$$

Question 6

A parallel plate capacitor is filled with a dielectric of constant $$\kappa. $$How does the capacitance change?

Accepted Answer

It increases by a factor of $$\kappa$$: $$C = \kappa C_0$

Question 7

Two capacitors, $$C_1 = 4$$\ \mu\mathrm{F}$$ and C_2$ = 8\ \mu\mathrm{F}$$, are connected in series. What is their equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{C_1$ C_2$}{C_1$ + C_2$} = 2.67\ \mu\mathrm{F}$$

Question 8

A $$10\ \mu\mathrm{F}$$ and a $$20\ \mu\mathrm{F}$$ capacitor are connected in parallel across a $$100\ \mathrm{V}$$ battery. What is the total charge stored?

Accepted Answer

$$Q = (C_1$ + C_2$) V = 3000\ \mu\mathrm{C}$$

Question 9

A parallel plate capacitor with area $$A = 2.0\ \mathrm{m}^2$$ and separation $$d = 2.0\ \mathrm{mm} is $$filled with a dielectric of $$\kappa = 5. $$What is its capacitance?

Accepted Answer

$$C = \dfrac{\kappa \$$varepsilon_0$$ A}{d} = 4.43 	imes 10$$^{-8}$$\ \mathrm{F}$$

Question 10

If the electric potential in a region is given by $$V(x, y, z) = 5x^2$y - 3yz$$, what is the $$z-$$component of the electric field $$E_z$$?

Accepted Answer

$$E_z = -\dfrac{\partial V}{\partial z} = 3y$$

Question 11

A $$6\ \mu\mathrm{F}$$ and a $$12\ \mu\mathrm{F}$$ capacitor are connected in parallel. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = 18\ \mu\mathrm{F}$$

Question 12

A parallel plate capacitor is charged and then disconnected from the battery. If a dielectric is inserted between the plates, what happens to the stored energy?

Accepted Answer

The stored energy decreases

Question 13

Which of the following best describes the effect of a dielectric on the electric field between the plates of a capacitor?

Accepted Answer

The electric field is reduced by a factor of $$\kappa$$

Question 14

A $$10\ \mu\mathrm{F}$$ capacitor is charged to $$100\ \mathrm{V}$$ and then disconnected from the battery. If the dielectric is removed, what happens to the voltage across the plates?

Accepted Answer

The voltage increases

Question 15

Which of the following is the correct expression for the energy stored in a capacitor?

Accepted Answer

$$U = \dfrac{1}{2} C (\Delta V)^2$

Question 16

A $$5\ \mu\mathrm{F}$$ and a $$10\ \mu\mathrm{F}$$ capacitor are connected in series across a $$100\ \mathrm{V}$$ battery. What is the voltage across the $$10\ \mu\mathrm{F}$$ capacitor?

Accepted Answer

$$V_{10} = 33.3\ \mathrm{V}$$

Question 17

Which of the following statements about capacitance is correct?

Accepted Answer

Capacitance depends only on the geometry of the conductors and the dielectric material

Week 3 Lec. 3

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