Q: A test charge $$q is $$moved by a displacement $$d\vec{r} in a $$non-uniform electric field $$\vec{E}(\vec{r})$$, as shown in the figure. Which of the following correctly expresses the work done by the field on the charge as it moves from point A to point B?

$$W_{A \to B} = q \int_A^B d\vec{r} \cdot \vec{E}(\vec{r})$$

Q: The resistivity $$\rho of a $$metal at temperature $$T is $$given by $$\rho(T) = \$$rho_0$$ (1 + \alpha (T - T_0$)). If$$ $$\$$rho_0$$ = 1.72 \times 10$$^{-8}$$ \, \Omega \cdot m$$ for copper at $$T_0 = 20$$^\circ C$$ and $$\alpha = 0.00393/\, ^\circ C$$, what is the resistivity at T = 100$$^\circ C$$?

$$\rho = 1.72 \times 10$$^{-8}$$ \times (1 + 0.00393 \times 80) \, \Omega \cdot m$$

Question 1

A test charge $$q is $$moved by a displacement $$d\vec{r} in a $$non-uniform electric field $$\vec{E}(\vec{r})$$, as shown in the figure. Which of the following correctly expresses the work done by the field on the charge as it moves from point A to point B?

Accepted Answer

$$W_{A 	o B} = q \int_A^B d\vec{r} \cdot \vec{E}(\vec{r})$$

Question 2

Which of the following statements about equipotential surfaces is correct?

Accepted Answer

The electric field is always perpendicular to equipotential surfaces.

Question 3

The electric potential $$V(\vec{r})$$ due to a point charge $$Q at $$the origin is given by which of the following expressions?

Accepted Answer

$$V(\vec{r}) = k_e \frac{Q}{r}$$

Question 4

A parallel-plate capacitor has plate area $$A$$ and separation $$d. $$What is the capacitance $$C of $$this capacitor in vacuum?

Accepted Answer

$$C = \$$epsilon_0$$ \frac{A}{d}$$

Question 5

A cylindrical capacitor consists of two coaxial cylinders of length $$L$$, inner radius $$a$$, and outer radius $$b. $$What is the capacitance of this system?

Accepted Answer

$$C = \frac{2\pi \$$epsilon_0$$ L}{\ln(b/a)}$$

Question 6

If two capacitors $$C_1$$ and $$C_2$$ are connected in series, what is the equivalent capacitance $$C$$?

Accepted Answer

$$\frac{1}{C} = \frac{1}{C_1$} + \frac{1}{C_2$}$$

Question 7

A dielectric slab of thickness $$a$$ and dielectric constant $$K is $$inserted between the plates of a parallel-plate capacitor of plate separation $$d$$ ($$a \u003c d). $$What is the new capacitance?

Accepted Answer

$$C = \frac{\$$epsilon_0$$ A}{d-a + a/K}$$

Question 8

The energy stored in a charged capacitor of capacitance $$C$$ and voltage $$V is $$given by:

Accepted Answer

$$U = \frac{1}{2} C V^2$

Question 9

A resistor of resistance $$R is $$connected to a battery of emf $$\mathcal{E}$$ and internal resistance $$r. $$What is the current $$I in $$the circuit?

Accepted Answer

$$I = \frac{\mathcal{E}}{R + r}$$

Question 10

The resistivity $$ho of a $$metal at temperature $$T is $$given by $$ho(T) = \$$rho_0$$ (1 + \alpha (T - T_0$)). If$$ $$\$$rho_0$$ = 1.72 	imes 10$$^{-8}$$ \, \Omega \cdot m$$ for copper at $$T_0 = 20$$^\circ C$$ and $$\alpha = 0.00393/\, ^\circ C$$, what is the resistivity at T = 100$$^\circ C$$?

Accepted Answer

$$ho = 1.72 	imes 10$$^{-8}$$ 	imes (1 + 0.00393 	imes 80) \, \Omega \cdot m$$

Question 11

A wire of length $$L$$ and cross-sectional area $$A$$ has resistance $$R. If $$the wire is stretched to three times its original length (with constant volume), what is its new resistance?

Accepted Answer

$$R_{new} = 9R$$

Question 12

In a circuit, two resistors $$R_1$$ and $$R_2$$ are connected in parallel. What is the equivalent resistance $$R_{eq}$$?

Accepted Answer

$$\frac{1}{R_{eq}} = \frac{1}{R_1$} + \frac{1}{R_2$}$$

Question 13

A capacitor of capacitance $$C is $$charged to a voltage $$V$$ and then disconnected from the battery. It is then connected across a resistor $$R. $$What is the time constant $$	au$$ for the discharge of the capacitor?

Accepted Answer

$$	au = RC$$

Question 14

A parallel-plate capacitor is filled with a dielectric of constant $$K. $$How does the capacitance change compared to the same capacitor without the dielectric?

Accepted Answer

It increases by a factor of $$K.$$

Question 15

The energy density $$u of $$the electric field in a parallel-plate capacitor is given by:

Accepted Answer

$$u = \frac{1}{2} \$$epsilon_0$$ E^2$

Question 16

A battery with emf $$\mathcal{E}$$ and internal resistance $$r is $$connected to a load resistor $$R. $$For what value of $$R is $$the power delivered to the load maximized?

Accepted Answer

$$R = r$$

Question 17

A point charge $$q is $$placed at the center of a hollow conducting spherical shell of inner radius $$a$$ and outer radius $$b$$, which carries a total charge $$Q. $$What is the surface charge density $$\sigma_{in} on $$the inner surface of the shell?

Accepted Answer

$$\sigma_{in} = -\frac{q}{4\pi a^2$}$$

Electric Potential, Capacitance, and DC Circuits: Structured Study Notes

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