Q: A $-1\ \mu\mathrm{C}$ charge experiences a $10^7\ \mathrm{N}$ electric force in a certain electric field. What force would a proton experience in the same field?

$F_p = (1.6 \times 10^{-19}\ \mathrm{C}) \dfrac{10^7\ \mathrm{N}}{-1.0 \times 10^{-6}\ \mathrm{C}} = -1.6\ \mathrm{pN}$

Q: A charge is spread uniformly over the surface of a spherical balloon of radius $70\ \mathrm{cm}$, resulting in an electric field of $26\ \mathrm{kN/C}$ at the balloon's surface. What is the net charge on the balloon?

$Q = \dfrac{E R^2}{k} = \dfrac{26 \times 10^3\ \mathrm{N/C} \times (0.70\ \mathrm{m})^2}{9.0 \times 10^9\ \mathrm{N \cdot m^2/C^2}} = 1.4\ \mu\mathrm{C}$

Q: A sphere of radius $10\ \mathrm{cm}$ carries a $40\ \mu\mathrm{C}$ charge distributed uniformly throughout its volume. It is surrounded by a concentric shell of radius $20\ \mathrm{cm}$, also uniformly charged with $40\ \mu\mathrm{C}$. What is the electric field at $r = 5.0\ \mathrm{cm}$ from the center?

$E = \dfrac{k Q r}{R^3} = \dfrac{9.0 \times 10^9 \times 40 \times 10^{-6} \times 0.05}{(0.10)^3} = 1.8 \times 10^6\ \mathrm{N/C}$

Q: The potential difference between the surface of a $3.0$-cm-diameter power line and a point $1.0$ m distant is $3.9$ kV. What is the line charge density on the power line?

$\lambda = \dfrac{2 \pi \varepsilon_0 V}{\ln(r_2/r_1)} = \dfrac{2 \pi \times 8.85 \times 10^{-12} \times 3900}{\ln(1.0/0.015)} = -2.0 \times 10^{-7}\ \mathrm{C/m}$

Q: The potential difference across a cell membrane is $65\ \mathrm{mV}$. On the outside are $1.5 \times 10^7$ singly ionized potassium atoms. Assuming an equal negative charge inside, what is the membrane's capacitance?

$C = \dfrac{Q}{V} = \dfrac{(1.5 \times 10^7)(1.6 \times 10^{-19})}{65 \times 10^{-3}} = 3.7 \times 10^{-11}\ \mathrm{F}$

Q: Four capacitors are arranged such that $C_1 = 3.3\ \mathrm{pF}$, $C_2 = 2.7\ \mathrm{pF}$, $C_3 = 1.2\ \mathrm{pF}$, and $C_4 = 1.8\ \mathrm{pF}$. If $C_3$ and $C_4$ are in parallel, and their combination is in series with $C_2$, what is the equivalent capacitance?

$C_{eq} = \left[\dfrac{1}{C_2} + \dfrac{1}{C_3 + C_4}\right]^{-1} = 1.4\ \mathrm{pF}$

Q: The maximum safe current in a 12-gauge ($2.1$-mm-diameter) copper wire is $20\ \mathrm{A}$. What is the current density?

$J = \dfrac{I}{A} = \dfrac{20}{\pi (1.05 \times 10^{-3})^2} = 5.8 \times 10^6\ \mathrm{A/m^2}$

Q: A cylindrical piece of material $2.4$ cm long and $2.0$ mm in diameter is connected to a $9$-V battery, resulting in a $2.6$-mA current. What is the resistivity of the material?

$\rho = \dfrac{V \pi r^2}{I L} = \dfrac{9 \times \pi \times (1.0 \times 10^{-3})^2}{2.6 \times 10^{-3} \times 0.024} = 0.47\ \Omega \cdot \mathrm{m}$

Question 1

An electron at Earth's surface experiences a gravitational force $mg$. At what distance from a proton would the electrical force between them equal this gravitational force? (Given: $k = 9.0 	imes 10^9\ \mathrm{N \cdot m^2/C^2}$, $e = 1.60 	imes 10^{-19}\ \mathrm{C}$, $m_e = 9.11 	imes 10^{-31}\ \mathrm{kg}$, $g = 9.8\ \mathrm{m/s^2}$)

Accepted Answer

$r = \sqrt{\dfrac{ke^2}{m_e g}} = 5.1\ \mathrm{m}$

Question 2

A $-1\ \mu\mathrm{C}$ charge experiences a $10^7\ \mathrm{N}$ electric force in a certain electric field. What force would a proton experience in the same field?

Accepted Answer

$F_p = (1.6 	imes 10^{-19}\ \mathrm{C}) \dfrac{10^7\ \mathrm{N}}{-1.0 	imes 10^{-6}\ \mathrm{C}} = -1.6\ \mathrm{pN}$

Question 3

Given $q_1 = 25\ \mu\mathrm{C}$ and $q_2 = 20\ \mu\mathrm{C}$, if the force on $q_1$ points in the $-x$ direction, what is the value of $q_3$?

Accepted Answer

$q_3 = -20\ \mu\mathrm{C}$

Question 4

If the magnitude of the middle charge in a field line diagram is $3\ \mu\mathrm{C}$, and the outer charges are $+3\ \mu\mathrm{C}$ each, what is the net charge shown?

Accepted Answer

$+3\ \mu\mathrm{C}$

Question 5

A $6.8\ \mu\mathrm{C}$ charge and a $-4.7\ \mu\mathrm{C}$ charge are inside an uncharged sphere. What is the electric flux through the sphere?

Accepted Answer

$\Phi = \dfrac{q_{enclosed}}{\varepsilon_0} = \dfrac{6.8\ \mu\mathrm{C} - 4.7\ \mu\mathrm{C}}{8.85 	imes 10^{-12}\ \mathrm{C^2/(N \cdot m^2)}} = 240\ \mathrm{kN \cdot m^2/C}$

Question 6

A charge is spread uniformly over the surface of a spherical balloon of radius $70\ \mathrm{cm}$, resulting in an electric field of $26\ \mathrm{kN/C}$ at the balloon's surface. What is the net charge on the balloon?

Accepted Answer

$Q = \dfrac{E R^2}{k} = \dfrac{26 	imes 10^3\ \mathrm{N/C} 	imes (0.70\ \mathrm{m})^2}{9.0 	imes 10^9\ \mathrm{N \cdot m^2/C^2}} = 1.4\ \mu\mathrm{C}$

Question 7

A sphere of radius $10\ \mathrm{cm}$ carries a $40\ \mu\mathrm{C}$ charge distributed uniformly throughout its volume. It is surrounded by a concentric shell of radius $20\ \mathrm{cm}$, also uniformly charged with $40\ \mu\mathrm{C}$. What is the electric field at $r = 5.0\ \mathrm{cm}$ from the center?

Accepted Answer

$E = \dfrac{k Q r}{R^3} = \dfrac{9.0 	imes 10^9 	imes 40 	imes 10^{-6} 	imes 0.05}{(0.10)^3} = 1.8 	imes 10^6\ \mathrm{N/C}$

Question 8

A charge $q$ is at the center of a spherical shell of radius $R$ carrying charge $2q$ uniformly over its surface. What is the electric field strength at a distance $r < R$ from the center?

Accepted Answer

$E = \dfrac{k q}{r^2}$

Question 9

A particle gains $1.50\ \mathrm{fJ}$ of energy when it moves through a potential difference of $3.12\ \mathrm{kV}$. Is it singly, doubly, triply, or quadruply ionized?

Accepted Answer

Singly ionized

Question 10

The potential difference between the surface of a $3.0$-cm-diameter power line and a point $1.0$ m distant is $3.9$ kV. What is the line charge density on the power line?

Accepted Answer

$\lambda = \dfrac{2 \pi \varepsilon_0 V}{\ln(r_2/r_1)} = \dfrac{2 \pi 	imes 8.85 	imes 10^{-12} 	imes 3900}{\ln(1.0/0.015)} = -2.0 	imes 10^{-7}\ \mathrm{C/m}$

Question 11

A $+Q$ charge lies at the origin and $-3Q$ at $x = a$. Find the point(s) on the $x$-axis where the electric potential is zero.

Accepted Answer

$x = \dfrac{a}{2}$

Question 12

The potential difference across a cell membrane is $65\ \mathrm{mV}$. On the outside are $1.5 	imes 10^7$ singly ionized potassium atoms. Assuming an equal negative charge inside, what is the membrane's capacitance?

Accepted Answer

$C = \dfrac{Q}{V} = \dfrac{(1.5 	imes 10^7)(1.6 	imes 10^{-19})}{65 	imes 10^{-3}} = 3.7 	imes 10^{-11}\ \mathrm{F}$

Question 13

Four capacitors are arranged such that $C_1 = 3.3\ \mathrm{pF}$, $C_2 = 2.7\ \mathrm{pF}$, $C_3 = 1.2\ \mathrm{pF}$, and $C_4 = 1.8\ \mathrm{pF}$. If $C_3$ and $C_4$ are in parallel, and their combination is in series with $C_2$, what is the equivalent capacitance?

Accepted Answer

$C_{eq} = \left[\dfrac{1}{C_2} + \dfrac{1}{C_3 + C_4}\right]^{-1} = 1.4\ \mathrm{pF}$

Question 14

The maximum safe current in a 12-gauge ($2.1$-mm-diameter) copper wire is $20\ \mathrm{A}$. What is the current density?

Accepted Answer

$J = \dfrac{I}{A} = \dfrac{20}{\pi (1.05 	imes 10^{-3})^2} = 5.8 	imes 10^6\ \mathrm{A/m^2}$

Question 15

A cylindrical piece of material $2.4$ cm long and $2.0$ mm in diameter is connected to a $9$-V battery, resulting in a $2.6$-mA current. What is the resistivity of the material?

Accepted Answer

$ho = \dfrac{V \pi r^2}{I L} = \dfrac{9 	imes \pi 	imes (1.0 	imes 10^{-3})^2}{2.6 	imes 10^{-3} 	imes 0.024} = 0.47\ \Omega \cdot \mathrm{m}$

Question 16

A Taser delivers $100\ \mu\mathrm{C}$ of charge to a victim in a pulse lasting $5.0\ \mathrm{ms}$, with a potential difference of $1200\ \mathrm{V}$ across the body. What is the instantaneous current during the pulse?

Accepted Answer

$I = \dfrac{Q}{\Delta t} = \dfrac{100 	imes 10^{-6}}{5.0 	imes 10^{-3}} = 0.020\ \mathrm{A}$

Electrostatics and Electric Fields: Key Concepts and Applications

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