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 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: 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 $$r \u003c R$$ 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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