Q: An arc of charge of length $$S$$ carries total charge $$+Q$$, uniformly distributed. Which of the following is true for the $$x-$$component of the total electric field at the origin?

$$E_x = \$$int_0$$^{\pi/2} \dfrac{k_e \lambda}{R^2$} \cos\theta d\theta$$

Question 1

Four identical point charges ($$q = 10.0\ \mu\mathrm{m}) $$are located at the corners of a rectangle: $$A=(0,0)$$, $$B=(0,60\ \mathrm{cm})$$, $$C=(15\ \mathrm{cm},0)$$, and $$D=(15\ \mathrm{cm},60\ \mathrm{cm}). $$What is the magnitude of the electric field at point $$P=(0,30\ \mathrm{cm})$$ due to the charge at $$C$$?

Accepted Answer

$$E = \dfrac{1}{4\pi\$$varepsilon_0$$} \dfrac{q}{(\sqrt{15^2$ + 30^2$}\ \mathrm{cm})^2$}$$

Question 2

An arc of a non-conducting charged ring subtends an angle $$2\$$theta_0$$ = 80^\circ$$ and is centered on the origin. The radius of the ring is $$R$$ and the total charge on the ring is $$q. $$What is the $$y-$$component of the electric field at the origin due to a charge element $$dq at $$angle $$	heta$$?

Accepted Answer

$$dE_y = \dfrac{1}{4\pi\$$varepsilon_0$$} \dfrac{dq}{R^2$} \sin	heta$$

Question 3

Three point charges are fixed at $$A=(-2.0\ \mathrm{cm}, 1.5\ \mathrm{cm})$$, $$B=(+2.5\ \mathrm{cm}, 0)$$, and $$C=(0, -2.5\ \mathrm{cm})$$ with $$Q_A=+6.0\ \mu\mathrm{C}$$, $$Q_B=-2.0\ \mu\mathrm{C}$$, $$Q_C=+1.0\ \mu\mathrm{C}. $$What is the direction of the net electric field at the origin?

Accepted Answer

The net field points in the direction of the vector sum of the individual fields from each charge.

Question 4

A negative charge is moving in the $$+x$$ direction in an electric field that points in the $$+x$$ direction. Which of the following statements is true?

Accepted Answer

The charge is slowing down.

Question 5

An infinitely long line of charge with charge density $$\lambda = -4.0\ \mathrm{nC/m} is $$placed along the $$z-$$axis. A particle with charge $$q=+2e$$ and mass $$m=6.7\$$times10^{-26}$$\ \mathrm{kg} is $$detected at point $$A=(0, -2.0\ \mathrm{mm}, 0)$$ and its velocity is $$\vec{v}_i in $$the $$-x$$ direction. What is the magnitude of the electric field at point $$A$$ due to the line of charge?

Accepted Answer

$$E = \dfrac{1}{2\pi\$$varepsilon_0$$} \dfrac{|\lambda|}{r}$$

Question 6

Which of the following is NOT a correct form of Gauss' Law?

Accepted Answer

$$\oint \vec{E} \cdot d\vec{A} = 4\pi k_e q_{enc}$$

Question 7

A square surface of area $$A=32.5\ \mathrm{cm}^2 is $$placed in a uniform electric field $$E=4.2\$$times10^3$$\ \mathrm{N/C} at an $$angle $$	heta=27^\circ to $$the horizontal. What is the electric flux through the area?

Accepted Answer

$$\Phi_E = EA\cos	heta$$

Question 8

Two charges, $$Q_1 = 3.0$$\ \mu\mathrm{C}$$ and Q_2$ = 7.0\ \mu\mathrm{C}$$, are placed along the $$x-$$axis and separated by $$a = 7.5\ \mathrm{mm}. $$What is the magnitude of the electric force that $$Q_1$$ exerts on $$Q_2$$?

Accepted Answer

$$F = \dfrac{1}{4\pi\$$varepsilon_0$$} \dfrac{|Q_1$ Q_2$|}{a^2$}$$

Question 9

An electric dipole with dipole moment $$p = 4.2 	imes 10$$^{-14}$$\ \mathrm{C\cdot m} is $$placed in a uniform electric field $$\vec{E} = 7.1 	imes 10^6$\ \mathrm{N/C}\ \hat{i}. $$What is the magnitude of the torque exerted on the dipole if the angle between $$\vec{p}$$ and $$\vec{E} is$$ $$90^\circ$$?

Accepted Answer

$$	au = pE\sin	heta$$

Question 10

A line of charge of length $$2L$$ carrying charge $$+Q is $$oriented vertically. What is the expression for the total electric field at point $$P$$ located at the origin, a distance $$a$$ from the line?

Accepted Answer

$$E = \dfrac{2k_e Q}{a\sqrt{a^2$ + L^2$}}\ \hat{i}$$

Question 11

Two charges, $$Q_1 = -1.0$$\ \mu\mathrm{C}$$ at x_1$ = -1.0\ \mathrm{m}$$ and $$Q_2 = -9.0$$\ \mu\mathrm{C}$$ at x_2$ = +1.0\ \mathrm{m}$$, are placed along the $$x-$$axis. At which point along the $$x-$$axis is the net electric field equal to zero?

Accepted Answer

$$x = +\dfrac{\sqrt{3} - 1}{2}a$$

Question 12

Which of the following statements about electric field lines is correct?

Accepted Answer

Electric field lines point in the direction of acceleration of a positive charge.

Question 13

An arc of charge of length $$S$$ carries total charge $$+Q$$, uniformly distributed. Which of the following is true for the $$x-$$component of the total electric field at the origin?

Accepted Answer

$$E_x = \$$int_0$$^{\pi/2} \dfrac{k_e \lambda}{R^2$} \cos	heta d	heta$$

Question 14

A square vertical surface of area $$A=32.5\ \mathrm{cm}^2 is $$placed in an electric field $$E=4.2\$$times10^3$$\ \mathrm{N/C} at an $$angle $$	heta=27^\circ$$ with the horizontal. What is the electric flux through the area?

Accepted Answer

$$\Phi_E = EA\cos	heta$$

Question 15

A nitrogen particle ($$N^{3-}) of $$mass $$m_N = 2.32 	imes 10$$^{-26}$$\ \mathrm{kg} is $$launched in a uniform electric field with initial velocity $$\vec{v} = (4.0\hat{i} - 7.5\hat{j}) 	imes 10^3$\ \mathrm{m/s}. $$The particle experiences acceleration $$\vec{a} = -1.39 	imes 10$$^{12}$$\ \mathrm{m/s^2$}\ \hat{i}. $$What is the electric field?

Accepted Answer

$$\vec{E} = \dfrac{m_N \vec{a}}{q}$$

Question 16

A solid sphere of radius $$R$$ carries charge $$|Q|$$ and the electric field as a function of distance $$r$$ from its center is measured. If the field inside the sphere increases linearly with $$r$$, what can you conclude about the material?

Accepted Answer

The sphere is an insulator.

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