Q: A point charge $$q_1 = +4.0$$\ \mu\mathrm{C}$$ is placed at one corner of an equilateral triangle with side length l = 0.40$$\ \mathrm{m}$$. What is the magnitude of the electric field at the location of q_3$ due to q_1$?

$$E_1$$ = \dfrac{$$kq_1$$}{$$l^2$$} = 2.25 \times $$10^5$$\ \mathrm{N/C}$$

Q: Given $$q_2 = +2.0$$\ \mu\mathrm{C}$$ at another corner of the triangle, what is the magnitude of the electric field at q_3$ due to q_2$?

$$E_2$$ = \dfrac{$$kq_2$$}{$$l^2$$} = 1.12 \times $$10^5$$\ \mathrm{N/C}$$

Q: A point charge $$Q = +6.0\ \mu\mathrm{C} is $$fixed in space. What is the electric potential at $$r = 0.50\ \mathrm{m}$$ from the charge?

$$V = \dfrac{kQ}{r} = 1.08 \times 10^5$\ \mathrm{V}$$

Q: A proton is released from rest at $$r = 0.50\ \mathrm{m}$$ and moves to $$r = 1.50\ \mathrm{m}$$ from a $$+6.0\ \mu\mathrm{C}$$ charge. What is the change in its electric potential energy?

$$\Delta U = kq_pQ\left(\dfrac{1}{r_f} - \dfrac{1}{r_i}\right) = -1.15 \times 10$$^{-14}$$\ \mathrm{J}$$

Q: What is the speed of the proton at $$r = 1.50\ \mathrm{m}$$ after being released from rest at $$r = 0.50\ \mathrm{m}$$ near a $$+6.0\ \mu\mathrm{C}$$ charge?

$$v_f = \sqrt{\dfrac{2K_f}{m_p}} = 3.71 \times 10^6$\ \mathrm{m/s}$$

Q: A dipole consists of charges $$+4.0\ \mu\mathrm{C}$$ and $$-4.0\ \mu\mathrm{C}$$ separated by $$5.0\ \mathrm{cm}. $$What is the magnitude of the dipole moment?

$$p = qd = 2.00 \times 10$$^{-7}$$\ \mathrm{C\cdot m}$$

Q: If the dipole is placed in a uniform electric field $$E = 3.0 \times 10^4$\ \mathrm{N/C}$$, making an angle of $$30^\circ$$ with the field, what is the magnitude of the torque on the dipole?

$$\tau = pE\sin\phi = 3.00\ \mathrm{mN\cdot m}$$

Q: What is the electric field at $$r = 0.30\ \mathrm{m}$$ from the center of the shell?

$$E = \dfrac{kQ}{r^2$} = 7.99 \times 10^5$\ \mathrm{N/C}$$

Question 1

A point charge $$q_1 = +4.0$$\ \mu\mathrm{C}$$ is placed at one corner of an equilateral triangle with side length l = 0.40$$\ \mathrm{m}$$. What is the magnitude of the electric field at the location of q_3$ due to q_1$?

Accepted Answer

$$E_1$$ = \dfrac{$$kq_1$$}{$$l^2$$} = 2.25 	imes $$10^5$$\ \mathrm{N/C}$$

Question 2

Given $$q_2 = +2.0$$\ \mu\mathrm{C}$$ at another corner of the triangle, what is the magnitude of the electric field at q_3$ due to q_2$?

Accepted Answer

$$E_2$$ = \dfrac{$$kq_2$$}{$$l^2$$} = 1.12 	imes $$10^5$$\ \mathrm{N/C}$$

Question 3

What is the angle between the electric fields $$E_1$$ and $$E_2 at $$the location of $$q_3 in $$the equilateral triangle?

Accepted Answer

$$120^\circ$$

Question 4

Calculate the magnitude of the net electric field at $$q_3$$ due to $$q_1$$ and $$q_2$$ using the cosine rule.

Accepted Answer

$$E_{\mathrm{net}} = \sqrt{E_1$^2 + E_2$^2 - 2E_1E_2\cos(120^\circ)} = 2.96 	imes 10^5$\ \mathrm{N/C}$$

Question 5

A point charge $$Q = +6.0\ \mu\mathrm{C} is $$fixed in space. What is the electric potential at $$r = 0.50\ \mathrm{m}$$ from the charge?

Accepted Answer

$$V = \dfrac{kQ}{r} = 1.08 	imes 10^5$\ \mathrm{V}$$

Question 6

A proton is released from rest at $$r = 0.50\ \mathrm{m}$$ and moves to $$r = 1.50\ \mathrm{m}$$ from a $$+6.0\ \mu\mathrm{C}$$ charge. What is the change in its electric potential energy?

Accepted Answer

$$\Delta U = kq_pQ\left(\dfrac{1}{r_f} - \dfrac{1}{r_i}ight) = -1.15 	imes 10$$^{-14}$$\ \mathrm{J}$$

Question 7

What is the speed of the proton at $$r = 1.50\ \mathrm{m}$$ after being released from rest at $$r = 0.50\ \mathrm{m}$$ near a $$+6.0\ \mu\mathrm{C}$$ charge?

Accepted Answer

$$v_f = \sqrt{\dfrac{2K_f}{m_p}} = 3.71 	imes 10^6$\ \mathrm{m/s}$$

Question 8

A dipole consists of charges $$+4.0\ \mu\mathrm{C}$$ and $$-4.0\ \mu\mathrm{C}$$ separated by $$5.0\ \mathrm{cm}. $$What is the magnitude of the dipole moment?

Accepted Answer

$$p = qd = 2.00 	imes 10$$^{-7}$$\ \mathrm{C\cdot m}$$

Question 9

If the dipole is placed in a uniform electric field $$E = 3.0 	imes 10^4$\ \mathrm{N/C}$$, making an angle of $$30^\circ$$ with the field, what is the magnitude of the torque on the dipole?

Accepted Answer

$$	au = pE\sin\phi = 3.00\ \mathrm{mN\cdot m}$$

Question 10

How much work must an external agent do to rotate the dipole from $$30^\circ to$$ $$90^\circ in $$the field $$E = 3.0 	imes 10^4$\ \mathrm{N/C}$$?

Accepted Answer

$$W_{\mathrm{ext}} = pE(\cos\phi_i - \cos\phi_f) = 5.20\ \mathrm{mJ}$$

Question 11

A thin spherical shell of radius $$R = 0.20\ \mathrm{m}$$ carries a total charge $$Q = 8.0\ \mu\mathrm{C}$$ uniformly distributed. What is the electric field at $$r = 0.10\ \mathrm{m}$$ from the center?

Accepted Answer

$$E = 0$$

Question 12

What is the electric field at $$r = 0.30\ \mathrm{m}$$ from the center of the shell?

Accepted Answer

$$E = \dfrac{kQ}{r^2$} = 7.99 	imes 10^5$\ \mathrm{N/C}$$

Question 13

What is the electric flux through a spherical Gaussian surface of radius $$0.50\ \mathrm{m}$$ concentric with the shell?

Accepted Answer

$$\Phi_E = \dfrac{q_{\mathrm{encl}}}{\$$epsilon_0$$} = 9.04 	imes 10^5$\ \mathrm{N\cdot m^2$/C}$$

Question 14

Which of the following statements about the electric field inside a uniformly charged thin spherical shell is correct?

Accepted Answer

The electric field is zero everywhere inside the shell.

Question 15

A proton is released from rest near a positive point charge. Which of the following best describes the change in the proton's electric potential energy as it moves away from the charge?

Accepted Answer

The electric potential energy decreases.

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