AP Physics C

Coulomb’s Law

Free Response Problems

1. Four equal and positive charges +q are arranged as shown on figure 1.

a. Calculate the net electric field at the center of square.

b. Calculate the net electric force on the positive charge Q placed at the

center of square.

Two positive charges are replaced with equal negative charges, as shown in

figure 2.

c. Calculate the net electric field at the center of square.

d. Calculate the net electric force on the positive charge Q placed at the

center of the square.

2. Two positive charges of equal magnitude Q are placed at the corners of equilateral triangle with a side length r, as shown above.

a. What is the magnitude and direction of the electric field at point C due to charge Q located at point A?

b. What is the magnitude and direction of the electric field at point C due to charge Q located at point B?

c. What is the magnitude and direction of the net electric field at point C due to two charges?

d. What is the net electric force on a test charge q placed at point C?

3. Two charges, one is positive and the other is negative, of equal magnitude Q are placed at the corners of equilateral triangle with a side length r, as shown above.

a. What is the magnitude and direction of the electric field at point C due to charge Q located at point A?

b. What is the magnitude and direction of the electric field at point C due to charge Q located at point B?

c. What is the magnitude and direction of the net electric field at point C due to two charges?

d. What is the net electric force on a test charge q placed at point C?

4. A parallel-plate capacitor is connected to a battery with a constant voltage

of 120 V. Each plate has a length of 0.1 m and they are separated by a

distance of 0.05 m. An electron with an initial velocity of 2.9*107 m/s is

moving horizontally and enters the space between the plates. Ignore

gravitation.

a. What is the direction of the electric field between the plates?

b. Calculate the magnitude of the electric field between the plates.

c. Describe the electron’s path when it moves between the plates.

d. What is the direction and magnitude of its acceleration?

e. Will the electron leave the space between the plates?

5. In an oil-drop experiment, two parallel conducting plates are connected to

a power supply with a constant voltage of 100 V. The separation between

the plates is 0.01 m. A 4.8*10-16 kg oil drop is suspended in the region

between the plates. Use g = 10 m/s2.

a. What is the direction of the electric field between the plates?

b. What is the magnitude of the electric field between the plates?

c. What is the sign and magnitude of the electric charge on the oil

drop when it stays stationary?

The mass of the drop is reduced to 3.2*10-16 kg because of vaporization.

d. What is the acceleration of the drop?

6. Two small spheres with masses m = 20.0 g are suspended at the ends of

two silk strings of length L = 1.5 m. Both spheres are charged with an equal

amount of positive charge q. Each string makes an angle θ = 30 ̊ with the

vertical.

a. On the diagram below, draw free-body diagram showing all

the applied forces on each sphere.

b. Determine the magnitude of charge q on each sphere.

c. Due to electric discharge, the strings reduce the angle from the

vertical to θ = 20 ̊. Determine the amount of charge that left

each sphere due to the electric discharge.

d. If one of the strings is cut, find the instantaneous acceleration

of the sphere that was hung from this string.

7. A nonconducting ring of radius R lies in the yz – plane. The ring is charged

with a positive charge Q that is uniformly distributed on the ring.

a. Find the x-component of the electric field along the x-axis.

b. What are the y- and z- components of the electric field along x-

axis?

c. Find the distance x for which the electric field is a maximum.

d. Find the maximum value of the electric field.

e. On the axes below, graph the x-component of the electric field

as a function of distance x.

8. A uniformly charged disk has a radius R = 0.5 m and carries a total charge Q

= 16×10-9 C.

a. What is the surface charge density on the disk?

b. What is the magnitude and direction of the electric field at

point P which lies on the x-axis at a distance x = 0.4 m from the

disk?

c. On the axes below, graph the electric field as a function of

distance x.

d. If instead of the disk you have an infinite sheet with the same

surface charge density, is the electric field at point P the same,

less, or greater that the electric field of the disk?

9. A positive charge Q is uniformly distributed along a thin rod of length L. The

left end of the rod is placed at a distance r from the origin 0.

a. What is the magnitude and direction of the electric field at the

origin due to the charged rod?

b. What is the electric force applied by the rod on a positive

charge q placed at the origin?

c. If charge q is released from the origin, describe its motion.

10. A positive charge is distributed uniformly around a semicircle of radius R =

0.2 m. The linear charge density is λ = 4.0×10-6 C/m.

a. Determine the total charge on the semicircle.

b. Determine the magnitude and direction of the electric field at

the center O of the semicircle.

A test charge q = 3 ×10-9 C is placed at the center O of the

semicircle.

c. Determine the magnitude and direction of the electric force

applied by the semicircle on the test charge q.

d. If the test charge is released at point O, describe its motion for

a long time after the release.

11. A thin rod of length 2a is charged uniformly with a positive charge. The

linear charge density is λ. Point P is a distance y meters above the rod.

a. Find the net charge Q on the rod

b. Find the expression for the electric field due to the charged rod

at point P on the perpendicular bisector of the rod.

An oil drop with a mass of m is placed at point P. The oil drop

then begins to stay in place.

c. Determine the expression for the charge of the oil drop that is

required to keep the drop suspended in the electric field due

to the rod.

d. The oil drop loses some mass because of evaporation. Describe

its motion in the electric field due to the charged rod.

Free response answers.

1. a. 0𝑁

𝐶

b. 0 𝑁

c. 4√2𝑘𝑞

𝑑2

d. 4√2𝑘𝑞2

𝑑2

2. a. 𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑜𝑛𝑔 𝑎𝑛 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑙𝑖𝑛𝑒 𝐴𝐶 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐶

b. 𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑜𝑛𝑔 𝑎𝑛 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑙𝑖𝑛𝑒 𝐵𝐶 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐶

c. 2√2𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑢𝑝

d. 2√2𝑘𝑄𝑞

𝑟2

3. a. 𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑜𝑛𝑔 𝑎𝑛 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑙𝑖𝑛𝑒 𝐴𝐶 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐶

b. 𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑡𝑜𝑤𝑎𝑟𝑑𝑠 𝐵 𝑏𝑒𝑔𝑖𝑛𝑖𝑛𝑔 𝑎𝑡 𝐶 𝑎𝑙𝑜𝑛𝑔 𝑙𝑖𝑛𝑒 𝐶𝐵

c. 2√2𝑘𝑄

𝑟2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑟𝑖𝑔ℎ𝑡

d. 2√2𝑘𝑄𝑞

𝑟2

4. a. 𝑈𝑝

b. 2400𝑁

𝐶

c. 𝐼𝑡 𝑤𝑖𝑙𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑒 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑𝑠 𝑡𝑜𝑤𝑎𝑟𝑑𝑠 𝑡ℎ𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑝𝑙𝑎𝑐𝑒.

d. 4.22 ∗ 1014 𝑚

𝑠2 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑑𝑜𝑤𝑛

e. 𝑌𝑒𝑠

FT

mg mg

FE FE

5. a. 𝐷𝑜𝑤𝑛

b. 10,000𝑁

𝐶

c. 4.8 ∗ 10−19 𝐶 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒

d. 5𝑚

𝑠

6. a.

b. 5.36 𝜇𝐶

c. 2.48 𝜇𝐶

d. 10.63𝑚

𝑠2

7. a. 𝑅𝑄

4𝜋𝜀𝑜(𝑅2+𝑥2)32

b. 0 𝑎𝑛𝑑 0

c. 𝑅

√2

d. 𝑄

6√3𝑅2𝜋𝜀

e.

FT

8. a. 2.03 ∗ 10−8 𝐶

𝑚2

b. 𝜎

2𝜀𝑜(1 −

𝑥

√𝑅2+𝑥2)

c.

d. 𝐼𝑡 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟

9. a. λ

4πεo∗ (

1

𝑟−

1

𝑟+𝐿)

b. λq

4πεo∗ (

1

𝑟−

1

𝑟+𝐿)

c. 𝑇ℎ𝑒 𝑐ℎ𝑎𝑟𝑔𝑒 𝑤𝑖𝑙𝑙 𝑚𝑜𝑣𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑤𝑖𝑡ℎ 𝑎𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑛𝑑 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔

𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛.

10. a. 𝑄 = 4.0 ∗ 10−6 ∗ 𝜋𝑟

b. 359509.70𝑁

𝐶

c. . 001 𝑁 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑟𝑖𝑔ℎ𝑡

d. 𝑇ℎ𝑒 𝑡𝑒𝑠𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 𝑚𝑜𝑣𝑒𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑤𝑖𝑡ℎ 𝑎𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑛𝑑 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔

𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

11. a. 𝑄 = λL

b. λa

2πεo𝑦√𝑎2+𝑦2

c. 𝑄 =mg∗2πεo𝑦√𝑎2+𝑦2

λa

d. 𝑇ℎ𝑒 𝑜𝑖𝑙 𝑤𝑖𝑙𝑙 𝑚𝑜𝑣𝑒 𝑢𝑝 𝑤𝑖𝑡ℎ 𝑎𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑛𝑑 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛