AP Physics 1 Exam Question

Question: An electron with a mass of 9.11 x 10^-31 kg is placed in an electric field generated by four point charges arranged in a square as shown below:

Q1(+q)                   Q2(-2q)
   |_________d_________|
   |                  |
   |                  |
   |                  |
   |                  |
   |                  |
   Q3(-q)               Q4(+3q)

The charges are located at the corners of a square with side length d = 0.1 meters. Q1 and Q4 have positive charges and Q2 and Q3 have negative charges. Q1 has a charge +q, Q2 has a charge -2q, Q3 has a charge -q, and Q4 has a charge +3q. Determine the electric potential energy (in joules) of the electron at the center of the square.

Answer:

To determine the electric potential energy of the electron at the center of the square, we need to calculate the electric potential generated by each point charge and then sum them up.

The electric potential (V) at a point due to a point charge can be calculated using the formula:

V = k * |q| / r

Where:

• V is the electric potential,
• k is the electrostatic constant which has a value of approximately 8.99 x 10^9 N m^2/C^2,
• |q| is the magnitude of the charge, and
• r is the distance from the charge to the point where the potential is being calculated.

Step 1: Calculate the electric potential due to Q1 (+q) at the center of the square.

The distance from Q1 to the center is d/2 (since it is a square). Using the formula, we have:

V1 = k * |q| / (d/2)
   = (8.99 x 10^9 N m^2/C^2) * |q| / (0.1 m / 2)
   = 1.798 x 10^11 N m^2/C

Step 2: Calculate the electric potential due to Q2 (-2q) at the center of the square.

The distance from Q2 to the center is d/2 (since it is a square). Using the formula, we have:

V2 = k * |-2q| / (d/2)
   = (8.99 x 10^9 N m^2/C^2) * |-2q| / (0.1 m / 2)
   = 3.596 x 10^11 N m^2/C

Step 3: Calculate the electric potential due to Q3 (-q) at the center of the square.

The distance from Q3 to the center is d/2 (since it is a square). Using the formula, we have:

V3 = k * |-q| / (d/2)
   = (8.99 x 10^9 N m^2/C^2) * |-q| / (0.1 m / 2)
   = 8.995 x 10^10 N m^2/C

Step 4: Calculate the electric potential due to Q4 (+3q) at the center of the square.

The distance from Q4 to the center is d/2 (since it is a square). Using the formula, we have:

V4 = k * |3q| / (d/2)
   = (8.99 x 10^9 N m^2/C^2) * |3q| / (0.1 m / 2)
   = 2.6985 x 10^11 N m^2/C

Step 5: Calculate the total electric potential energy at the center of the square by summing up the individual electric potentials:

V_total = V1 + V2 + V3 + V4
        = 1.798 x 10^11 N m^2/C + 3.596 x 10^11 N m^2/C + 8.995 x 10^10 N m^2/C + 2.6985 x 10^11 N m^2/C
        = 8.0925 x 10^11 N m^2/C

Therefore, the electric potential energy of the electron at the center of the square is 8.0925 x 10^11 joules.