Electric Charge and Field - AP Physics 2 Exam Question

A small metallic sphere with a charge of +2 μC experiences an electric force of 0.5 N. The sphere is in a region of space where there is a uniform electric field of magnitude 1.2 x 10^4 N/C. Determine the following:

a) The direction of the electric force on the charge. b) The mass of the sphere if it experiences an acceleration of 2 m/s² when released from rest. c) The net force acting on the sphere if an external electric field of magnitude 1.5 x 10^4 N/C is applied in the opposite direction. d) The electric potential energy of the sphere in its initial position.

Provide your answers with detailed explanations and calculations.

Answer:

a) The direction of the electric force on the charge can be determined by the direction of the electric field. Since the charge is positive (+2 μC), it experiences a force in the same direction as the electric field. Hence, the electric force on the charge is in the direction of the electric field.

b) To find the mass of the sphere, we can use Newton's second law, which relates force, mass, and acceleration. The formula is:

F = ma

Given: Force (F) = 0.5 N, Acceleration (a) = 2 m/s²

Rearranging the formula, we have:

m = F / a

Substituting the given values:

m = 0.5 N / 2 m/s² = 0.25 kg

Therefore, the mass of the sphere is 0.25 kg.

c) When an external electric field is applied in the opposite direction, the net force on the sphere will be the vector sum of the forces due to the external field and the initial field.

Given: Initial field strength = 1.2 x 10^4 N/C External field strength = 1.5 x 10^4 N/C (opposite direction)

The net force on the sphere can be calculated using the formula:

F_net = qE_initial + qE_external

where q is the charge on the sphere, E_initial is the initial electric field, and E_external is the external electric field.

Substituting the given values, we have:

F_net = (2 μC) (1.2 x 10^4 N/C) + (2 μC) (-1.5 x 10^4 N/C) = (2 x 10^(-6) C) (1.2 x 10^4 N/C) + (2 x 10^(-6) C) (-1.5 x 10^4 N/C) = 2.4 x 10^(-2) N - 3.0 x 10^(-2) N = -0.6 x 10^(-2) N

Therefore, the net force on the sphere when the external electric field is applied in the opposite direction is -0.6 x 10^(-2) N.

d) The electric potential energy (U) of the sphere can be calculated using the formula:

U = qV

where q is the charge on the sphere, and V is the electric potential at its initial position.

The electric potential can be determined using the formula:

V = E_initial * d

where E_initial is the initial electric field and d is the distance from the initial position.

Given: Charge (q) = +2 μC, Electric field (E_initial) = 1.2 x 10^4 N/C

The distance (d) is not given, but let's assume it is 1 meter for simplicity.

Substituting the given values, we have:

V = (1.2 x 10^4 N/C) * 1 m = 1.2 x 10^4 V

Now, calculating the electric potential energy:

U = (2 μC) * (1.2 x 10^4 V) = (2 x 10^(-6) C) * (1.2 x 10^4 V) = 2.4 x 10^(-2) J

Therefore, the electric potential energy of the sphere in its initial position is 2.4 x 10^(-2) J.

Note: The direction of the force, as determined in part (a), is essential while calculating the potential energy based on the charge's sign.