A small 15.0 g plastic ball is suspended by a 30.0 cm string in a uniform electric field of
3000 N/C, as shown in the picture. If the ball is in equilibrium when the string makes a 45° angle with the vertical as indicated, what is the net charge on the ball?
To determine the net charge on the ball, we need to apply the principles of electrostatics and equilibrium. Here's how we can go about solving this problem:
1. Identify the forces acting on the plastic ball:
- Gravity (mg), acting vertically downward
- Electric force (Fe), acting in the direction of the electric field
- Tension (T) in the string, acting along the string
2. Break down the forces into their components:
- The gravitational force (mg) can be decomposed into two components as follows:
- mg_cosθ acting parallel to the string
- mg_sinθ acting perpendicular to the string, in the opposite direction to the tension
3. Establish equilibrium conditions:
- For the ball to be in equilibrium, the forces in the vertical and horizontal directions must cancel out.
4. Set up the equations:
- In the vertical direction:
- T - mg_cosθ = 0 (Equation 1)
- In the horizontal direction:
- Fe - mg_sinθ = 0 (Equation 2)
5. Solve the equations:
- Rearrange Equation 1 to solve for T:
- T = mg_cosθ
- Rearrange Equation 2 to solve for Fe:
- Fe = mg_sinθ
6. Substituting the given values:
- m = 15.0 g
- g (acceleration due to gravity) = 9.8 m/s^2
- θ = 45°
- T = (0.015 kg)(9.8 m/s^2)cos(45°)
= 0.100 N (approximately)
- Fe = (0.015 kg)(9.8 m/s^2)sin(45°)
= 0.100 N (approximately)
- Since the ball is in equilibrium, the tension and electric force must be equal in magnitude.
7. Calculate the net charge on the ball:
- The electric force between two charges can be expressed as Fe = k*q/Q*r^2, where k is the electrostatic constant, q and Q are the charges, and r is the distance between the charges.
- Rearrange the equation to solve for the charge q:
- q = (Fe * Q * r^2) / (k)
- Substitute the given values:
- Fe = 0.100 N
- Q = ?
- r = 0.3 m (30.0 cm)
- k = 9 x 10^9 N•m^2/C^2 (electrostatic constant)
- q = (0.100 N * Q * (0.3 m)^2) / (9 x 10^9 N•m^2/C^2)
= (0.009 N•m^2/C * Q) / (9 x 10^9 N•m^2/C^2)
= 0.001 Q (approximately)
- Therefore, the net charge on the ball is 0.001 Q (where Q represents the charge).
Thus, to determine the net charge on the plastic ball, we need to know the charge (Q) that corresponds to 0.001 times the magnitude of the electric force. Unfortunately, the charge cannot be determined with the given information.