A small plastic ball of mass m = 1.50 g is suspended by a string of length L = 15.5 cm in a uniform electric field, as shown in the figure below. If the ball is in equilibrium when the string makes a θ = 19.0° angle with the vertical as indicated, what is the net charge on the ball?
To find the net charge on the ball, we can use the concept of electrostatic equilibrium. In electrostatic equilibrium, the sum of the forces acting on an object is zero.
In this case, the force acting on the ball is the tension in the string, which is balanced by the electric force exerted on the ball. We can break down these forces and determine their magnitudes:
- Tension in the string: T = mgcosθ
- Electric force: Fe = qEsinθ
Where:
- T is the tension in the string
- m is the mass of the ball
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- θ is the angle the string makes with the vertical
- Fe is the electric force
- q is the net charge on the ball
- E is the magnitude of the electric field
Since the ball is in equilibrium, the tension in the string is equal to the electric force:
T = Fe
Substituting the expressions for T and Fe, we get:
mgcosθ = qEsinθ
We can rearrange this equation to solve for the net charge q:
q = (mgcosθ) / (Esinθ)
Now we can plug in the known values and solve for q.
Given:
m = 1.50 g = 0.0015 kg
L = 15.5 cm = 0.155 m
θ = 19.0°
First, let's calculate the electric field E.
The electric field E can be determined using the equation:
E = V / L
To find the electric potential difference V, we need more information. If you provide the voltage across the region, we can calculate the electric field E and then find the net charge q using the equation mentioned earlier.