2. Two equally charged balls, each of mass 0.10 gm, are suspended from the same point by threads 13 cm long. The balls come to rest 10 cm apart due to electrostatic repulsion. Determine the charge on each ball.
To determine the charge on each ball, we can use Coulomb's Law which states that the electrostatic force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
First, let's find the electrostatic force between the two balls:
1. Convert the mass of each ball from grams to kilograms:
- Mass of each ball = 0.10 gm = 0.10 × 10^(-3) kg = 1 × 10^(-4) kg
2. Calculate the gravitational force acting on each ball:
- Gravitational force = mass × gravitational acceleration
- Gravitational acceleration on Earth ≈ 9.8 m/s^2
- Gravitational force on each ball = (1 × 10^(-4) kg) × (9.8 m/s^2) = 9.8 × 10^(-5) N
3. Since the balls are at rest, the electrostatic force between them is equal in magnitude to the gravitational force acting on each ball. Therefore, the electrostatic force is 9.8 × 10^(-5) N.
4. The distance between the two balls is given as 10 cm. Convert this to meters:
- Distance = 10 cm = 10 × 10^(-2) m = 0.1 m
5. Apply Coulomb's Law to find the charges on the balls:
- Coulomb's Law: electrostatic force = (1 / 4πε₀) × (q1 × q2) / r^2
(where q1 and q2 are the charges, r is the distance between the balls, and ε₀ is the permittivity of free space)
- Rearranging the equation:
(q1 × q2) = (4πε₀ × electrostatic force × r^2)
6. Let's calculate the charge on each ball:
- We can assume the charges on both balls are the same since they repel each other.
(q × q) = (4πε₀ × 9.8 × 10^(-5) N × (0.1 m)^2)
- ε₀ is the permittivity of free space and its value is approximately 8.854 × 10^(-12) C^2/N·m^2.
- Substitute the known values:
(q × q) = (4π × 8.854 × 10^(-12) C^2/N·m^2 × 9.8 × 10^(-5) N × (0.1 m)^2)
- Simplify to find the charge on each ball:
q = √((4π × 8.854 × 10^(-12) C^2/N·m^2 × 9.8 × 10^(-5) N × (0.1 m)^2))
Calculating this expression will give the charge on each ball.