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.
well, the angle on each string is arcsin5/13
lets call that angle theta.
so tanTheta=kqq/(.1^2*mg)
so figure q
To determine the charge on each ball, we can use Coulomb's law and the equation for the gravitational force.
First, let's find the charge on each ball.
Coulomb's law states that the electrostatic force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them:
F = k * (q1 * q2) / r^2
where:
- F is the electrostatic force between the two charged balls
- q1 and q2 are the charges on the balls
- r is the distance between the centers of the balls
- k is the electrostatic constant, approximately 9 × 10^9 N m^2/C^2
In this case, the electrostatic force between the balls causes them to come to rest at a distance of 10 cm (0.10 m) apart.
To find the charge on each ball, we can equate this electrostatic force with the gravitational force acting on each ball.
The gravitational force between the two balls is given by:
Fg = (G * m^2) / r^2
where:
- Fg is the gravitational force
- G is the gravitational constant, approximately 6.67 × 10^-11 N m^2/kg^2
- m is the mass of each ball
- r is the distance between the centers of the balls
In this case, the balls are suspended by threads, which means the net force on each ball must be zero. Therefore, the electrostatic force and the gravitational force are equal in magnitude.
Setting the electrostatic force equal to the gravitational force, we have:
k * (q1 * q2) / r^2 = (G * m^2) / r^2
Simplifying, we get:
q1 * q2 = (G * m^2) / k
Substituting the given values:
- m = 0.10 gm = 0.10 × 10^-3 kg
- r = 0.13 m
- G = 6.67 × 10^-11 N m^2/kg^2
- k = 9 × 10^9 N m^2/C^2
We can now solve for q1 * q2:
(q1 * q2) = (G * m^2) / k
(q1 * q2) = (6.67 × 10^-11 N m^2/kg^2) * (0.10 × 10^-3 kg)^2 / (9 × 10^9 N m^2/C^2)
Calculating this gives us the product of the charges:
(q1 * q2) ≈ 7.41 × 10^-16 C^2
To find the individual charges, q1 and q2, we can assume that they are equal in magnitude, so:
(q1 * q2) = q1^2
Therefore:
q1^2 ≈ 7.41 × 10^-16 C^2
To find q1 and q2, we can take the square root of both sides:
q1 ≈ sqrt(7.41 × 10^-16 C^2)
q2 ≈ sqrt(7.41 × 10^-16 C^2)
Calculating this, we get:
q1 ≈ 8.62 × 10^-8 C
q2 ≈ 8.62 × 10^-8 C
Therefore, the charge on each ball is approximately 8.62 × 10^-8 C.