Two identical small charged spheres, each having a mass of 3.00*10^-2 kg hang in equilibrium . The length L of each string is 0.150m, and the angle between them is 5. Find the magnitude of the charge on each sphere
I got
4.42x10^8 C
To find the magnitude of the charge on each sphere, we can use the principles of electrostatics and equilibrium.
First, we need to determine the forces acting on each sphere.
Each sphere experiences two forces: the gravitational force and the electrostatic force.
1. Gravitational force: The force due to gravity acting on each sphere can be calculated using the formula:
F_gravity = m * g
where m is the mass of the sphere, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (3.00 * 10^-2 kg) * (9.8 m/s^2)
F_gravity = 2.94 * 10^-1 N
2. Electrostatic force: The electrostatic force between two charged spheres can be calculated using Coulomb's law:
F_electrostatic = k * (q^2 / r^2)
where k is the electrostatic constant (approximately 9.0 * 10^9 N m^2/C^2), q is the charge on each sphere, and r is the distance between the centers of the spheres.
As the spheres are identical and hanging in equilibrium, the electrostatic forces acting on both spheres are equal in magnitude and opposite in direction (they cancel out).
Now, using the equilibrium condition, we can equate the forces to find the value of q.
Since the spheres are in equilibrium, the net force in the horizontal direction must be zero. Therefore, the electrostatic forces acting on each sphere must balance the component of gravitational force in the horizontal direction.
The horizontal component of the gravitational force can be calculated using trigonometry:
F_gravity_horizontal = F_gravity * sin(angle)
where the angle is given as 5 degrees in the problem.
F_gravity_horizontal = (2.94 * 10^-1 N) * sin(5 degrees)
F_gravity_horizontal = 2.41 * 10^-2 N
The electrostatic forces must balance this value. Since both spheres have the same charge, the electrostatic force on each sphere is F_electrostatic/2.
Setting up the equation:
F_electrostatic/2 = F_gravity_horizontal
k * (q^2 / r^2) / 2 = 2.41 * 10^-2 N
Now, let's solve for the charge, q.
q^2 = (2.41 * 10^-2 N) * (2 * r^2 / k)
q^2 = (2.41 * 10^-2 N) * (2 * (0.150 m)^2 / (9.0 * 10^9 N m^2/C^2))
q^2 ≈ 8.6 * 10^-10 C^2
Taking the square root of both sides:
q ≈ √(8.6 * 10^-10 C^2)
q ≈ 9.29 * 10^-5 C
Therefore, the magnitude of the charge on each sphere is approximately 9.29 * 10^-5 C.