Two small spheres, each with mass m=3g and charge q, are suspended from a point by threads of length L=0.22m. What is the charge on each sphere if the threads make an angle theta of 15 degrees with the vertical?

To find the charge on each sphere, we can make use of the properties of electric forces and the force of gravity.

Let's start by analyzing the forces acting on each sphere. Since the spheres are suspended and are in equilibrium, the forces acting on them must balance out.

1. Electric force:
The electric force between the two spheres can be calculated using Coulomb's law:
F_electric = k * (q^2) / r^2
where k is the electrostatic constant, q is the charge on each sphere, and r is the distance between them.
Note that this force is attractive if the spheres have opposite charges (different signs) and repulsive if they have the same charge (same signs).

2. Force of gravity:
The force of gravity acting on each sphere can be calculated using the formula:
F_gravity = m * g
where m is the mass of each sphere (3g) and g is the acceleration due to gravity.

Now, let's consider the forces acting on one of the spheres:
1. Vertical force component:
This component opposes the force of gravity and is given by F_vertical = F_gravity * cos(theta).

2. Tension force component:
This component provides the necessary force to keep the sphere in circular motion and is given by F_tension = Tension * sin(theta).
Since the sphere is in equilibrium, these two forces must balance each other:
F_vertical = F_tension

Now, let's plug in the appropriate values and solve for the charge on each sphere.

1. Vertical force component:
F_vertical = F_gravity * cos(theta)
F_vertical = (m * g) * cos(theta)
F_vertical = (0.003kg * 9.8m/s^2) * cos(15 degrees)

2. Tension force component:
F_tension = Tension * sin(theta)
F_tension = Tension * sin(15 degrees)

Since the tension force Tension is the same for both spheres and we are assuming the threads have negligible mass, we can equate the two forces:
F_vertical = F_tension
(m * g) * cos(theta) = Tension * sin(theta)

Now, both the tension and the mass of the spheres can be canceled out since they are the same for both spheres, leading to the following equation:
g * cos(theta) = sin(theta)

Now we can solve for theta:
tan(theta) = g
theta = arctan(g)

Given that g is approximately 9.8 m/s^2, we can calculate theta:
theta = arctan(9.8)
theta ≈ 82.82 degrees

By comparing this value with the given angle theta (15 degrees), we see that the vertical force component dominates. Therefore, we can neglect the tension force component.

Now, we can solve for the charge on each sphere by considering the electric force between them:
F_electric = k * (q^2) / r^2

Since the spheres have the same charge (q) and the electric force is attractive, we have:
F_electric = k * (q^2) / r^2 = F_vertical

Now, we can rearrange the equation to solve for q:
q^2 = (F_vertical * r^2) / k
q^2 = [(m * g * cos(theta)) * r^2] / k
q^2 = [(0.003kg * 9.8m/s^2 * cos(15 degrees)) * (0.22m)^2] / (9 * 10^9 N m^2 / C^2)

Finally, we can solve for q:
q^2 ≈ 2.352 * 10^-11 C^2
q ≈ ± 4.85 * 10^-6 C

Therefore, the charge on each sphere is approximately ± 4.85 * 10^-6 C.