Two 25.0 g spheres are hanging from lightweight strings that are each 35.0 cm in length. Each has the same charge.

They repel each other and make an angle of 5.00 to the vertical. What is the
magnitude of the charge on each sphere?

To find the magnitude of the charge on each sphere, we can use Coulomb's Law and the concept of electrical forces.

Coulomb's Law states that the magnitude of the electrical force between two charged objects is given by the formula:

F = k * (q1 * q2) / r^2

Where:
- F is the electrical force between the two objects,
- k is a constant (usually written as 8.99 x 10^9 Nm^2/C^2) known as the Coulomb's constant,
- q1 and q2 are the charges on the objects, and
- r is the distance between the centers of the objects.

In this case, both spheres are hanging from lightweight strings and they repel each other, making an angle of 5.00 degrees to the vertical. Since the strings are lightweight, the gravitational force can be neglected, and we can consider only the electrical force.

The electrical force between the spheres causes them to repel, resulting in an equilibrium position where the angle is 5.00 degrees to the vertical. This means that the electrical force is equal to the component of the weight of each sphere along the line connecting the two spheres.

To start, let's calculate the weight of each sphere using the formula:

Weight = mass * gravity

Given that the mass of each sphere is 25.0 g and the acceleration due to gravity is approximately 9.8 m/s^2, we can convert the mass from grams to kilograms and calculate the weight:

Weight = (25.0 g) * (9.8 m/s^2)
Weight = 0.025 kg * 9.8 m/s^2
Weight = 0.245 N

Next, let's calculate the component of the weight of each sphere along the line connecting the two spheres. We can use the following formula:

Component of weight = Weight * sin(angle)

Given that the angle is 5.00 degrees, we can calculate the component of the weight:

Component of weight = 0.245 N * sin(5.00 degrees)
Component of weight ≈ 0.021 N

Since the electrical force between the spheres is equal to the component of the weight, we can set up the following equation:

F = 0.021 N = k * (q1 * q2) / r^2

The distance between the centers of the spheres is given as 35.0 cm, which we convert to meters:

r = 35.0 cm = 0.35 m

Now, we can rearrange the equation to solve for the product of the charges:

(q1 * q2) = (F * r^2) / k
(q1 * q2) = (0.021 N) * (0.35 m)^2 / (8.99 x 10^9 Nm^2/C^2)

Inserting the values and evaluating the expression, we get:

(q1 * q2) ≈ 0.0172 C^2

Since both spheres have the same charge, we can express this as:

q^2 = 0.0172 C^2

Now, we can solve for the magnitude of the charge (q) by taking the square root of both sides:

q = √(0.0172 C^2)
q ≈ 0.131 C

Therefore, the magnitude of the charge on each sphere is approximately 0.131 C.

To find the magnitude of the charge on each sphere, we can use Coulomb's Law. Coulomb's Law states that the force of repulsion between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

First, let's calculate the force of repulsion between the two spheres. We can use the equation:

F = k(q₁q₂)/r²

where:
F is the force of repulsion,
k is Coulomb's constant (k = 8.99 x 10^9 Nm²/C²),
q₁ and q₂ are the charges on the spheres,
and r is the distance between the centers of the spheres.

Since the spheres are hanging and making an angle of 5.00 degrees to the vertical, we need to consider the component of gravitational force that acts in the direction of repulsion between the spheres. This component can be calculated as:

Fg = mg * sinθ

where:
Fg is the component of gravitational force,
m is the mass of each sphere (25.0 g = 0.025 kg),
g is the acceleration due to gravity (9.8 m/s²),
and θ is the angle between the string and the vertical (in this case, 5.00 degrees).

Now, we know that the force of repulsion is equal to the component of gravitational force acting in the repulsion direction:

F = Fg

Substituting the equations for F and Fg:

k(q₁q₂)/r² = mg * sinθ

Simplifying and solving for q₁q₂:

q₁q₂ = (mgr² * sinθ) / k

Since both spheres have the same charge, we can assume q₁ = q₂ = q. Therefore:

q² = (mgr² * sinθ) / k

Finally, solving for q:

q = √((mgr² * sinθ) / k)