Two 2.0g spheres are attached to each end of a silk thread 1.20m long. The spheres are given identical charges and the midpoint of the thread is then suspended from a point on the ceiling. The spheres come to rest in equilibrium, with their centers 15cm apart. What is the magnitude of the charge on each sphere?

To find the magnitude of the charge on each sphere, we can use the principle of electrostatic equilibrium. In this situation, the electrostatic force between the two spheres attracts them towards each other and the tension in the silk thread balances this force, resulting in equilibrium.

We can start by calculating the electrostatic force between the spheres using Coulomb's law:

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

Where:
- F is the electrostatic force between the spheres
- k is the electrostatic constant (9.0 x 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges on the spheres
- r is the distance between the centers of the spheres

In this case, the distance between the centers of the spheres is given as 15 cm, equal to 0.15 m.

Next, we need to calculate the tension in the silk thread. Since the spheres are in equilibrium, the tension in the thread must balance the electrostatic force between them.

Using Newton's second law, we know that the net force acting on an object in equilibrium is zero. In this case, the net force is the vector sum of the electrostatic force between the spheres and the tension in the thread.

Since the thread is in tension, the thread pulls on both spheres equally, resulting in two equal and opposite tension forces. These tension forces form a right-angled triangle with the electrostatic force between the spheres.

Using Pythagoras' theorem, we can relate the electrostatic force and the tension force:

F_tension^2 + F_tension^2 = F_electrostatic^2

Simplifying this equation, we get:

2 * F_tension^2 = F_electrostatic^2

Now we can proceed to solve for the magnitude of the charge on each sphere.

1. Calculate the electrostatic force between the spheres:
F_electrostatic = k * (q1 * q2) / r^2

2. Calculate the tension force in the silk thread:
F_tension = F_electrostatic / sqrt(2)

3. Apply Pythagoras' theorem to relate the electrostatic force and the tension force:
2 * F_tension^2 = F_electrostatic^2

Substitute the expressions for F_electrostatic and F_tension:
2 * (F_electrostatic / sqrt(2))^2 = F_electrostatic^2

Simplify the equation:
2 * (F_electrostatic^2 / 2) = F_electrostatic^2

Cancel out the common factors:
F_electrostatic^2 = F_electrostatic^2

This equation is true, which means that the charges on the spheres do not affect each other. Thus, each sphere has the same magnitude of charge.

Therefore, to find the magnitude of the charge on each sphere, you can calculate the electrostatic force using Coulomb's law and then solve for q1 or q2.

Let's assume q is the magnitude of the charge on each sphere. Using the given values:

F_electrostatic = k * (q * q) / r^2

We can rearrange this equation to solve for q:

q^2 = (F_electrostatic * r^2) / k

Substituting the values:
q^2 = (F_electrostatic * 0.15^2) / (9.0 x 10^9)

Now solve for q:

q = sqrt((F_electrostatic * 0.15^2) / (9.0 x 10^9))

Plug in the known values to calculate the magnitude of the charge on each sphere.

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