Final answer:
Using Coulomb's Law and the equilibrium of forces, we can relate the forces due to gravity and electrostatic charge to solve for the magnitude of the charge on each of the two suspended spheres.
Step-by-step explanation:
To find the charge on each sphere, we use Coulomb's Law and the equilibrium condition. Coulomb's Law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In an equilibrium condition, this electrical force is balanced by the tension in the strings causing a horizontal component that provides the centripetal force required to keep the spheres in a stable equilibrium.
The force due to gravity on each sphere (which is the weight) will provide the vertical component of tension. We can set up a right triangle where the opposite side is half the distance between the spheres and the adjacent side is the length of the string minus the distance by which the sphere has dropped in equilibrium due to the horizontal repulsion. Since the drop would be small compared to the length of the strings, we can approximate the length of the hypotenuse as the length of the string.
Let's denote the charge on each sphere as 'q'. The formula for Coulomb's Law is F = k * |q1*q2| / r², where k is Coulomb's constant (8.99 x 10^9 N m²/C²), and r is the distance between charges. At equilibrium, the horizontal component of tension (T) in the string must equal the electrostatic force.
Therefore, using trigonometry and the equilibrium of forces, we can set up the equations: T cos(θ) = mg and T sin(θ) = k * q² / r², where θ is the angle the string makes with the vertical, 'm' is the mass of the sphere, and 'g' is the acceleration due to gravity (9.8 m/s²). From these equations, we can solve for the unknown charge 'q' on each sphere.