Question

Two equally charged insulating balls each weigh 0.18 g and hang from a common point by identical threads 45 cm long. The balls repel each other so that the separation between their centers is 6.4 cm. What is the magnitude of the charge on each ball?

Answer 1

The repulsive force between them is proportional to their charge, so first you have to find how much force it takes to repel two balls so that they hang 6.4 cm apart.

We'll start by finding the weight of each ball. The mass is 0.18 g (or 1.8x10^-4 kg in mks units), so the weight is given by:

Fw = mg = (1.8x10^-4 kg)(9.81 m/s²) = 1.77x10^-3 N

The weight of each ball is analogous to the vertical component of the tension in each string. The tension in each string is a force directed along the length of the string and is the hypotenuse of a right triangle. In order to solve this problem we have to know the horizontal component of the tension, and that means we'll have to use trig.

The next thing we're going to do is determine the angle each string makes with an imaginary vertical line. That's easy enough to do using trig functions. We know that each string is 45 cm long and forms the hypotenuse of a right triangle which also has a horizontal leg of 3.2 cm (half of the distance between the balls). It follows that the angle between each string and the imaginary vertical can be found by using the inverse sine function:

sin(θ) = (3.2 cm) / (45 cm) = 0.0711

sin-1(0.0711) = 4.08º

Now that we know the angle we can return to the forces. The vertical component of the tension is 1.77x10^-3 N. What is the horizontal component?

tan(4.08º) = horiz.comp. / 1.77x10^-3 N

horizontal component = 1.26^-4 N

Each ball is being repelled by a force of 1.26x10^-4 N, which is being exerted by the other ball. The repulsive force between the balls is 1.26x10^-4 N.

Now that we know the force we can use Coulomb's Law to calculate the charge on the balls:

F = k Q1 Q2 / r²

1.26x10^-4 N = (8.988x10^9 Nm²/C²) (Q1) (Q2) / (0.064 m)²

Given that Q1 = Q2, we can say that (Q1) (Q2) = (Q1)²

1.26x10^-4 N = (8.988x10^9 Nm²/C²) (Q1)² / (0.064 m)²

Q1 = Q2 = 7.58x10^-9 C, or 7.58 nC