A sphere has a surface uniformly charged with 1.00 coulombs. At what distance from its centre is the potential 5.00 megavolts?
We can call the given charge spread over the sphere 1.00 coulombs 𝑄. And the potential of 5.00 megavolts, we’ll call 𝑉. We want to solve for the distance from the centre of the sphere, at which its potential is equal to 𝑉. We’ll call that distance 𝑑. Let’s start with a sketch of the sphere.
We have a sphere with charge 𝑄 spread evenly over its surface. We don’t know the sphere’s radius. But we do know that at some point external to the sphere, at distance 𝑑 from its centre, the potential is 𝑉, given as 5.00 megavolts.
To solve for that distance 𝑑, we can consider the mathematical relationship for the electric potential outside of a uniformly charged sphere. That potential 𝑉 is equal to the charge 𝑄 spread over the surface of the sphere divided by four 𝜋𝜀 nought, where 𝜀 nought is the permittivity of free space times the distance from the centre of the sphere 𝑟.
For our purposes, we’ll treat 𝜀 nought as having an exact value of 8.85 times 10 to the negative 12th farads per metre. So the electric potential 𝑉 at distance 𝑑 away from the centre of the sphere equals the charge spread over the sphere over four 𝜋𝜀 nought 𝑑.
Rearranging this equation to solve for 𝑑, we find it’s 𝑄 over four 𝜋𝜀 nought 𝑉. For each of these three variables, we have values. So we’re now ready to plug in and solve for 𝑑. When we do, we’re careful to use units of volts in our expression for the potential difference.
Entering these values on our calculator, we find that to three significant figures 𝑑 is 1.80 kilometres. That’s the distance from the centre of the sphere, at which 𝑉 equals 5.00 megavolts.