A small spherical pith ball of radius 0.36 centimeters is painted with a silver paint and then negative 8.0 microcoulombs of charge is applied to it. The charged pith ball is put at the center of a gold spherical shell of radius 3.0 centimeters and outer radius 3.3 centimeters. Find the electric potential at the surface of the gold shell with respect to zero potential at infinity.
Considering the situation, our first question might be what exactly is a pith ball? A pith ball is a small, lightweight sphere, typically coated with a conducting material. Because of its conducting outer shell, a pith ball can be given a net charge or the charge on its conducting surface can be polarized due to the influence of a nearby charge.
In our scenario, we have negative 8.0 microcoulombs of charge applied to the pith ball. Since the ball is covered in silver paint, which is a conductor, that means that charge distributes evenly over the surface of the pith ball. It’s not all bunched in one place, but is distributed uniformly across the pith ball surface.
This charged pith ball is then put at the center of the spherical shell made of gold, another conductor. We’re told what the inner and outer radii of that shell are. We’ll call the outer radius 𝑟 sub 𝑜. And we want to solve for the electric potential at the surface of the gold shell.
We know that when it comes to electric potential, that value is always measured in reference to some other point. That is, we can never say what the electric potential is at a given point without reference to some other point; it’s not an absolute measure.
We’re told to imagine that if we go infinitely far away from our spherical shell, say this is infinitely far away, then the potential at that point is zero. So that’s our reference; in comparison to which, we’ll solve for the potential at the surface of this gold shell.
Taking another look at our gold shell and our pith ball, we see that we want to solve for the potential at a location outside of the pith ball radius. That’s a very helpful bit of information because the pith ball we know has a uniformly distributed charge across its surface. Because we want to calculate the effect of all that charge at a point outside the pith ball, that means we can model the charge as though it were all concentrated at the center of the pith ball.
In other words, we’ll pretend that there’s a point charge of negative 8.0 microcoulombs at the center of the pith ball. And it’s that charge that helps to create an electric potential at the surface of the gold shell. The reasons we can make this assumption are number one: we have a uniform distribution of charge across the pith ball and number two: the location where we’re solving for the electric potential is outside that spherical distribution.
Knowing all this then, we want to solve for the electric potential created by a point charge some distance away from that charge, where our reference point is that potential is zero at infinity. The mathematical relationship for this is that 𝑉, the electric potential, is equal to 𝑘, Coulomb’s constant, times the charge 𝑄 all divided by the distance 𝑟 between the charge and the point at which we want to compute potential.
Recall that Coulomb’s constant can be approximated as 8.99 times 10 to the ninth newton meter squared per coulomb squared. Applying this relationship to our particulars, we know our charge is negative 8.0 times 10 to the negative sixth coulombs; that’s the charge on the pith ball.
And our radius value we’ll use is the distance between the center of the pith ball and the outer part of the gold spherical shell. We’re given that distance in centimeters, but we want to convert it to units of meters. We know that 3.3 centimeters convert to 0.33 meters. And now, lastly, we’ll insert the value for the constant 𝑘.
We’re now all ready to calculate this potential. And notice something interesting: we’ll end up with a negative value. That makes sense because remember that our reference point at infinity is that potential of zero. And we have a negative charge distributed over this pith ball.
When we crunched the numbers, we find an answer to two significant figures of negative 2.2 million volts. That’s the electric potential at the surface of the gold shell with respect to zero potential at infinity.