A charge of positive 3.0 microcoulombs is put on the surface of a hollow aluminum spherical ball which has a radius of 5.0 centimeters. The charge is in equilibrium at the outer surface of the ball. Determine the magnitude of the electric field produced by this charge at a point 1.0 centimeters from the center of the aluminum ball. Determine the magnitude of the electric field produced by this charge at a point 10 centimeters from the center of the aluminum ball.
We’re told the magnitude of charge spread over the surface of the sphere 3.0 microcoulombs and also that the sphere has a radius of 5.0 centimeters. We want to solve for the magnitude of the electric field at two locations: one, at point 1.0 centimeters from the center of the ball and secondly, at a point 10 centimeters from that center. We’ll call these values 𝐸 of 1.0 centimeter and 𝐸 of 10 centimeters, respectively.
Let’s start out with a sketch of the ball. We have a metal sphere whose radius, which we can call 𝑟, is 5.0 centimeters. Spread out evenly on the surface of the sphere, is an amount of charge we can call 𝑄, where 𝑄 is positive 3.0 microcoulombs. We want to know the electric field created by this ball at two locations. The first location is one centimeter away from the center of the ball. That’s 𝐸 of 1.0 centimeter. And the second location is at a spot 10 centimeters away from the ball center, where we’ll solve for 𝐸 of 10 centimeters.
To start solving for the electric field one centimeter away from the ball center, let’s consider a cross section of the charged sphere. A cross section of this metal sphere shows charge evenly distributed around its circumference. As we consider different points inside the sphere and what their electric field may be, we can imagine the charges on the surface of the sphere creating electric field lines. Say the charge on the top creates an electric field line pointing downward. But then we see that there is also a charge on the bottom of the sphere which creates an electric field line pointing upward. And these two field lines effectively cancel one another out. The same thing happens with any other pair of charges in this cross section. Any charge we could pick, with its resulting electric field lines coming off of it, is counteracted equally and oppositely by a charge on the other side of the sphere. The net effect of all this is that all the electric field lines inside the sphere cancel one another out. The field inside the sphere at any location is zero.
This tells us that the electric field 1.0 centimeters away from the sphere’s center, which is still inside the sphere, is simply zero newtons per coulomb.
Now when we move outside the sphere to a location 10 centimeters away from its center, the story is different. In this case, the electric field lines from the charges on the sphere do not cancel one another out. And in fact, the relationship for the electric field outside of a sphere is equal to the charge on the sphere itself over four 𝜋 𝜖 naught, where 𝜖 naught is the permittivity of free space, times the distance from the center of the sphere to the point under consideration, 𝑟 squared. In this exercise, we’ll treat the constant 𝜖 naught as exactly 8.85 times 10 to the negative 12th farads per meter. So the electric field 𝐸 of 10 centimeters equals 𝑄 over four 𝜋 𝜖 naught times 10 centimeters quantity squared.
Plugging in the given values for 𝑄 and 𝜖 naught and changing our distance to units of meters, when we enter these values on our calculator, we find that, to two significant figures, 𝐸 is 2.7 times 10 to the sixth newtons per coulomb. That’s the electric field strength 10 centimeters away from the center of the sphere.