CALC. A metal sphere with radius r_a is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius r_b. There is charge +q on the inner sphere and charge -q on the outer spherical shell. (a) Calculate the potential V(r) for (i) r < r_a; (ii) r_a < r < r_b; (iii) r > r_b. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take V to be zero when r is infinite. (b) Show that the potential of the inner sphere with respect to the outer is V_ab=q/(4πϵ_0 ) (1/r_a -1/r_b). (c) Use E_r=-∂V/∂r=(-∂/∂r) (1/(4πϵ_0 ) q/r)=[1/(4πϵ_0 )](q/r^2) and the result from part (a) to show that the electric field at any point between the spheres has magnitude E(r)=[V_ab/(1/r_a -1/r_b )](1/r^2) (d) Use E_r = [1/(4πϵ_0 )](q/r^2) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance r from the center, where r > r_b. (e) Suppose the charge on the outer sphere is not -q but a negative charge of different magnitude, say -Q. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.

Video transcript

Hey everyone. So this problem is working with electric potential. Let's see what they're giving us, what they're asking from us. As we work through this problem, I'm going to just draw a diagram to kind of help us visualize what's going on here. So we know we have a hollow brass fear that has a certain radius and a certain charge. The there is a solid aluminum ball of a different radius given by R. A. And that aluminum ball also has a charge. And so we know that the aluminum ball is being held in place inside of this brass sphere. And then it gives us this equation for electric field magnitude. And it asks us to use the expression for electric potential outside of the brass sphere to obtain an expression for that electric field magnitude Outside of the brass fear. So that this point outside of the brass sphere, I am going to mark as our sub over Outside. So the first thing we need to do for this problem is recall that electric potential is given by B equals K Q over R. So we can look at the three electric potentials for each of these spheres. For V A N V B We have a charge. So these are non-0 terms, we have a charge, we have a given radius. Okay, for the outside um point though as are are not goes to infinity our potential goes to zero. Right? So as this denominator term reaches infinity, this whole term for electric potential goes to zero. So be not approaches zero. Alright, so from there we can plug this back into the equation that they gave us for the electric field magnitude where we are deriving K. Q over R. Which we just said for this outside point, which is what they're asking us about. And the problem is 00 is a constant. When you derive a constant, it's zero. So the answer is that the electric field magnitude in this situation is zero. We look at our potential options, answer choices and it aligns with D. So that is the answer for this problem. We'll see you in the next video.