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.

Verified step by step guidance

1. Identify the regions of interest based on the radius r: (i) r < r_a, (ii) r_a < r < r_b, and (iii) r > r_b. Use the principle of superposition to calculate the potential V(r) in each region by summing the contributions from the inner and outer spheres.
2. For r < r_a, only the potential due to the inner sphere affects this region as the outer shell's influence is nullified within it. Use the formula for the potential due to a point charge, V = kq/r, where k = 1/(4πϵ_0).
3. For r_a < r < r_b, the potential is constant throughout this region because the electric field inside a conductor is zero, which implies that the potential is the same at any point inside the shell. Calculate this constant potential using the potential at r_a due to both the inner sphere and the induced charge on the inner surface of the outer shell.
4. For r > r_b, the potential is due to both the inner sphere and the outer shell. Since the shell is grounded, its potential contribution at a distance r can be calculated using the formula for the potential outside a spherical shell.
5. To find the potential difference V_ab between the inner and outer spheres, use the expression V_ab = q/(4πϵ_0) (1/r_a - 1/r_b). This represents the work done per unit charge in moving a test charge from the outer sphere to the inner sphere without any acceleration.