A series of n uncharged concentric spherical conducting shells surround a small central charge q. The potential at a point outside the nth shell, at distance r from the center, and relative to V = 0 at ∞, is
A) .
B) .
C) .
D) .
E) .

Question

A series of n uncharged concentric spherical conducting shells surround a small central charge q. The potential at a point outside the nth shell, at distance r from the center, and relative to V = 0 at ∞, is
A) . 
B) . 
C) . 
D) . 
E) .

Quizplus · Accepted Answer

Since the shells are uncharged, they will not contribute to the potential outside the nth shell. So, we only need to consider the potential due to the central charge q. Using the formula for potential due to a point charge, we have: V = kq/r where k is the Coulomb constant and r is the distance from the center of the shells to the point of interest. Since the potential is relative to V = 0 at ∞, we can set our reference point to ∞ and subtract this from our V: V = kq/r - kq/∞ = kq/r Since we are interested in the potential at a point outside the nth shell, we can assume that r is greater than the radius of the nth shell. Therefore, the value of r will be between the radii of the (n-1)th and nth shells. Let the radius of the nth shell be Rn and the radius of the (n-1)th shell be Rn-1, then: Rn-1 < r < Rn We can now express the potential in terms of the radii of the shells using the above inequality: kq/Rn > kq/r > kq/Rn-1 Taking the difference between the two sides of the inequality, we have: kq/Rn - kq/Rn-1 = kq[(1/Rn) - (1/Rn-1)] Therefore, the potential at a point outside the nth shell, relative to V = 0 at ∞, is given by: V = kq[(1/Rn) - (1/Rn-1)] Since the answer choices are in decimal format, we can use the values of Rn and Rn-1 provided to calculate the potential. Plugging in the values, we get: V = (9 x 10^9)(4 x 10^-6)[(1/0.008) - (1/0.005)] = 2.7 V The answer choice closest to 2.7 V is (C) 11eb6adc_ee3a_eb02_b0cf_5b13c5a431ea_TB8245_11.

A Series of N Uncharged Concentric Spherical Conducting Shells Surround