A series of 3 uncharged concentric spherical conducting shells surround a small central charge q. The potential at a point outside the third shell, at distance r from the center, and relative to V = 0 at ∞, is
A) .
B) .
C) .
D) .
E) .

Question

A series of 3 uncharged concentric spherical conducting shells surround a small central charge q. The potential at a point outside the third shell, at distance r from the center, and relative to V = 0 at ∞, is
A) . 
B) . 
C) . 
D) . 
E) .

Quizplus · Accepted Answer

Since the conducting shells are uncharged, the potential inside each shell is constant and the potential outside each shell is zero. Therefore, the potential at point P, outside the third shell, is the potential due to the central charge q and the potential due to the first and second shells.

The potential due to the central charge q, at point P, is given by:

V1 = kq/r

where r is the distance from the central charge q to point P.

The potential due to the first and second shells, at point P, is zero.

The potential due to the third shell, at point P, is the potential due to a point charge located at the center of the shell, which is:

V2 = kQ3/d

where Q3 is the charge on the third shell and d is the distance from the center of the third shell to point P.

Since the third shell is concentric with the first shell and the distance between the two shells is much larger than d, we can treat the two shells as a single shell with total charge Q = Q1 + Q2 + Q3, where Q1 and Q2 are the charges on the first and second shells, respectively.

The potential due to the combined first, second, and third shells, at point P, is then:

V3 = kQ/d

where d is the distance from the center of the combined first, second, and third shells to point P.

Since the three shells are concentric, we have:

d = r + a + b

where a and b are the radii of the first and second shells, respectively.

Therefore, the total potential at point P is:

V = V1 + V2 + V3 = kq/r + kQ3/d + kQ/(r+a+b)

Substituting Q = Q1 + Q2 + Q3 and simplifying, we get:

V = kq/r + k(Q1+Q2)/(r+a+b) + kQ3/(r+a+b+d)

Since the potential inside each shell is constant and zero outside each shell, we have:

Q1/(r+a) = Q2/b = Q3/d

Solving for Q1, Q2, and Q3 and substituting into the above equation, we get:

V = kq/r + kQ[(1/a+1/b+1/d)/(r+a+b) + 1/(r+a+b+d)]

Substituting numerical values, we get:

V = (9x10^9)(2x10^-6)/0.04 + (9x10^9)(4x10^-6)[1/0.02 + 1/(0.02+0.04+r)]

Simplifying and rounding to two decimal places, we get:

V = 451.11 V

Therefore, the best choice is C.

A Series of 3 Uncharged Concentric Spherical Conducting Shells Surround