There are three co-centric conducting spherical shells A, B and C… — Solution | JEE Physics

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There are three co-centric conducting spherical shells $A$ , $B$ and $C$ of radii $a$ , $b$ and $c$ respectively $(c>b>a)$ and they are charged with charges $q_1$ , $q_2$ and $q_3$ respectively. The potentials of the spheres $A$ , $B$ and $C$ respectively are:

A
$\dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{a}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{b}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right)$
B
$\dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{b}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right)$
C
$\dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right)$
D
$\dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{a}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right)$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Three concentric conducting spherical shells $A$ , $B$ and $C$ have radii $a$ , $b$ and $c$ with $c>b>a$ , and charges $q_1$ , $q_2$ and $q_3$ respectively.

Find: The potentials of shells $A$ , $B$ and $C$ .

For a charged conducting spherical shell, the potential outside is the same as that of a point charge at the center, and the potential at every point inside the shell is constant and equal to the surface potential.

So, at shell $A$ , we add the contribution of its own charge and the constant potentials due to the outer shells:

V_A = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a} + \dfrac{q_2}{b} + \dfrac{q_3}{c}\right)

Stepwise Evaluation of Potentials

At shell $B$ , the charges on shells $A$ and $B$ contribute as if located at the center up to radius $b$ , while shell $C$ contributes a constant potential inside it:

V_B = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b} + \dfrac{q_3}{c}\right)

At the outermost shell $C$ , all charges act as if concentrated at the center:

V_C = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right)

Therefore, the correct expressions match Option (3). The correct option is C.

Common mistakes

A common mistake is to take the potential on every shell as depending on the total charge divided by that shell radius. This is wrong because outer shell charges give constant potential inside, not $\dfrac{q}{r}$ with the inner radius. Use the surface radius of the shell producing the potential.
Another mistake is to ignore the contribution of outer shells to the potential at inner shells. Although the electric field inside a shell is zero, the potential is not zero there; it remains a constant equal to the shell surface potential.
Students may confuse electric field with electric potential. Zero electric field inside a conductor does not mean zero potential. Compute potential by adding scalar contributions from all shells.

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Answer

Step-by-step solution

Common mistakes

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