Question

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. $\dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{a}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right),\; \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right)$
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Hint:

For concentric conducting shells, always remember: potential at a shell equals the sum of potentials due to all charges, with outer shell charges contributing constant potential inside.

Answer 1

The Correct Option is C

Step 1: Understanding potential due to spherical shells.
The electric potential at any point due to a charged conducting spherical shell is equal to the potential due to a point charge placed at its center.
For points inside a conducting shell, the potential remains constant and is equal to the potential at the surface of the shell.
Step 2: Potential of sphere $A$ (radius $a$).
Sphere $A$ lies inside both spheres $B$ and $C$. Hence, its potential is the sum of:
Potential due to its own charge $q_1$ at radius $a$,
Potential due to charge $q_2$ of sphere $B$ at radius $b$,
Potential due to charge $q_3$ of sphere $C$ at radius $c$.
\[ V_A = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a} + \dfrac{q_2}{b} + \dfrac{q_3}{c}\right) \] Step 3: Potential of sphere $B$ (radius $b$).
At sphere $B$, the contribution comes from:
Charge $q_1$ and $q_2$ acting as if located at the center up to radius $b$,
Charge $q_3$ contributing its surface potential at radius $c$.
\[ V_B = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b} + \dfrac{q_3}{c}\right) \] Step 4: Potential of sphere $C$ (radius $c$).
At the outermost sphere $C$, all charges behave as a point charge at the center.
\[ V_C = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \] Step 5: Final conclusion.
The correct expressions for the potentials of spheres $A$, $B$ and $C$ correspond to Option (3).