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

To determine the potentials of the spherical shells, we need to analyze the configuration of the system: three concentric spherical shells with radii \(a\), \(b\), and \(c\) are charged with \(q_1\), \(q_2\), and \(q_3\) respectively.

The potential \(V\) due to a charged conducting sphere with charge \(Q\) and radius \(r\) at a point outside its surface is given by:

\(V = \dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{Q}{r}\)

Where \(4\pi\varepsilon_0\) is a constant. For the innermost shell \(A\), the potential at its surface \(V_A\) is simply the potential due to all charges. This is because all charges contribute to the potential on the surface of \(A\), which lies within the radius of all the shells:

1. Potential of Sphere A:

The potential at the surface of sphere \(A\) is affected by charges \(q_1\), \(q_2\), and \(q_3\). Thus, it is given by:

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

1. Potential of Sphere B:

For the sphere \(B\), it is at radius \(b\), so the potential here due to the inner shell \(A\) and itself (\(B\)), plus any contribution from \(C\), gives:

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

1. Potential of Sphere C:

For the outermost sphere \(C\), the potential is due to all enclosed charges \(q_1\), \(q_2\), and \(q_3\) at radius \(c\):

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

Thus, the correct set of potentials for spheres \(A\), \(B\), and \(C\) is:

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

This matches the correct answer provided.