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A charge Q is distributed over three concentric spherical shells of radii a,b,c (a<b<c) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where r<a, would be:

  • Option 1)

  • Option 2)

  • Option 3)

  • Option 4)

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Potential Due to 3 Concentric Spheres -

The figure shows three conducting concentric shell of radii a, b and c (a < b < c) having charges Qa, Qb and Qc respectively

- wherein

 

Potential at A;

V_{A}= \frac{1}{4\pi \varepsilon _{0}}\left [ \frac{Q_{a}}{a}+\frac{Q_{b}}{b} +\frac{Q_{c}}{c}\right ]

Potential at B;

V_{B}= \frac{1}{4\pi \varepsilon _{0}}\left [ \frac{Q_{a}}{b}+\frac{Q_{b}}{b} +\frac{Q_{c}}{c}\right ]

Potential at C;

V_{C}= \frac{1}{4\pi \varepsilon _{0}}\left [ \frac{Q_{a}}{c}+\frac{Q_{b}}{c} +\frac{Q_{c}}{c}\right ]

V_{P}=V_{A}+V_{B}+V_{C}\\\\=\frac{KQ_{a}}{a}+\frac{KQ_{b}}{b}+\frac{KQ_{c}}{c}\\\\\\Q=sigmaA \\\\A=\pi r^{2}\\\\(r_{a}=r_{b}=r_{c})\\\\\\Q_{a} :Q_{b}:Q_{c}::a^{2}:b^{2}:c^{2}\\\\\\Q_{a}=\left ( \frac{a^{2}}{a^{2}+b^2+c^{2}} \right )Q

Q_{b}= \left ( \frac{b^{2}}{a^{2}+b^{2}+c^{2}} \right )Q\:

Q_{c}= \left ( \frac{c^{2}}{a^{2}+b^{2}+c^{2}} \right )Q\:

V= K \left ( \frac{Q_{a}}{a} +\frac{Q_{b}}{b}+\frac{Q_{c}}{c}\right )\\\\=\frac{1}{4\pi \varepsilon _{0}}\times Q\times \left [ \frac{\left ( a+b+c \right )}{\left ( a^{2}+b^{2}+c^{2} \right )} \right ]

 

 

 


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Option 2)

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