Three concentric metallic shells A, B and C of radii a,b and c (a<b<c) have surface charge densities +\sigma, -\sigma and +\sigma respectively as shown.

(a) Obtain the expressions for the potential of three shells A, B and C.

(b) If shells A and C are at the same potential, obtain the relation between a, b and c.

 

 
 
 
 
 

Answers (1)
S safeer

(a) We know,

The potential of sphere 'A' (V_{A}) is given as,

V_{A}= \frac{KQ_{A}}{a}-\frac{KQ_{b}}{b}+\frac{KQ_{C}}{C}

Where, K= \frac{1}{4\pi \epsilon _{0}}

then,

V_{A}= K\left [ \frac{Q_{A}}{a}-\frac{Q_{b}}{b}+\frac{Q_{C}}{c} \right ]

Such that, \sigma = \frac{Q}{A}= \frac{Q_{A}}{4\pi a^{2}}, then Q_{A}= \sigma \times 4\pi a^{2}

Similarly,  Q_{b}= \sigma \times 4\pi b^{2}

                 Q_{c}= \sigma \times 4\pi c^{2}

Hence, 

V_{A}= \frac{1}{4\pi \epsilon _{0}}\left [ \frac{4\pi a^{2}\sigma}{a}-\frac{4\pi b^{2}\sigma }{b}+\frac{4\pi c^{2}\sigma }{c} \right ]

V_{A}= \frac{\sigma }{ \epsilon _{0}}\left [ a-b+c \right ]

Similarly, the potential of sphere 'B' \left (V_{B} \right ) is given as;

V_{B}= \frac{KQ_{A}}{b}-\frac{KQ_{b}}{b}+\frac{KQ_{C}}{C}

\becauseK= \frac{1}{4\pi \epsilon _{0}}

V_{B}= \frac{1}{4\pi \epsilon _{0}}\left [ \frac{Q_{A}}{b}-\frac{Q_{b}}{b}+\frac{Q_{C}}{c} \right ]

Such that, \sigma = \frac{Q}{A}= \frac{Q_{A}}{4\pi b^{2}}, then Q_{A}= \sigma \times 4\pi b^{2}

Similarly,  Q_{b}= \sigma \times 4\pi b^{2}

                 Q_{c}= \sigma \times 4\pi c^{2}

V_{B}= \frac{1}{4\pi \epsilon _{0}}\left [ \frac{4\pi a^{2}\sigma}{b}-\frac{4\pi b^{2}\sigma }{b}+\frac{4\pi c^{2}\sigma }{c} \right ]

V_{B}= \frac{\sigma }{ \epsilon _{0}}\left [ \frac{a^{2}}{b}-b+c \right ]

V_{B}= \frac{\sigma }{ \epsilon _{0}}\left [ \frac{a^{2}-b^{2}}{b}+c \right ]

Similarly for \left (V_{C} \right )

V_{C}= \frac{KQ_{A}}{c}-\frac{KQ_{b}}{c}+\frac{KQ_{C}}{c}

V_{C}= K\left [ \frac{Q_{A}}{c}-\frac{Q_{b}}{c}+\frac{Q_{C}}{c} \right ]

V_{C}= \frac{1}{4\pi \epsilon _{0}}\left [ \frac{4\pi a^{2}\sigma}{c}-\frac{4\pi b^{2}\sigma }{c}+\frac{4\pi c^{2}\sigma }{c} \right ]

V_{C}= \frac{\sigma }{ \epsilon _{0}}\left ( \frac{a^{2}-b^{2}+c^{2}}{c} \right )

 

(b) we know,

V_{A}= \frac{\sigma }{ \epsilon _{0}}\left [ a-b+c \right ]

V_{C}= \frac{\sigma }{ \epsilon _{0}}\left ( \frac{a^{2}-b^{2}+c^{2}}{c} \right )

then, the given condition if shell A and C are equal;

V_{A}=V_{C}

On putting the values, we get

a-b+c = \frac{a^{2}-b^{2}+c^{2}}{c}

\left (a-b+c \right ) c=a^{2}-b^{2}+c^{2}

ac-bc= a^{2}-b^{2}

c(a-b)= a^{2}-b^{2}

such that a^{2}-b^{2}= (a+b)(a-b)

c = \frac{a^{2}-b^{2}}{a-b}

c= a+b  - this is the obtained relation between a, b and c.

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