question_answer

Three concentric spherical metallic shells A, B and C of radii a, b and c \[\left( a<b<c \right)\] have surface charge densities\[\sigma \], -\[\sigma \] and \[\sigma \] respectively. If the shells A and C are at same potential, then the correct relation between a, b and c is-

A) \[a+c=b\]

B) \[b+c=a\]

C) \[a-b=c\]                      

D) \[a+b=c\]

Correct Answer: D

Solution :

Potential of shell A is, \[{{V}_{A}}\,\,=\,\,\frac{1}{4\pi {{\in }_{0}}}\,\left( \frac{4\pi {{a}^{2}}\sigma }{a}\frac{-4\pi {{b}^{2}}\sigma }{b}\frac{+4\pi {{c}^{2}}\sigma }{c} \right)\] \[=\,\,\,\frac{\sigma }{{{\in }_{0}}}\,\left( a-b+c \right)\] Potential of shell C is, \[{{V}_{C}}\,\,=\,\,\frac{1}{4\pi {{\in }_{0}}}\,\left( \frac{4\pi {{a}^{2}}\sigma }{c}\frac{-4\pi {{b}^{2}}\sigma }{c}\frac{+4\pi {{c}^{2}}\sigma }{c} \right)\] \[=\,\,\,\frac{\sigma }{{{\in }_{0}}}\,\left( \frac{{{a}^{2}}}{c}-\frac{{{b}^{2}}}{c}+c \right)\] \[As\,\,\,{{V}_{A}}\,\,=\,\,{{V}_{C}}\] \[\therefore \,\,\,\,\frac{\sigma }{{{\in }_{0}}}\,\,\,(a-b+c)=\frac{\sigma }{{{\in }_{0}}}\left( \frac{{{a}^{2}}}{c}\frac{-{{b}^{2}}}{c}+c \right)\] \[or\,\,\,a-b\,\,=\,\,\frac{(a-b)(a+b)}{c}\,\,or\,\,a+b=c\]