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Three concentric spherical conductors of...

Three concentric spherical conductors of radii a, 2a, and 3a have charges `-Q, +2Q`, and `-4Q`, respectively. If r is the distance of the point under consideration from the center of the spheres, then find the electric field and potential due to the given configuration, for the values
(i) `r lt a`
(ii) `a lt r lt 2a`
(iii) `2 a lt r lt 3 a`
`r gt 3 a`.
.

Text Solution

Verified by Experts

(i) `r lt a`
`V = v_(1, "in") + V_(2,"in") + V_(3,"in")`
=`(1)/(4 pi epsilon_0) ((- Q))/a + (1)/(4 pi epsilon_0) ((2 Q))/(2 a) + (1)/(4 pi epsilon_0) ((-4 Q))/(3 a)`
=`(1)/(4 pi epsilon_0) Q/a[- 1 + 1 -(4)/(3)] = (1)/(4 pi epsilon_0)Q/a[-(4)/(3)] = -Q/(3 pi epsilon_0 a)`
.
(ii) `a,r lt 2 a`
`V = V_(1,out) + V_(2,out) + V_(3,out)`
=`(1)/(4 pi epsilon_0) ((-Q))/(r) + (1)/(4 pi epsilon_0) ((2 Q))/(2 a) + (1)/(4 pi epsilon_0)((-4Q))/(3 a)`
=`(1)/(4 pi epsilon_0) Q [-(1)/(r) + (1)/a -(4)/(3 a)] = (1)/(4 pi epsilon_0) Q [-(1)/(r) - (1)/(3 a)]`
=`- (1)/(4 pi epsilon_0) [(1)/(r) + (1)/(3 a)]`
(iii) `2a lt r, 3a`
`V = V_(1,out) + V_(2, out) + V_(3,"in")`
=`(1)/(4 pi epsilon_0)((- Q))/(r) + (1)/(4 pi epsilon_0)((2 Q))/(r) + (1)/(4 pi epsilon_0) ((-4Q))/(3 a)`
=`Q/(4 pi epsilon_0)[-(1)/(r) + (2)/(r) - (4)/(3 a)] = Q/(4 pi epsilon_0)[(1)/(r) - (4)/(3 a)]`
(iv) `r gt 3a`
`V = V_(1,out) + V_(2, out) + V_(3, out)`
=`(1)/(4 pi epsilon_0)((-Q))/(r) + (1)/(4 pi epsilon_0)((2 Q))/(r) + (1)/(4pi epsilon_0) ((-4 Q))/(r)`
=`(1)/(4 pi epsilon_0) Q/(r) [-1 + 2- 4] = -(3)/(4 pi epsilon_0)Q/(r)`.
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