A metal ball of radius `a` is surrounded by a thin concentric metal shell of radius `b`. The space between these electrodes is filled up with a poorly conducting homogenous medium of resistivity `rho`. Find the resistance of the interelectrode gap. Analyse the obtained solution at `b rarr oo`.

Explanation

(a) Taking an element of a spherical shell of radius `r` and thickness `dr`.
`dA = 4 pi r^2`
`l = dr`
`dR = (rho l)/(dA) = (rho dr)/(4 pi r^2)`
`R = (rho)/(4 pi) int_a^b r^-2 dr`
=`(rho)/(4 pi) |(r^-1)/(-1)|_a^b = (rho)/(4 pi) |-(1)/(r)|_a^b`
=`-(rho)/(4 pi)|(1)/(r)|_a^b = -(rho)/(4 pi) |(1)/(b) -(1)/(a)| = (rho)/(4 pi) ((b-a))/(ab)`
`R = (rho(b - a))/(4 pi ab)`
When `b rarr oo`
`R = -(rho)/(4 pi) ((1)/(oo) -(1)/(a)) = (rho)/(4 pi a)`
(b) (i) Area of cross-section `A = pi (b^2 -a^2)`
`R = (rho L)/(A) = (rho L)/(pi(b^2 - a^2))`
(ii) Taking an element a cylindrical shell of radius `r` and thickness `dr`
`A = 2 pi r L`
`dR = (rho dr)/(2 pi rL)`
`R = (rho)/(2 pi L) int_a^b (dr)/(r) = (rho)/(2 pi L) |log_e r|_a^b`
=`(rho)/(2 pi L) (log_e b- log_e a) = (rho log_e(b//a))/(2 pi L)`
( c) `r_x = r_1 + (r_2 -r_1)/(L)x`
=`r_1 + lamda x`,
where `lamda = (r_2 - r_1)/(L)`
`dR = (rho dx)/(pi r_x^2) = (rho dx)/(pi(r_1 + lamdax)^2)`
`R = (rho)/(pi) int_0^L (dx)/((r_1 + lamda x)^2)`
=`(rho)/(pi) |((r_1 + lamda x)^-1)/((-1)(lamda))|_0^L`
=`-(rho)/(pi lamda)|(1)/(r_1 + lamdax)|_0^L`
=`-(rho)/(pi lamda)|(1)/(r_1 + lamda L)-(1)/(r_1)|`
=`-(rho)/(pi lamda) |(1)/(r_2) -(1)/(r_1)| = (rho)/(pi lamda) ((r_2 - r_1))/(r_1 r_2)`
=`(rho L)/(pi r_1 r_2)`
In case of frustum of cone, take area as
`A = sqrt(A_1 A_2) = sqrt(pi r_1^2. pi r_2^2) = pi r_1 r_2`
(d) Taking an element of area `A` and separation between plates `dx`.
Conductivity at distance `x` from the left end.
`sigma_x = sigma_1 + (sigma_2 - sigma_1)/(L) x = sigma_1 + lamda x`
where `lamda = (sigma_2 - sigma_1)/(L)`
`dR = (rho_x dx)/(A) = (dx)/(sigma_x A) = (dx)/((sigma_1 + lamda x)A)`
`R = (1)/(A) int_0^L (dx)/((sigma_1 + lamdax))`
=`(1)/(A) (|log_e (sigma_1 + lamda x)|_0^L)/(lamda)`
=`(1)/(A lamda) [log_e(sigma_1 + lamda L)-log_e (sigma_1)]`
=`(1)/(A lamda) [log_e (sigma_2) - log_e (sigma_1)]`
=`(1)/(A((sigma_2 - sigma_1))/(L)) log_e((sigma_2)/(sigma_1))`
=`(L log_e (sigma_2 //sigma_1))/(A(sigma_2 - sigma_1))`.




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