Question

The space between two conducting concentric spheres of radii `a` and `b (a lt b)` is filled up with homongeneous poorly conducting medium. The capacitance of such a system equals `C`. Find the resistivity of the medium if the potential difference between the spheres, when they are disconnected from an external voltage , decreases `eta`-fold during the time interval `Delta t`.

Answer 1

In our system, resistance of the medium `R = (rho)/(4pi) [(1)/(a) - (1)/(b)]`
where `rho` is the resistivity of the medium
The current `i = (varphi)/(R) = (varphi)/((rho)/(4pi) [(1)/(a) - (1)/(b)])`
Also, `i = (-dq)/(dt) = - (d(C varphi))/(dt) = -C (d varphi)/(dt)`, as capacitance is constant. ....(2)
So, equating (1) and (2) we get,
`(varphi)/((rho)/(4pi) [(1)/(a)-(1)/(b)]) = -C (d varphi)/(dt)`
or, `-int (d varphi)/(varphi) = (Delta t)/((C rho)/(4pi) [(1)/(a) - (1)/(b)])`
or, In `eta = (Delta t 4pi ab)/(C rho (b-a))`
Hence, resistivity of the medium,
`rho = (4pi Delta t ab)/(C (b-a) In eta)`