The radii of spherical capacitor electrodes are equal to `a` and `b`, with `a lt b`. The interlectordes `epsilon` and resistivity `rho`. Inititally the capacitor is not charged. At the moment `t = 0` the internal electorde gets a charge `q_(0)` Find:
(a) the times variation of the charge on the internal elecrtordes,
(b) the amount of the heat generated during the spreading of the charge.

Explanation

(a) Let us mentally isolatate a thin spherical layer with inner and outer radii `r` and `r + dr` respective. Lince of current at all the points of this layer are perpendicular to it and therefore such a layer can be treated as a spherical conductor of thickness `dr` and cross sectional area `4pi r^(3)`. Now we know that resistance,
`dR = rho (dr)/(S(r)) = rho (dr)/(4pi r^(2))` ....(1)
Intergating expression (1) between teh Hints,
`int_(0)^(R) dR = int_(R)^(b) rho (dr)/(4pi r^(2))` or, `R = (rho)/(4pi) [(1)/(a) - (1)/(b)]` ......(2)
Capacitance of the network, `C = (4pi epsilon_(0) epsilon)/([(1)/(a) - (1)/(b)])` ......(3)
and `q = C varphi` [where `q` is the charge at any arbitary moment] ....(4)
also, `varphi = ((-dq)/(dt)) R`as capacitor is discharging, ....(5)
From Eqs.(2), (3),(4) and (5) we get,
`q = (4pi epsilon_(0) epsilon)/([(1)/(a) - (1)/(b)]) ([- (dq)/(dt)] rho [(1)/(a) - (1)/(b)])/(4pi)` or, `(dq)/(q) = (dt)/(rho epsilon epsilon_(0))`
Intergating `int_(q_(0))^(q) - (dq)/(q) = (1)/(rho epsilon_(0) epsilon) int_(0)^(t) dt = (dt)/(rho epsilon epsilon_(0))`
Hence `q = q_(0) e^((-t)/(rho epsilon_(0) epsilon))`
(b) From energy conservation heat generated,m during the spreading of the charge,
`H = U_(i) - U_(f)` [because `A_("cell") = 0`]
`= (1)/(2) (q_(0)^(2))/(4pi epsilon_(0) epsilon) [(1)/(a) - (1)/(b)] - 0 = (q_(0)^(2))/(8pi epsilon_(0) epsion) (b-a)/(ab)`