A conductor has a temperature-independent resistance `R` and a total heat capacity `C`. At the momenet `t = 0` it is connected to a `dc` voltage `V`. Find the time dependence of a conductor's temperature `T` assuming the thermal power dissipated into surrounding space to very as `q = k(T - T_(0))`, where `k` is constant, `T_(0)` is the enviroment temperature (equal to the conductor's temperature at the initial moment).

Explanation

The equaction of heat balance is
`(V^(2))/(R) - k (T - T_(0)) = C (dT)/(dt)`
Put `T - T_(0) = xi`
So, `C xi + k xi = (V^(2))/(R)` or, `xi + (k)/(C) xi = (V^(2))/(C R)`
or, `(d)/(dt) (xi e^(kt//c)) = (V^(2))/(C R) e^(ky//c)`
or, `xi e^(kt//c) = (V^(2))/(kR) e^(kt//c) + A`
where `A` is a constant Clearly
`xi = 0` at `t = 0`, so `A = (V^(2))/(kR)` and hence,
`T = T_(0) + (V^(2))/(kR) (1 - e^(-kt//C))`