In a circuit shown in fig. the capacitance of each capacitor is equal to `C` and the resistance, to `R` One of the capacitor was connected to a voltage `V_(0)` and then at the moment `t = 0` was shorted by means of the swich `Sw`. Find:
(a) a current `I` in the circuit as a function of time `t`:
(b) the amount of generated heat provided a dependence `I(t)` is known.

Explanation

Let, at any moment of time, charge flown be `q` then current `i = (dq)/(dt)`
Applying loop rule in the circuit, `Delta varphi = 0`, we get :
`(dq)/(dt) IR - ((C V_(0) - q))/(C) + (q)/(C) = 0`
or, `(dq)/(C V_(0) - 2q) = (1)/(R C) dt`
So, In `(C V_(0) - 2q)/(CV_(0)) = 2 (t)/(RC)` for `0 le t le t`
or, `q = (C V_(0))/(2) (1 - e^((-2t)/(R C)))`
Hence, `i = (dq)/(dt) = (C V_(0))/(2) (2)/(R C) e^(2t//R C) = (V_(0))/(R) e^(-2t//R C)`
Now, heat liberated,
`Q = int_(0)^(oo) i^(2) R dt = (V_(0)^(2))/(R^(2)) R int_(0)^(oo) e^((-4t)/(R C)) dt = (1)/(4) C V_(0)^(2)`