What amount of heat will be generated in the circuit shown in Fig. after the swich `Sw` is shifted from position 2 ?

Answer

`Q = (1)/(2) C epsilon_(2)^(2)`
It is remarkable that the result obtained is independent of `epsilon_(1)`]

Explanation

When S is at position (1)
By KVL

`rArr epsilon_(1) - epsilon_(2) - (q)/(C) = 0`
`rArr q = (epsilon_(1)-epsilon_(2))C`
Energy stored `= (1)/(2) C(epsilon_(1)-epsilon_(2))^(2) = (q^(2))/(2C)`
When switch 'S' in at position (2)

By KVL ,
`epsilon_(1) = (q+C)/(C)` Put q from (1)
`epsilon_(1)=((epsilon_(1)-epsilon_(2))C+Q)/(C)Q=epsilon_(2)C`....(2)
Energy stored `= ((Q+q)^(2))/(2C)`
`:.` Work done by battery `epsilon_(1)`
`W = epsilon_(1)Q = epsilon_(1) epsilon_(2)C`
`:.` Heat produced
`H = W - Delta U`
`:. H = epsilon_(1)epsilon_(2) C-(U_(f)-U)`
`H=epsilon_(1)epsilon_(2)-[((Q+q)^(2))/(2C)-(q^(2))/(2C)]`
Put Q and q from (1) & (2)
`:. H = (1)/(2) C epsilon_(2)^(2)`