What amount of heat in micro joule will be generated in the circuit , after the switch is shifted from position 1 to position 2 ? $\left(\mathrm{C}={\mathrm{C}}_0=2\mathrm{\mu F},\mathrm{E}=\sqrt{6}\mathrm{V}\right)$

As only the cell with emf E1 is responsible for redistribution of the charge. 

The charge on the condensers in position 1 are as shown.

$\frac{\mathrm{q}}{\mathrm{C}}=\frac{{\mathrm{q}}_0}{{\mathrm{C}}_0}=\frac{\mathrm{q}+{\mathrm{q}}_0}{\mathrm{C}+{\mathrm{C}}_0}$

$\text{ and }\left(\mathrm{q}+{\mathrm{q}}_0\right)\left(\frac{1}{\mathrm{C}+{\mathrm{C}}_0}+\frac{1}{\mathrm{C}}\right)=\xi \text{ or, }\mathrm{q}+{\mathrm{q}}_0=\frac{\mathrm{C}\left(\mathrm{C}+{\mathrm{C}}_0\right)\xi }{{\mathrm{C}}_0+2\mathrm{C}}$

$\text{ Hence, }\mathrm{q}=\frac{{\mathrm{C}}^2\xi }{{\mathrm{C}}_0+2\mathrm{C}}\text{ and }{\mathrm{q}}_0=\frac{{\mathrm{CC}}_0\xi }{{\mathrm{C}}_0+2\mathrm{C}}$

After the switch is thrown to position 2, the charges charge as shown in figure. A charge $q_{\mathrm{o}}$ has flown in the right loop through the two condensers and a charge $q_{\mathrm{o}}$ through the cell, Because of the symmetry of the problem there is no change in the energy stored in the condensers. Thus
H (Heat produced) = Energy delivered by the cell