Initially, switch S is connected to position 1 for a long time (figure). The net amount of heat generated in the circuit after it is shifted to position 2 is

$\begin{array}{l}{\mathrm{U}}_{\mathrm{i}}=\frac{1}{2}{\mathrm{C\epsilon }}_1^2\\ {\mathrm{U}}_{\mathrm{f}}=\frac{1}{2}{\mathrm{C\epsilon }}_2^2,\mathrm{\Delta U}=\frac{1}{2}\mathrm{C}\left({\mathrm{\epsilon }}_2^2-{\mathrm{\epsilon }}_1^2\right)\\ {\mathrm{Q}}_{\mathrm{in}}=+{\mathrm{C\epsilon }}_1,{\mathrm{Q}}_{\mathrm{f}}=-{\mathrm{C\epsilon }}_2,\mathrm{\Delta Q}=\left|{\mathrm{Q}}_{\mathrm{f}}-{\mathrm{Q}}_{\mathrm{i}}\right|=\mathrm{C}\left({\mathrm{\epsilon }}_2+{\mathrm{\epsilon }}_1\right)\end{array}$
Work done by battery ${\mathrm{W}}_{\mathrm{b}}={\mathrm{\epsilon }}_2\mathrm{\Delta Q}=\mathrm{C}\left({\mathrm{\epsilon }}_2+{\mathrm{\epsilon }}_1\right){\mathrm{\epsilon }}_2$
Heat generated  $={\mathrm{W}}_{\mathrm{b}}-\mathrm{\Delta U}$
$\begin{array}{l}={\mathrm{C\epsilon }}_2^2+{\mathrm{C\epsilon }}_1{\mathrm{\epsilon }}_2-\frac{1}{2}\mathrm{C}\left({\mathrm{\epsilon }}_2^2-{\mathrm{\epsilon }}_1^2\right)\\ =\frac{1}{2}\mathrm{C}\left({\mathrm{\epsilon }}_2^2+{\mathrm{\epsilon }}_1^2+2{\mathrm{\epsilon }}_1{\mathrm{\epsilon }}_2\right)=\frac{1}{2}\mathrm{C}{\left({\mathrm{\epsilon }}_1+{\mathrm{\epsilon }}_2\right)}^2\end{array}$