d
\(R_1\) અવરોધમાંથી \(I_1\) અને \(R_2\) અવરોધમાંથી \(I_1\) પ્રવાહ પસાર થાય છે. તેમજ તેનો આંતરિક અવરોધ \(r\) છે.

\(\therefore \,\,\varepsilon \,\, = \,\,{I_1}\,({R_1} + r)\,\,\, \Rightarrow \,\,{I_1}\,\, = \,\,\frac{\varepsilon }{{{R_1} + r}}\)

\( \Rightarrow \,\,I_1^2{R_1}t\,\, = \,\,\frac{{{\varepsilon ^2}{R_1}t}}{{{{({R_1} + r)}^2}}}\,\, = \,\,{H_1}\)

\(\therefore \,\,\,\varepsilon \,\, = \,\,{I_2}\,({R_2} + r)\,\,\)   \( \Rightarrow \,\,{I_2}\,\, = \,\,\frac{\varepsilon }{{{R_2} + r}}\,\)

\( \Rightarrow \,\,I_2^2{R_2}t\,\, = \,\,\frac{{{\varepsilon ^2}{R_2}t}}{{{{({R_2} + r)}^2}}}\,\, = \,\,{H_2}\)

પરંતુ \({{\text{H}}_{\text{1}}}\, = \,\,{H_2}\)

\(\therefore \,\,\frac{{{\varepsilon ^2}{R_1}t}}{{{{({R_1} + r)}^2}}}\,\, = \,\,\frac{{{\varepsilon ^2}{R_2}t}}{{{{({R_2} + r)}^2}}}\)

\( \Rightarrow \,\,\frac{{{R_1}}}{{{{({R_1} + r)}^2}}}\,\, = \,\,\frac{{{R_2}}}{{{{({R_2} + r)}^2}}}\)

\(\therefore \,\,{R_1}{({R_2} + r)^2}\,\, = \,\,{R_2}{({R_1} + r)^2}\,\)

\(\,\therefore \,\,\,\,\,\,{R_1}(R_2^2 + 2{R_2}.r\, + \,{r^2})\,\, = \,\,{R_2}(R_1^2 + 2{R_1}.\,r + {r^2})\)

\(\therefore \,\,{R_1}R_2^2\, + \,2{R_2}{R_2}r + {R_1}{r^2}\,\, = \,\,{R_2}R_1^2 + {R_1}{R_2}r + {R_2}{r^2}\,\,\,\)

\(\therefore \,\,\,\,\,({R_1} - {R_2}){r^2}\,\, = \,\,{R_2}{R_1}\,({R_1} - {R_2})\)

\(\therefore \,\,{r^2}\,\, = \,\,{R_1}{R_2}\,\, \Rightarrow \,\,r\,\, = \,\,\sqrt {{R_1} + {R_2}} \)