Question

Two unequal resistances, R₁ and R_2, are connected across two identical batteries of emf ε and internal resistance r (figure). Can the thermal energies developed in R₁ and R₂ be equal in a given time? If yes, what will be the condition?

Answer 1

For the given time t, let the currents passing through the resistance R₁ and R₂ be i₁ and i₂ , respectively.
Applying Kirchoff's Voltage Law to circuit-1, we get:

$$\begin{gathered} \epsilon -i_{1}r-i_1{R}_1=0 \\ \Rightarrow i_1=\frac{\epsilon }{r+R_1} \end{gathered}$$
Similarly, the current in the other circuit,
$i_2=\frac{\epsilon }{r+R_2}$
The thermal energies through the resistances are given by

$$\begin{gathered} i_1^2{R}_{1}t=i_2^2{R}_{2}t \\ {\left(\frac{\epsilon }{r+R_1}\right)}^2{R}_{1}t={\left(\frac{\epsilon }{r+R_2}\right)}^2{R}_{2}t \\ \frac{R_1}{{\left(r+R_1\right)}^2}=\frac{R_2}{{\left(r+R_2\right)}^2} \\ \frac{\left(r^2+{R_1}^2+2r{R}_1\right)}{R_1}=\frac{\left(r^2+{R_2}^2+2r{R}_2\right)}{R_2} \\ \frac{r^2}{R_1}+R_1=\frac{r^2}{R_2}+R_2 \\ {r}^2\left(\frac{1}{R_1}-\frac{1}{R_2}\right)=R_2-R_1 \\ {r}^2\times \frac{R_2-R_1}{R_1{R}_2}=R_2-R_1 \\ {r}^2=R_1{R}_2 \\ \Rightarrow r=\sqrt{R_1{R}_2} \end{gathered}$$