Question

Two unequal resistances, R₁ and R_2, are connected across two identical batteries of emf ε and internal resistance r (see the 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:-

\[\epsilon -  i_1 r -  i_1  R_1  = 0\]

\[ \Rightarrow  i_1  = \frac{\epsilon}{r + R_1}\]

Similarly, the current in the other circuit,

\[i_2 = \frac{\epsilon}{r + R_2}\]

The thermal energies through the resistances are given by

\[i_1^2  R_1 t =  i_2^2  R_2 t\]

\[ \left( \frac{\epsilon}{r + R_1} \right)^2  R_1 t =  \left( \frac{\epsilon}{r + R_2} \right)^2  R_2 t\]

\[\frac{R_1}{\left( r + R_1 \right)^2} = \frac{R_2}{\left( r + R_2 \right)^2}\]

\[\frac{\left( r^2 + {R_1}^2 + 2r R_1 \right)}{R_1} = \frac{\left( r^2 + {R_2}^2 + 2r R_2 \right)}{R_2}\]

\[\frac{r^2}{R_1} +  R_1  = \frac{r^2}{R_2} +  R_2 \]

\[ r^2 \left( \frac{1}{R_1} - \frac{1}{R_2} \right) =  R_2  -  R_1 \]

\[ r^2  \times \frac{R_2 - R_1}{R_1 R_2} =  R_2  -  R_1 \]

\[ r^2  =  R_1  R_2 \]

\[ \Rightarrow r = \sqrt{R_1 R_2}\]