Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Two Unequal Resistances, R1 And R2, Are Connected Across Two Identical Batteries of Emf ε And Internal Resistance R (See the Figure). - Physics

Short Note

Two unequal resistances, R1 and R2, 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?

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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

\[i_1^2  R_1 t =  i_2^2  R_2 t\]

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

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

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

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

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

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

\[ r^2  =  R_1  R_2 \]

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

  Is there an error in this question or solution?
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 11 Thermal and Chemical Effects of Current
Short Answers | Q 2 | Page 217
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