Verified Solution • 2025–26 Session moderate

QPage 217: Q2: HC Verma: Concepts of Physics - Volume 2 Thermal and Chemical Effects of Electric Current Page 217: Q2 Solution: Two unequal resistances, and , are connected across two…

Two unequal resistances, R1R_1 and R2R_2, are connected across two identical batteries of emf ε\varepsilon and internal resistance r see the figure). Can the thermal energies developed in R1R_1 and R2R_2 be equal in a given time? If yes, what will be the condition?

Step-by-Step Verified Solution

**Answer:** Yes, the thermal energies can be equal. The condition is r=R1R2r = \sqrt{R_1 R_2}. **Step-by-Step Explanation:** For circuit 1: i1=εr+R1i_1 = \frac{\varepsilon}{r + R_1}. Heat in R1R_1: H1=i12R1tH_1 = i_1^2 R_1 t. For circuit 2: i2=εr+R2i_2 = \frac{\varepsilon}{r + R_2}. Heat in R2R_2: H2=i22R2tH_2 = i_2^2 R_2 t. Setting H1=H2H_1 = H_2:
ε2R1(r+R1)2=ε2R2(r+R2)2\frac{\varepsilon^2 R_1}{(r + R_1)^2} = \frac{\varepsilon^2 R_2}{(r + R_2)^2}
Since R1≠R2R_1 \neq R_2, solving gives r2=R1R2r^2 = R_1 R_2, so r=R1R2r = \sqrt{R_1 R_2} [citation:12].

All Questions in this Chapter

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