Two ideal batteries of emf V_1 and V_2 and th | vedclass

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(C) Let the potential at the junction where $R_1, R_2, R_3$ meet be $V_O$. For the current in $R_2$ to be zero,the potential difference across $R_2$ m

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$(A, B, D)$

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(C) Let the potential at the junction where $R_1, R_2, R_3$ meet be $V_O$. For the current in $R_2$ to be zero,the potential difference across $R_2$ must be zero. Since one end of $R_2$ is connected to the common junction and the other to the negative terminal of $V_1$ (which we can take as reference potential $0$),the potential at the junction $V_O$ must be $0$.Using Kirchhoff's Current Law $(KCL)$ at the junction $O$:$\frac{V_O - V_1}{R_1} + \frac{V_O - 0}{R_2} + \frac{V_O - (-V_2)}{R_3} = 0$Setting $V_O = 0$ for zero current in $R_2$:$\frac{-V_1}{R_1} + 0 + \frac{V_2}{R_3} = 0$$\frac{V_1}{R_1} = \frac{V_2}{R_3} \Rightarrow \frac{V_1}{V_2} = \frac{R_1}{R_3}$Now check the options:$(A)$ $V_1=V_2, R_1=R_3 \Rightarrow \frac{V_1}{V_2} = 1, \frac{R_1}{R_3} = 1$. (Correct)$(B)$ $V_1=V_2, R_1=R_3 \Rightarrow \frac{V_1}{V_2} = 1, \frac{R_1}{R_3} = 1$. (Correct)$(C)$ $V_1=2V_2, R_1=R_3/2 \Rightarrow \frac{V_1}{V_2} = 2, \frac{R_1}{R_3} = 1/2$. (Incorrect)$(D)$ $V_1/V_2 = 1/2, R_1/R_3 = (R_2/2)/R_2 = 1/2$. (Correct)Thus,options $(A, B, D)$ are correct.

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