The series combination of two batteries, both of the same emf 10 V, but different internal resistance of 20 $Ω$ and 5 $Ω$ , is connected to the parallel combination of two resistors 30 $Ω$ and x $Ω$ . The voltage difference across the battery of internal resistance 20 $Ω$ is zero, the value of x(in $Ω$ ) is ------

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Detailed Solution

$\begin{array}{l} F o r f i r s t b a t t e r y, \\ V_{1} = ε_{1} - i . r_{1} \\ \Rightarrow 0 = 10 - i \times 20 (∵ V_{1} = 0) \\ \Rightarrow i = 0.5 A \\ F o r s e c o n d b a t t e r y, \\ V_{2} = ε_{2} - i r_{2} = 10 - 0.5 \times 5 \\ \Rightarrow V_{2} = 7.5 V \\ A s p e r K i r c h h o f f s r u l e \\ i = i_{1} + i_{2} \\ \Rightarrow 0.5 = \frac{7.5}{30} + \frac{7.5}{x} \\ \Rightarrow 0.5 = 0.25 + \frac{7.5}{x} \\ \Rightarrow \frac{7.5}{x} = 0.25 \\ \Rightarrow x = 30 \end{array}$