Under what condition can the current passing | vedclass

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(B) Initially,the current in the circuit is $I_1 = \frac{E_1 + E_2}{R + r_1 + r_2}$.When the battery of emf $E_2$ is short-circuited,the new current i

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$E_1r_2 > E_2 (R + r_1)$

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(B) Initially,the current in the circuit is $I_1 = \frac{E_1 + E_2}{R + r_1 + r_2}$.When the battery of emf $E_2$ is short-circuited,the new current in the circuit is $I_2 = \frac{E_1}{R + r_1}$.For the current to increase,we must have $I_2 > I_1$,which implies:$\frac{E_1}{R + r_1} > \frac{E_1 + E_2}{R + r_1 + r_2}$.Cross-multiplying gives:$E_1(R + r_1 + r_2) > (E_1 + E_2)(R + r_1)$.Expanding both sides:$E_1R + E_1r_1 + E_1r_2 > E_1R + E_1r_1 + E_2R + E_2r_1$.Subtracting $E_1R + E_1r_1$ from both sides:$E_1r_2 > E_2(R + r_1)$.

Under what condition can the current passing through the resistance $R$ be increased by short-circuiting the battery of emf $E_2$? The internal resistances of the two batteries are $r_1$ and $r_2$ respectively.

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