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Mixed Combination of Cells

Under what co...

Question

Under what conditions current passing through the resistance $R$ can 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.

E_{2} r_{1} > E_{1} (R + r_{2})

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E_{1} r_{2} < E_{2} (r_{1} + R)

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E_{2} r_{2} < E (R + r_{2})

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E_{1} r_{1} > E_{2} (r_{1} + R)

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Solution

The correct option is B $E_{1} r_{2} < E_{2} (r_{1} + R)$
The current through the circuit before the battery of emf $E_{2}$ is short circuited, is
$i_{1} = \frac{E_{1} + E_{2}}{r_{1} + r_{2} + R}$ ...(i)
After short circuiting the battery of emf $E_{2}$ , current through the resistance $R$ would be
$i_{2} = \frac{E_{1}}{R + r_{1}}$ ...(ii)
Since, from equation (i) and (ii), we get
$i_{2} > i_{1}$
$∴ \frac{E_{1}}{R + r_{1}} > \frac{E_{1} + E_{2}}{R + r_{1} + r_{2}}$
This gives $E_{1} r_{2} > E_{2} (R + r_{1})$

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Under what conditions current passing through the resistance R can be increased by short circuiting the battery of emf E2? The internal resistances of the two batteries are r1 and r2 respectively.

Under what conditions current passing through the resistance $R$ can 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.