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Parallel Combination of Cells

For the batte...

Question

For the batteries shown in the figure, $R_{1}, R_{2},$ and $R_{3}$ are the internal resistances of $E_{1}, E_{2},$ and $E_{3}$ respectively. Then, which of the following is/are correct?

A

The equivalent internal resistance

R

of the system is given by

\frac{(R_{1} R_{2} R_{3})}{(R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}) .}

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B

If

E_{3} = \frac{(E_{1} R_{2} + E_{2} R_{1})}{(R_{1} + R_{2})}

, the equivalent emf of the batteries will be equal to

E_{3}

.

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C

The equivalent e.m.f. of the battery is equal to

= \frac{E_{1} + E_{2} + E_{3}}{3}

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D

The equivalent e.m.f. of the battery not only depends upon values of

E_{1}, E_{2}

and

E_{3}

but also depends upon values of

R_{1}, R_{2}

and

R_{3}

.

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Solution

The correct option is D The equivalent e.m.f. of the battery not only depends upon values of $E_{1}, E_{2}$ and $E_{3}$ but also depends upon values of $R_{1}, R_{2}$ and $R_{3}$ .
The equivalent internal resistance diagram of the circuit can be shown as
Equivalent internal resistance is given by
$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$
$⟹ \frac{1}{R} = \frac{R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}}{R_{1} R_{2} R_{3}}$

$R = \frac{R_{1} R_{2} R_{3}}{R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}}$

According to question, we need to take
$E_{e q} = E_{3}$
$E_{e q} = \frac{\sum \frac{E}{R}}{\sum \frac{1}{R}} = \frac{\frac{E_{1}}{R_{1}} + \frac{E_{2}}{R_{2}} + \frac{E_{3}}{R_{3}}}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}}$ ---(i)
$⟹ E_{3} = \frac{\frac{E_{1}}{R_{1}} + \frac{E_{2}}{R_{2}} + \frac{E_{3}}{R_{3}}}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}}$

$⟹ \frac{E_{3}}{R_{1}} + \frac{E_{3}}{R_{2}} + \frac{E_{3}}{R_{3}} = \frac{E_{1}}{R_{1}} + \frac{E_{2}}{R_{2}} + \frac{E_{3}}{R_{3}}$

$∴ E_{3} = \frac{E_{1} R_{2} + E_{2} R_{1}}{R_{1} + R_{2}}$

From equation (i) we can say that the equivalent emf of the battery depends not only on values of $E_{1}, E_{2}$ and $E_{3}$ but also depends on values of $R_{1}$ , $R_{2}$ and $R_{3}$ .

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