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 R 1 R 2 R 3- R 1 R 2- R 2 R 3- R 3 R 1 -B- If E 3- E 1 R 2- E 2 R 1- R 1- R 2- the equivalent emf of the batteries will be equal to E 3-C- The equivalent e-m-f- of the battery is equal to - E 1- E 2- E 3-3D- The equivalent e-m-f- of the battery not only depends upon values of E1- E2 and E3 but also depends upon values of R 1- R 2 and R 3-

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Byju's Answer

Standard XII

Physics

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?

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

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

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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 options are
A The equivalent internal resistance $R$ of the system is given by $\frac{(R_{1} R_{2} R_{3})}{(R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}) .}$
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 $E_{3}$ .
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 $R_{3}$ .

Suggest Corrections

Similar questions

Q. 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?

Q. In the circuit shown in figure, emf of the batteries are

E_{1} = 3 V, E_{2} = 2 V, E_{3} = 1 V

and their internal resistances are

R = r_{1} = r_{2} = r_{3} = 1 Ω

. The potential difference between the points

A

and

B

, and the current through branch containing resistor

r_{2}

will be respectively

Q. Two resistances

R_{1}

and

R_{2}

are joined as shown in the figure to two batteries of e.m.f.

E_{1}

and

E_{2}

. If

E_{2}

is short-circuited, the current through

R_{1}

Q. In the circuit shown in Figure,

E_{1} = E_{2} = E_{3} = 2 V

and

R_{1} = R_{2} = 4 Ω

. The current flowing between points

A

and

B

through battery

E_{2}

A single battery fo emf E and internal resistance r is equivalent to a parallel combination of two batteries of emfs $E_{1} = 2 V$ and $E_{2} = 1.5 V$ and internal resistances $r_{1} = 1 Ω$ and $r_{2} = 2 Ω$ respectively, with polarities as shown in the figure.

(IIT-JEE 1997)

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For the batteries shown in the figure, R1,R2, and R3 are the internal resistances of E1,E2, and E3 respectively. Then, which of the following is/are correct?

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?