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Series and Parallel Combination of Resistors

Two ideal bat...

Question

Two ideal batteries of emf $V_{1}$ and $V_{2}$ and three resistances $R_{1}$ , $R_{2}$ and $R_{3}$ are connected as shown in the figure. The current in resistance $R_{2}$ would be zero if

A

V_{1} = V_{2}

and

R_{1} = R_{2} = R_{3}

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B

V_{1} = V_{2}

and

R_{1} = 2 R_{2} = R_{3}

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C

V_{1} = 2 V_{2}

and

2 R_{1} = 2 R_{2} = R_{3}

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D

2 V_{1} = V_{2}

and

2 R_{1} = R_{2} = R_{3}

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Solution

The correct options are
A $V_{1} = V_{2}$ and $R_{1} = 2 R_{2} = R_{3}$
B $2 V_{1} = V_{2}$ and $2 R_{1} = R_{2} = R_{3}$
C $V_{1} = V_{2}$ and $R_{1} = R_{2} = R_{3}$
Writing Kirchoff's laws for the top and bottom loops ,
$V_{1} - i_{1} R_{1} - (i_{1} - i_{2}) R_{2} = 0$
$V_{2} - i_{2} R_{3} + (i_{1} - i_{2}) R_{2} = 0$
If Current through $R_{2}$ is zero , $i_{1} = i_{2}$
Substituting in the above equations, we get:
$\frac{V_{1}}{V_{2}} = \frac{R_{1}}{R_{3}}$
Substituting the values given in the options and verifying,
options A , B and D satisfy this condition.

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