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Limitation of Dimensional Analysis

If the resist...

Question

If the resistances of values $R_{1}$ and $R_{2}$ are connected on both ends as shown in figure, current $I$ , flowing through resistance $R_{1}$ is given by

A

\frac{B l R_{2} (v_{1} r_{2} - v_{2} r_{1})}{R_{1} R_{2} (r_{1} + r_{2}) + r_{2} r_{1} (R_{1} + R_{2})}

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B

\frac{B l R_{2} (v_{1} r_{2} + v_{2} r_{1})}{R_{1} R_{2} (r_{1} + r_{2}) + r_{2} r_{1} (R_{1} + R_{2})}

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C

\frac{B l R_{2} (v_{1} r_{2} - v_{2} r_{1})}{R_{1} R_{2} (r_{1} - r_{2}) + r_{2} r_{1} (R_{1} + R_{2})}

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D

\frac{B l R_{2} (v_{1} r_{2} - v_{2} r_{1})}{R_{1} R_{2} (r_{1} + r_{2}) - r_{2} r_{1} (R_{1} + R_{2})}

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E

n o n e

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Solution

The correct option is A $\frac{B l R_{2} (v_{1} r_{2} - v_{2} r_{1})}{R_{1} R_{2} (r_{1} + r_{2}) + r_{2} r_{1} (R_{1} + R_{2})}$
As the conducting rods are moving in magnetic field with velocity $v$ in opposite direction, thus an $E M F$ is induced in each rod according to the Lenz law.
EMF induced in rod $M N$ $E_{1} = B v_{1} l$ .........(a)
EMF induced in rod $M^{'} N^{'}$ $E_{2} = B v_{2} l$ .......(b)

Combining the EMFs of the two rods, $\frac{1}{r_{e q}} = \frac{1}{r_{1}} + \frac{1}{r_{2}}$ $⟹ r_{e q} = \frac{r_{1} r_{2}}{r_{1} + r_{2}}$ .......(1)
Also equivalent EMF $E_{e q} = \frac{E_{2}}{r_{2}} r_{e q} - \frac{E_{1}}{r_{1}} r_{e q}$
Putting the values and on solving we get, $E_{e q} = \frac{E_{2} r_{1} - E_{1} r_{2}}{r_{1} + r_{2}}$ ............(2)
From Kirchhoff current law $i = i_{1} + i_{2}$
Using Nodal analysis $\frac{E_{e q} - V}{r_{e q}} = \frac{V - 0}{R_{1}} + \frac{V - 0}{R_{2}}$

$⟹ V = \frac{E_{e q}}{r_{e q}} \frac{R_{1} R_{2} r_{e q}}{[(R_{1} + R_{2}) r_{e q} + R_{1} R_{2}]}$ .............(3)

$∴$ $i_{1} = \frac{V}{R_{1}}$
From (1), (2) and (3), we get $i_{1} =$ $\frac{(E_{2} r_{1} - E_{1} r_{2})}{r_{1} r_{2}} \frac{R_{2} r_{1} r_{2}}{[(R_{1} + R_{2}) r_{1} r_{2} + (r_{1} + r_{2}) R_{1} R_{2}]}$

Using (a) and (b), $⟹ | i_{1} | =$ $\frac{| v_{2} r_{1} - v_{1} r_{2} | B l R_{2}}{[(R_{1} + R_{2}) r_{1} r_{2} + (r_{1} + r_{2}) R_{1} R_{2}]}$

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