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A rectangular...

Question

A rectangular loop with a sliding conductor of length $l$ is located in a uniform magnetic field perpendicular to the plane of the loop (figure). The magnetic induction is $B$ . The conductor has a resistance $R$ . The sides $A B$ and $C D$ have resistances $R_{1}$ and $R_{2}$ respectively. Find the current through the conductor during its motion to the right with a constant velocity $v$ .

A

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

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B

\frac{B l^{2} v}{R_{1} + R_{1} R_{2}}

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C

\frac{B l v (R_{1} + R_{2})}{R_{1} R_{2} + R (R_{1} + R_{2})}

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D

\frac{B l^{2} v (R_{1} + R_{2})}{R_{1} R_{2} + R (R_{1} + R_{2})}

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Solution

The correct option is A $\frac{B l v (R_{1} + R_{2})}{R_{1} R_{2} + R (R_{1} + R_{2})}$
$R_{1} a n d R_{2} a r e i n p a r a l l e l t h e t h i s c o m b i n a t i o n i s i n s e r i e s w i t h R .$
$T h u s R e s i s t a n c e = \frac{R_{1} R_{2}}{R_{1} + R_{2}} + R$
$E M F = B l v$
$i = \frac{B l v}{\frac{R_{1} R_{2}}{R_{1} + R_{2}} + R} = \frac{B l v (R_{1} + R_{2})}{R (R_{1} + R_{2}) + R_{1} R_{2}}$

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