A wire carrying current $l$ is bent in the shape $A B C D E F A$ as shown, where rectangle $A B C D A$ and $A D E F A$ are perpendicular to each other. If the sides of the rectangles are of length $a$ and $b$ , then the magnitude and direction of magnetic moment of the loop $A B C D E F A$ is

a b l, along (\frac{^j}{\sqrt{2}} + \frac{^k}{\sqrt{2}})

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\sqrt{2} a b l, along (\frac{^j}{\sqrt{2}} + \frac{^k}{\sqrt{2}})

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\sqrt{2} a b l, along (\frac{^j}{\sqrt{5}} + \frac{2^k}{\sqrt{5}})

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a b l, along (\frac{^j}{\sqrt{5}} + \frac{2^k}{\sqrt{5}})

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Solution

The correct option is B $\sqrt{2} a b l, along (\frac{^j}{\sqrt{2}} + \frac{^k}{\sqrt{2}})$
Magnetic moment of loop $A B C D$ ,
$M_{1} = area of loop \times current$
$_{1} = (a b l) (^j) (Here, a b = area of rectangle)$
Magnetic moment of loop $D E F A$ ,
$_{2} = (a b l) (^i)$
Net magnetic moment,
$- \to M =_{1} +_{2} \Rightarrow - \to M = a b l (^i +^j)$
$\Rightarrow | - \to M | = \sqrt{2} a b l (\frac{^j}{\sqrt{2}} + \frac{^k}{\sqrt{2}})$

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