A wire carrying current I is bent in the shape ABCDEFA as shown, where rectangle ABCDA and ADEFA are perpendicular to each other. If the sides of the rectangles are of lengths a and b, then the magnitude and direction of magnetic moment of the loop ABCDEFA is:

Moderate

A

$\sqrt{2} a b I, a l o n g (\frac{\hat{j}}{\sqrt{2}} + \frac{\hat{k}}{\sqrt{2}})$
B

$a b I, a l o n g (\frac{\hat{j}}{\sqrt{5}} + \frac{2 \hat{k}}{\sqrt{5}})$
C

$a b I, a l o n g (\frac{\hat{j}}{\sqrt{2}} + \frac{\hat{k}}{\sqrt{2}})$
D

$\sqrt{2} a b I, a lo n g (\frac{\hat{j}}{\sqrt{5}} + \frac{2 \hat{k}}{\sqrt{5}})$

Solution

Magnetic moment of loop ABCDA
$\bar{M_{1}} = I a b \hat{k}$
Magnetic moment of loop ADEFA
$\bar{M_{2}} = I a b \hat{j}$
Net magnetic moment
$\bar{M_{n e t}} = \bar{M_{1}} + \bar{M_{2}}$
$\bar{M_{n e t}} = I a b (\hat{j} + \hat{k})$
$|\bar{M_{n e t}}| = \sqrt{2} I a b$
Unit vector along magnetic moment direction= $\frac{(\hat{j} + \hat{k})}{\sqrt{2}}$

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