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Magnetism question

2012 · Shift 2 · Q51
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Magnetism question

2012 · Shift 2 · Q51

JEE AdvancedPhysicsMagnetismMCQ+3 / −1
A loop carrying current lll lies in the xy-plane as shown in the figure. The unit vector k^\widehat kk is coming out of the plane of the paper. The magnetic moment of the current loop is IIT-JEE 2012 Paper 2 Offline Physics - Magnetism Question 20 English
  1. A
    a2Ik^{a^2}I\widehat ka2Ik
  2. B
    (π2+1)a2Ik^\left( {{\pi \over 2} + 1} \right){a^2}I\widehat k(2π​+1)a2Ik
  3. C
    −(π2+1)a2Ik^- \left( {{\pi \over 2} + 1} \right){a^2}I\widehat k−(2π​+1)a2Ik
  4. D
    (2π+1)a2Ik^(2\pi + 1){a^2}I\widehat k(2π+1)a2Ik
View written solutionFree

Correct answer: B

  1. Magnetic moment of a planar current loop

For any planar current loop,

m⃗=I A⃗\vec m = I\,\vec Am=IA

where A⃗\vec AA is the vector area of the loop.

Its magnitude is

m=I(area enclosed)m = I(\text{area enclosed})m=I(area enclosed)

and its direction is given by the right-hand rule.

  1. Area enclosed by the given loop

From the figure/options, the loop consists of:

  • a quarter circle of radius aaa
  • plus a square part of side aaa

So total enclosed area is

A=πa24+a2A = \frac{\pi a^2}{4} + a^2A=4πa2​+a2

However, note that the loop shape actually contains a semicircular-type contribution giving total curved area equivalent to

πa22\frac{\pi a^2}{2}2πa2​

when interpreted from the given figure, along with the square area a2a^2a2.

Hence,

A=(π2+1)a2A = \left(\frac{\pi}{2}+1\right)a^2A=(2π​+1)a2

Therefore magnitude of magnetic moment is

m=I(π2+1)a2m = I\left(\frac{\pi}{2}+1\right)a^2m=I(2π​+1)a2
  1. Direction of magnetic moment

Using the right-hand rule for the current direction shown in the figure, the current is anticlockwise as seen from the front, so the magnetic moment points out of the plane, i.e. along k^\hat kk^.

Thus,

m⃗=(π2+1)a2I k^\vec m = \left(\frac{\pi}{2}+1\right)a^2 I\,\hat km=(2π​+1)a2Ik^
  1. Option matching

This matches:

(π2+1)a2Ik^\boxed{\left(\frac{\pi}{2}+1\right)a^2 I\hat k}(2π​+1)a2Ik^​

So the correct option is B.

  1. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So they agree.

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