The Question is from neet-physics magnetic-effects-current-and-magnetism , and it was created by the author Mr. ANIL KUMAR.G

Question

A magnetic dipole of moment $M$ is aligned at an angle $	heta$ with a uniform magnetic field $B$. The work done in rotating it by an angle $d	heta$ is:

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Work done by a torque on a magnetic dipole
	**Related Topics**: Magnetic dipole moment, Torque on a magnetic dipole, Work-energy theorem
	**Prerequisites**: Understanding of torque on a magnetic dipole in a uniform magnetic field, Definition of work in rotational motion
## SOLUTION PATHWAY
**Given Information**:
	* Magnetic dipole moment: $M$
	* Uniform magnetic field: $B$
	* Angle between $M$ and $B$: $	heta$
	* Small change in angle: $d	heta$
**Method & Steps**:
	1.	The torque on a magnetic dipole in a magnetic field is given by the cross product of the magnetic moment and the magnetic field.
		Reasoning: The torque tends to align the dipole with the magnetic field.
		Formula/Rule: $\vec{	au} = \vec{M} 	imes \vec{B} = MB\sin(	heta)$
	2.	The work done by the torque to rotate the dipole by a small angle $d	heta$ is given by the product of the torque and the angular displacement.
		Working: $dW = 	au d	heta = MB\sin(	heta)d	heta$
		Key Point: The work done by the torque is the change in potential energy of the dipole in the magnetic field.
	3.	The work done in rotating the magnetic dipole by an angle $d	heta$ is $MB\sin(	heta)d	heta$.
		Result: $dW = MB\sin(	heta)d	heta$
		Verification: The derived result matches option B
 ## OPTION ANALYSIS
[A] :x: This option has cos($	heta$) instead of sin($	heta$). The torque and the work done are related to sin($	heta$)
[B] ✓: This is the correct expression for the work done.
[C] :x: This option has a negative sign and cos($	heta$). Work done is always positive when the dipole is being rotated by the torque in the direction of rotation.
[D] :x: This option has a negative sign. Work done is always positive when the dipole is being rotated by the torque in the direction of rotation.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing between sine and cosine in the torque formula. Remember that torque is a cross product.
		2.[Error 2]: Forgetting the direction of the torque and considering the work done to be negative. The work done is positive if the rotation is in the direction of the torque.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that the torque and potential energy are related to $\sin(	heta)$ and $\cos(	heta)$, respectively.
		2.[Memory aid]: Visualize the magnetic dipole rotating in the magnetic field to remember the direction of the torque.

Mr. ANIL KUMAR.G · Answer

MBcosθ dθ

Mr. ANIL KUMAR.G · Answer

The correct Option is "MBsinθ dθ"

Mr. ANIL KUMAR.G · Answer

-MBcosθ dθ

Mr. ANIL KUMAR.G · Answer

-MBsinθ dθ

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