The Question is from neet-physics magnetic-effects-current-and-magnetism , and it was created by the author SUBHANI

Question

Two identical magnetic dipoles of magnetic moment 'm' are placed at a distance 'd' apart with their axes perpendicular to each other. The resultant magnetic moment of the combination is:

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
 **Primary Concept**: Vector Addition of Magnetic Moments
 **Related Topics**: Magnetic dipoles, vectors
 **Prerequisites**: Understanding of vector addition, particularly in two dimensions.
## SOLUTION PATHWAY
**Given Information**:
  Two magnetic dipoles with magnetic moment $\vec{m_1} = \vec{m}$ and $\vec{m_2} = \vec{m}$.
  The dipoles are perpendicular to each other.
  The distance between them is 'd' (this information is irrelevant to the resultant magnetic moment).
**Method & Steps**:
 1. Represent the magnetic moments as vectors. Since they are identical, their magnitudes are equal to 'm'.
  Reasoning: Magnetic moment is a vector quantity, possessing both magnitude and direction.
  Formula/Rule: None specifically needed for this step.
 2. Since the axes are perpendicular, we can represent the vectors as $\vec{m_1} = m\hat{i}$ and $\vec{m_2} = m\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along the x and y axes respectively.
  Working: The resultant magnetic moment $\vec{M}$ is given by the vector sum: $\vec{M} = \vec{m_1} + \vec{m_2} = m\hat{i} + m\hat{j}$
  Key Point: The direction of the resultant vector is important.  We are considering only the magnitude in this case.
 3. Calculate the magnitude of the resultant magnetic moment.
  Result: $|\vec{M}| = \sqrt{(m)^2 + (m)^2} = \sqrt{2m^2} = m\sqrt{2}$
  Verification: The Pythagorean theorem is used to find the magnitude of the resultant vector in a two-dimensional space.
 ## OPTION ANALYSIS
[A] :x: $2m$: This would be correct if the dipoles were parallel.
[B] :x: $m$: This would be correct if one dipole's moment were zero.
[C] ✓: $m\sqrt{2}$: Correctly applies vector addition for perpendicular vectors.
[D] :x: $m/\sqrt{2}$: This would be incorrect; it's not a valid outcome of vector addition in this case.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Treating magnetic moments as scalar quantities instead of vectors: Remember that magnetic moment has both magnitude and direction.
  2.Incorrectly applying vector addition:  Ensure you understand how to add vectors, especially in two or three dimensions.
 **Quick Tips**:
  1.Visualize the vectors: Drawing a diagram can help understand the vector addition.
  2.Remember the Pythagorean theorem: It's crucial for calculating the magnitude of the resultant vector when vectors are perpendicular.

SUBHANI · Answer

$2m$

SUBHANI · Answer

$m$

SUBHANI · Answer

The correct Option is "$m\sqrt{2}$"

SUBHANI · Answer

$m/\sqrt{2}$

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