The Question is from neet-physics magnetic-effects-current-and-magnetism magnetic-moment-and-resultant-magnetic-moment, and it was created by the author NAGA SAI

Question

A thin wire of length 'L' is bent into a planar loop of irregular shape. A current 'I' flows through it. The loop is placed in a uniform magnetic field 'B' with its plane perpendicular to the field.  If the magnetic moment of a circular loop of same perimeter and carrying same current is $\mu_c$, the magnetic moment ($\mu$) of the irregular loop satisfies:

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
 **Primary Concept**: Magnetic moment of a current loop
 **Related Topics**:  Area enclosed by a loop,  circular loop, irregular loop
 **Prerequisites**: Understanding of magnetic moment ($\mu = IA$), where I is current and A is the area.
## SOLUTION PATHWAY
**Given Information**:
  - Length of wire: L
  - Current: I
  - Magnetic field: B (uniform and perpendicular to the loop plane)
  - Magnetic moment of circular loop with perimeter L and current I: $\mu_c$
  - Magnetic moment of irregular loop with perimeter L and current I: $\mu$
**Method & Steps**:
 1. Consider a circular loop with perimeter L. Its radius $r$ is given by $2\pi r = L$, so $r = \frac{L}{2\pi}$.
  Reasoning: The area of this circular loop is $A_c = \pi r^2 = \pi (\frac{L}{2\pi})^2 = \frac{L^2}{4\pi}$.
  Formula/Rule: Magnetic moment $\mu = IA$.
 2. The magnetic moment of the circular loop is $\mu_c = I A_c = I \frac{L^2}{4\pi}$.
  Working: This is the maximum area that can be enclosed by a loop of length L.
  Key Point:  For a given perimeter, the circle encloses the maximum area.
 3. Now consider the irregular loop. Since the perimeter is the same (L), the area enclosed by the irregular loop ($A$) will always be less than the area enclosed by the circular loop ($A_c$).  Therefore, $A < A_c$.
  Result: The magnetic moment of the irregular loop, $\mu = IA$, will be less than $\mu_c = IA_c$, because $A < A_c$.  So $\mu < \mu_c$.
  Verification: The inequality $A < A_c$ directly leads to $\mu < \mu_c$.
 ## OPTION ANALYSIS
A :x:  This would only be true if the irregular loop somehow had the same area as the circular loop, which is impossible with the same perimeter.
B :x: This is incorrect; the irregular loop always has a smaller area.
C ✓: This is the correct option. The circle maximizes the area for a given perimeter.
D :x: The relationship is not dependent on the shape; it's always $\mu < \mu_c$ because the area is always smaller for any irregular shape with the same perimeter.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Assuming the area of the irregular loop is equal to the area of the circular loop: Remember that for a fixed perimeter, a circle has the maximum area.
  2.Overlooking the relationship between area and perimeter: The area is a function of the perimeter, and the circle optimizes this relationship.
 **Quick Tips**:
  1.  Remember that a circle maximizes the area for a given perimeter. This is a key concept in geometry and has implications for various physics problems.
  2. Visualize different shapes with the same perimeter to understand the area differences.

NAGA SAI · Answer

$\mu = \mu_c$

NAGA SAI · Answer

$\mu > \mu_c$

NAGA SAI · Answer

The correct Option is "$\mu < \mu_c$"

NAGA SAI · Answer

The relationship between $\mu$ and $\mu_c$ depends on the shape of the irregular loop.

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