There are two wires of same length and they are shaped as square and circle. Same current flows through them. If ratio of magnetic moment of square to that with circle is found to be `pi/n`. What is the value of n?

Explanation

To solve the problem, we need to find the ratio of the magnetic moments of a square and a circular wire of the same length, given that the same current flows through both wires. Let's break this down step by step.

### Step 1: Define the Length of the Wire
Let the length of the wire be \( L \).

### Step 2: Determine the Side Length of the Square
Since the wire is shaped into a square, the total length of the wire is equal to the perimeter of the square. The perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Thus, we can express the side length \( a \) of the square as:
\[ 4a = L \implies a = \frac{L}{4} \]

### Step 3: Calculate the Area of the Square
The area \( A_s \) of the square is given by:
\[ A_s = a^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \]

### Step 4: Calculate the Magnetic Moment of the Square
The magnetic moment \( M_s \) of a current-carrying loop is given by:
\[ M = I \times A \]
For the square, we have:
\[ M_s = I \times A_s = I \times \frac{L^2}{16} = \frac{IL^2}{16} \]

### Step 5: Determine the Radius of the Circle
For the circular wire, the length of the wire is equal to the circumference of the circle. The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Thus, we can express the radius \( r \) of the circle as:
\[ 2\pi r = L \implies r = \frac{L}{2\pi} \]

### Step 6: Calculate the Area of the Circle
The area \( A_c \) of the circle is given by:
\[ A_c = \pi r^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

### Step 7: Calculate the Magnetic Moment of the Circle
For the circle, the magnetic moment \( M_c \) is:
\[ M_c = I \times A_c = I \times \frac{L^2}{4\pi} = \frac{IL^2}{4\pi} \]

### Step 8: Find the Ratio of the Magnetic Moments
Now we can find the ratio of the magnetic moment of the square to that of the circle:
\[ \frac{M_s}{M_c} = \frac{\frac{IL^2}{16}}{\frac{IL^2}{4\pi}} = \frac{IL^2}{16} \times \frac{4\pi}{IL^2} = \frac{4\pi}{16} = \frac{\pi}{4} \]

### Step 9: Relate to Given Ratio
According to the problem, the ratio of the magnetic moment of the square to that of the circle is given as:
\[ \frac{\pi}{n} \]
From our calculation, we found that:
\[ \frac{M_s}{M_c} = \frac{\pi}{4} \]
Thus, we can equate:
\[ \frac{\pi}{n} = \frac{\pi}{4} \]

### Step 10: Solve for \( n \)
From the above equation, we can see that:
\[ n = 4 \]

### Final Answer
The value of \( n \) is \( 4 \).

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