A circular coil of wire n turns has a radius r and carries a current I . Its magnetic dipole moment is M .Now the coil is unwound and again rewound into a circular coil of half the initial radius and the same current is passed through it, then the dipole moment of this new coil is 1

Options

A. M/2
B. M/4
C. M
D. 2M

Explanation

To find the magnetic dipole moment of the new coil after it has been rewound with half the radius, we can follow these steps:

### Step 1: Understand the formula for magnetic dipole moment
The magnetic dipole moment \( M \) of a circular coil is given by the formula:
\[ M = n \cdot I \cdot A \]
where \( n \) is the number of turns, \( I \) is the current flowing through the coil, and \( A \) is the area of the coil.

### Step 2: Calculate the area of the initial coil
The area \( A \) of a circular coil with radius \( r \) is given by:
\[ A = \pi r^2 \]
Thus, the magnetic dipole moment of the initial coil can be expressed as:
\[ M = n \cdot I \cdot \pi r^2 \]

### Step 3: Determine the new radius and area of the coil
When the coil is rewound into a circular coil of half the initial radius, the new radius \( r' \) is:
\[ r' = \frac{r}{2} \]
Now, we can calculate the area \( A' \) of the new coil:
\[ A' = \pi (r')^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \cdot \frac{r^2}{4} = \frac{\pi r^2}{4} \]

### Step 4: Calculate the magnetic dipole moment of the new coil
Using the new area in the formula for magnetic dipole moment, we have:
\[ M' = n \cdot I \cdot A' = n \cdot I \cdot \frac{\pi r^2}{4} \]

### Step 5: Relate the new dipole moment to the initial dipole moment
Now, we can express \( M' \) in terms of \( M \):
\[ M' = n \cdot I \cdot \frac{\pi r^2}{4} = \frac{1}{4} (n \cdot I \cdot \pi r^2) = \frac{M}{4} \]

### Conclusion
Thus, the magnetic dipole moment of the new coil is:
\[ M' = \frac{M}{4} \]

### Final Answer
The dipole moment of the new coil is \( \frac{M}{4} \).

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