Magnetic field at the centre ofa circular loop of area A is B. Then magnetic moment of the loop will:

Options

A. `BA^(2)/mu_(0)pi`
B. `BA/mu_(0) sqrt(A)`
C. `BA/mu_(0)pi sqrt(A)`
D. `2BA/mu_(0)sqrt(A/pi)`

Answer

To find the magnetic moment of a circular loop given the magnetic field at its center and its area, we can follow these steps:

### Step 1: Understand the relationship between magnetic field and magnetic moment
The magnetic field \( B \) at the center of a circular loop carrying a current \( I \) can be expressed as:
\[ B = \frac{\mu_0 I}{2R} \]
where \( R \) is the radius of the loop and \( \mu_0 \) is the permeability of free space.

### Step 2: Relate the area of the loop to its radius
The area \( A \) of the circular loop can be expressed as:
\[ A = \pi R^2 \]
From this equation, we can express \( R \) in terms of \( A \):
\[ R = \sqrt{\frac{A}{\pi}} \]

### Step 3: Substitute \( R \) into the magnetic field equation
Now, substitute \( R \) back into the equation for \( B \):
\[ B = \frac{\mu_0 I}{2 \sqrt{\frac{A}{\pi}}} \]
This simplifies to:
\[ B = \frac{\mu_0 I \sqrt{\pi}}{2 \sqrt{A}} \]

### Step 4: Solve for current \( I \)
Rearranging the equation to solve for \( I \):
\[ I = \frac{2B \sqrt{A}}{\mu_0 \sqrt{\pi}} \]

### Step 5: Calculate the magnetic moment \( m \)
The magnetic moment \( m \) of the loop is given by:
\[ m = I \cdot A \]
Substituting the expression for \( I \):
\[ m = \left(\frac{2B \sqrt{A}}{\mu_0 \sqrt{\pi}}\right) \cdot A \]
This simplifies to:
\[ m = \frac{2B A \sqrt{A}}{\mu_0 \sqrt{\pi}} \]

### Conclusion
Thus, the magnetic moment \( m \) of the circular loop in terms of the magnetic field \( B \) and area \( A \) is:
\[ m = \frac{2B A \sqrt{A}}{\mu_0 \sqrt{\pi}} \]