Magnetic field at the centre of a circualr loop of area A carrying current I is B. What is the magnetic moment of this loop?

Options

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

Answer

To find the magnetic moment of a circular loop carrying current, we can follow these steps:

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

### Step 2: Relate area and radius
The area \( A \) of the circular loop is given by:
\[ A = \pi R^2 \]
From this, we can express the radius \( R \) in terms of the area \( A \):
\[ R = \sqrt{\frac{A}{\pi}} \]

### Step 3: Substitute \( R \) into the magnetic field equation
Substituting the expression for \( R \) into the magnetic field equation, we have:
\[ B = \frac{\mu_0 I}{2\sqrt{\frac{A}{\pi}}} \]
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
The magnetic moment \( M \) of the loop is given by the product of the current \( I \) and the area \( A \):
\[ M = I \cdot A \]
Substituting the expression for \( I \):
\[ M = \left(\frac{2B \sqrt{A}}{\mu_0 \sqrt{\pi}}\right) \cdot A \]
This simplifies to:
\[ M = \frac{2B A \sqrt{A}}{\mu_0 \sqrt{\pi}} \]

### Final Answer
Thus, the magnetic moment \( M \) of the circular loop is:
\[ M = \frac{2B A \sqrt{A}}{\mu_0 \sqrt{\pi}} \]