A circular loop has a radius of `5 cm` and it is carrying a current of `0.1 amp`. It magneitc moment is

Options

A. (a)`1.32xx10^(-4)amp-m^(2)`
B. (b)`2.62xx10^(-4)amp-m^(2)`
C. (c)`5.25xx10^(-4)amp-m^(2)`
D. (d)`7.85xx10^(-4)amp-m^(2)`

Answer

To find the magnetic moment of a circular loop carrying a current, we can use the formula:

\[ \text{Magnetic Moment} (M) = I \times A \]

where \(I\) is the current in amperes and \(A\) is the area of the circular loop in square meters.

### Step 1: Convert the radius from centimeters to meters
Given the radius \(R = 5 \, \text{cm}\):

\[ R = \frac{5}{100} = 0.05 \, \text{m} \]

### Step 2: Calculate the area of the circular loop
The area \(A\) of a circle is given by the formula:

\[ A = \pi R^2 \]

Substituting the value of \(R\):

\[ A = \pi (0.05)^2 \]

Calculating \(0.05^2\):

\[ 0.05^2 = 0.0025 \]

Now substituting this value into the area formula:

\[ A = \pi \times 0.0025 \]

Using \(\pi \approx \frac{22}{7}\):

\[ A \approx \frac{22}{7} \times 0.0025 \approx \frac{22 \times 0.0025}{7} \approx \frac{0.055}{7} \approx 0.007857 \, \text{m}^2 \]

### Step 3: Calculate the magnetic moment
Now we can substitute the values of current \(I\) and area \(A\) into the magnetic moment formula:

Given \(I = 0.1 \, \text{A}\):

\[ M = I \times A = 0.1 \times 0.007857 \]

Calculating this:

\[ M = 0.1 \times 0.007857 \approx 0.0007857 \, \text{A m}^2 \]

### Step 4: Convert to standard form
To express this in scientific notation:

\[ M \approx 7.857 \times 10^{-4} \, \text{A m}^2 \]

### Final Answer
Thus, the magnetic moment of the circular loop is:

\[ M \approx 7.85 \times 10^{-4} \, \text{A m}^2 \]