The magnetic field due to current carrying circular coil loop of radius 6 cm at a point on axis at a distance of 8 cm from the centre is `54 muT` . What is the value at the centre of loop ?

Options

A. `75 muT`
B. `125 muT`
C. `150 muT`
D. `250 muT`

Answer

To find the magnetic field at the center of a circular current-carrying coil loop, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The radius of the circular coil (r) = 6 cm = 0.06 m (convert to meters for standard SI units)
- The distance from the center to the point on the axis (x) = 8 cm = 0.08 m
- The magnetic field at the point on the axis (B) = 54 µT = 54 x 10^-6 T

### Step 2: Use the formula for the magnetic field at a point on the axis of a circular loop
The formula for the magnetic field (B) at a distance x from the center of a circular loop of radius r carrying current I is given by:

\[ B = \frac{\mu_0 I r^2}{2 (x^2 + r^2)^{3/2}} \]

Where:
- \(\mu_0\) is the permeability of free space (\(\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}\))

### Step 3: Rearrange the formula to find the current (I)
From the information given, we can rearrange the formula to solve for the current I:

\[ 54 \times 10^{-6} = \frac{(4\pi \times 10^{-7}) I (0.06)^2}{2 ((0.08)^2 + (0.06)^2)^{3/2}} \]

### Step 4: Calculate the denominator
First, calculate \((0.08)^2 + (0.06)^2\):

\[ (0.08)^2 = 0.0064 \quad \text{and} \quad (0.06)^2 = 0.0036 \]

\[ (0.08)^2 + (0.06)^2 = 0.0064 + 0.0036 = 0.0100 \]

Now, calculate \((0.0100)^{3/2}\):

\[ (0.0100)^{3/2} = (0.01)^{3/2} = 0.001 \quad \text{(since } 0.01 = 10^{-2}\text{)} \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:

\[ 54 \times 10^{-6} = \frac{(4\pi \times 10^{-7}) I (0.06)^2}{2 \times 0.001} \]

### Step 6: Solve for I
Now we can solve for I:

\[ 54 \times 10^{-6} = \frac{(4\pi \times 10^{-7}) I (0.0036)}{0.002} \]

Rearranging gives:

\[ I = \frac{54 \times 10^{-6} \times 0.002}{(4\pi \times 10^{-7}) \times 0.0036} \]

### Step 7: Calculate the magnetic field at the center of the loop
The magnetic field at the center of the loop is given by:

\[ B_{center} = \frac{\mu_0 I}{2r} \]

Substituting the value of I we found into this equation will give us the magnetic field at the center.

### Step 8: Final calculation
After substituting and simplifying, we find:

\[ B_{center} = \frac{(4\pi \times 10^{-7}) I}{2 \times 0.06} \]

Finally, substituting the value of I calculated from the previous steps will yield:

\[ B_{center} = 250 \, \mu T \]

### Conclusion
Thus, the magnetic field at the center of the loop is **250 µT**.