The ratio of the magnetic field at the centre of a current carrying circular coil to its magnetic moment is x. If the current and radius both are doubled the new ratio will become

Options

A. 2x
B. 4x
C. x/4
D. x/8

Answer

To solve the problem, we need to find the new ratio of the magnetic field at the center of a current-carrying circular coil to its magnetic moment when both the current and the radius are doubled.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field (B) at the Center of the Coil:**
The magnetic field at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 I}{2A} \]
where \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( A \) is the radius of the coil.

2. **Understanding the Magnetic Moment (M) of the Coil:**
The magnetic moment of a circular coil is given by:
\[ M = I \cdot \text{Area} = I \cdot \pi A^2 \]

3. **Finding the Initial Ratio (R):**
The initial ratio of the magnetic field to the magnetic moment is:
\[ R = \frac{B}{M} = \frac{\frac{\mu_0 I}{2A}}{I \cdot \pi A^2} \]
Simplifying this:
\[ R = \frac{\mu_0 I}{2A} \cdot \frac{1}{I \cdot \pi A^2} = \frac{\mu_0}{2\pi A^3} \]
Thus, the initial ratio is:
\[ R = \frac{\mu_0}{2\pi A^3} \]

4. **Doubling the Current and Radius:**
If the current \( I \) is doubled to \( 2I \) and the radius \( A \) is doubled to \( 2A \), we need to find the new magnetic field \( B' \) and magnetic moment \( M' \).

- New magnetic field:
\[ B' = \frac{\mu_0 (2I)}{2(2A)} = \frac{\mu_0 I}{2A} = B \]

- New magnetic moment:
\[ M' = (2I) \cdot \pi (2A)^2 = 2I \cdot \pi \cdot 4A^2 = 8I \cdot \pi A^2 \]

5. **Finding the New Ratio (R'):**
The new ratio \( R' \) is:
\[ R' = \frac{B'}{M'} = \frac{\frac{\mu_0 I}{2A}}{8I \cdot \pi A^2} \]
Simplifying this:
\[ R' = \frac{\mu_0 I}{2A} \cdot \frac{1}{8I \cdot \pi A^2} = \frac{\mu_0}{16\pi A^3} \]

6. **Relating the New Ratio to the Initial Ratio:**
We can express \( R' \) in terms of \( R \):
\[ R' = \frac{1}{8} \cdot \frac{\mu_0}{2\pi A^3} = \frac{R}{8} \]

### Conclusion:
The new ratio of the magnetic field at the center of a current-carrying circular coil to its magnetic moment when both the current and radius are doubled is:
\[ R' = \frac{R}{8} \]
Thus, if the initial ratio was \( x \), the new ratio will be:
\[ \frac{x}{8} \]