A long straight wire of radius `a` carries a steady current `i`. The current is uniformly distributed across its cross section. The ratio of the magnetis field at `(a)//(2) and (2a)` is

Options

A. `( 1//2)`
B. `(1//4)`
C. `4`
D. `1`

Explanation

To solve the problem of finding the ratio of the magnetic field at distances \( \frac{a}{2} \) and \( 2a \) from a long straight wire carrying a steady current \( i \), we will follow these steps:

### Step 1: Determine the magnetic field at \( r = \frac{a}{2} \) (inside the wire)
For a point inside a long straight wire, the magnetic field \( B_1 \) can be calculated using the formula:

\[ B = \frac{\mu_0 I_{\text{enc}}}{2 \pi r} \]

Where:
- \( \mu_0 \) is the permeability of free space,
- \( I_{\text{enc}} \) is the current enclosed by the amperian loop of radius \( r \),
- \( r \) is the distance from the center of the wire.

The current density \( J \) is uniform across the cross-section, given by:

\[ J = \frac{I}{\pi a^2} \]

The enclosed current \( I_{\text{enc}} \) for a radius \( r = \frac{a}{2} \) is:

\[ I_{\text{enc}} = J \cdot \text{Area} = J \cdot \pi \left(\frac{a}{2}\right)^2 = \frac{I}{\pi a^2} \cdot \pi \left(\frac{a^2}{4}\right) = \frac{I}{4} \]

Now substituting \( I_{\text{enc}} \) into the magnetic field formula:

\[ B_1 = \frac{\mu_0 \cdot \frac{I}{4}}{2 \pi \cdot \frac{a}{2}} = \frac{\mu_0 I}{4 \cdot \pi} \cdot \frac{2}{a} = \frac{\mu_0 I}{2 \pi a} \]

### Step 2: Determine the magnetic field at \( r = 2a \) (outside the wire)
For a point outside the wire, the magnetic field \( B_2 \) is given by:

\[ B = \frac{\mu_0 I}{2 \pi r} \]

Substituting \( r = 2a \):

\[ B_2 = \frac{\mu_0 I}{2 \pi (2a)} = \frac{\mu_0 I}{4 \pi a} \]

### Step 3: Calculate the ratio \( \frac{B_1}{B_2} \)
Now, we can find the ratio of the magnetic fields:

\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 I}{2 \pi a}}{\frac{\mu_0 I}{4 \pi a}} = \frac{4 \pi a}{2 \pi a} = 2 \]

### Conclusion
The ratio of the magnetic field at \( \frac{a}{2} \) to that at \( 2a \) is:

\[ \frac{B_1}{B_2} = 2 \]