The strength of the magnetic field at a point `r` near a long straight current carrying wire is `B`. The field at a distance `r/2` will be

Options

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

Answer

To find the strength of the magnetic field at a distance \( \frac{r}{2} \) from a long straight current-carrying wire, we can use the formula for the magnetic field around a long straight wire, which is given by:

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

where:
- \( B \) is the magnetic field strength,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the wire,
- \( r \) is the distance from the wire.

### Step-by-Step Solution:

1. **Identify the magnetic field at distance \( r \)**:
Given that the magnetic field at a distance \( r \) from the wire is \( B \), we can express this as:
\[ B = \frac{\mu_0 I}{2 \pi r} \]

2. **Determine the magnetic field at distance \( \frac{r}{2} \)**:
We want to find the magnetic field at a distance \( \frac{r}{2} \). Using the formula for magnetic field:
\[ B' = \frac{\mu_0 I}{2 \pi \left(\frac{r}{2}\right)} = \frac{\mu_0 I}{2 \pi \cdot \frac{r}{2}} = \frac{\mu_0 I}{\pi r} \]

3. **Relate \( B' \) to \( B \)**:
Now we can relate \( B' \) to \( B \):
\[ B' = \frac{\mu_0 I}{\pi r} = 2 \cdot \frac{\mu_0 I}{2 \pi r} = 2B \]

4. **Conclusion**:
Therefore, the magnetic field at a distance \( \frac{r}{2} \) is:
\[ B' = 2B \]

### Final Answer:
The strength of the magnetic field at a distance \( \frac{r}{2} \) from the wire is \( 2B \).

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