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 solve the problem, we need to determine the strength of the magnetic field at a distance \( \frac{r}{2} \) from a long straight current-carrying wire, given that the magnetic field at a distance \( r \) is \( B \).

### Step-by-Step Solution:

1. **Understand the Formula for Magnetic Field**:
The magnetic field \( B \) around a long straight current-carrying wire is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
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.

2. **Determine the Magnetic Field at Distance \( r \)**:
From the problem, we know that at distance \( r \), the magnetic field is \( B \):
\[ B = \frac{\mu_0 I}{2 \pi r} \]

3. **Calculate the Magnetic Field at Distance \( \frac{r}{2} \)**:
Now, we need to find the magnetic field at a distance \( \frac{r}{2} \):
\[ B' = \frac{\mu_0 I}{2 \pi \left(\frac{r}{2}\right)} \]
Simplifying this expression:
\[ B' = \frac{\mu_0 I}{2 \pi \cdot \frac{r}{2}} = \frac{\mu_0 I \cdot 2}{2 \pi r} = \frac{2 \mu_0 I}{2 \pi r} \]
Thus, we can express \( B' \) in terms of \( B \):
\[ B' = 2 \left(\frac{\mu_0 I}{2 \pi r}\right) = 2B \]

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

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

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