At a distance of `10 cm` from a long straight wire carrying current, the magnetic field is `0.04 T`. At the distance of `40 cm`, the magnetic field will be

Options

A. `0.01T`
B. `0.02 T`
C. `0.08 T`
D. `0.16 T`

Explanation

To solve the problem, we will use the formula for the magnetic field around a long straight wire carrying current, which is given by:

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

Where:
- \( B \) is the magnetic field,
- \( \mu_0 \) 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 Given Values:**
- At a distance \( r_1 = 10 \, \text{cm} = 0.1 \, \text{m} \), the magnetic field \( B_1 = 0.04 \, \text{T} \).
- We need to find the magnetic field \( B_2 \) at a distance \( r_2 = 40 \, \text{cm} = 0.4 \, \text{m} \).

2. **Understand the Relationship:**
- The magnetic field \( B \) is inversely proportional to the distance \( r \) from the wire. This means:
\[ B \propto \frac{1}{r} \]

3. **Set Up the Ratio:**
- Using the inverse proportionality, we can set up the following ratio:
\[ \frac{B_1}{B_2} = \frac{r_2}{r_1} \]

4. **Substitute the Known Values:**
- Substitute \( B_1 = 0.04 \, \text{T} \), \( r_1 = 0.1 \, \text{m} \), and \( r_2 = 0.4 \, \text{m} \):
\[ \frac{0.04}{B_2} = \frac{0.4}{0.1} \]

5. **Calculate the Right Side:**
- Calculate the right side:
\[ \frac{0.4}{0.1} = 4 \]

6. **Rearrange to Solve for \( B_2 \):**
- Rearranging gives us:
\[ B_2 = \frac{0.04}{4} \]

7. **Perform the Calculation:**
- Calculate \( B_2 \):
\[ B_2 = 0.01 \, \text{T} \]

### Final Answer:
The magnetic field at a distance of 40 cm from the wire will be \( 0.01 \, \text{T} \).

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