A long straight vertical conductor carries a current of 8A in the upward direction. What the magnitude of the resultant magnetic induction at a point in the horizonatal plane at a distance of 4 cm from the conductor towards South? (The horizontal compo-nent of earth's magnetic induction `=4xx10^(-5)T`)

Options

A. `2sqrt2xx10^(-5)T`
B. `4sqrt2xx10^(-5)T`
C. `4xx10^(-5)T`
D. `2xx10^(-5)T`

Answer

To solve the problem of finding the resultant magnetic induction at a point in the horizontal plane at a distance of 4 cm from a long straight vertical conductor carrying a current of 8 A, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Current (I) = 8 A (upward direction)
- Distance from the conductor (d) = 4 cm = 0.04 m (convert to meters)
- Horizontal component of Earth's magnetic induction (B2) = 4 × 10^(-5) T (towards the south)

2. **Determine the Magnetic Field (B1) due to the Conductor:**
The magnetic field (B1) at a distance d from a long straight conductor carrying current I is given by the formula:
\[ B_1 = \frac{\mu_0 I}{2 \pi d} \]
where \(\mu_0\) (permeability of free space) = \(4 \pi \times 10^{-7} \, \text{T m/A}\).

3. **Substitute the Values into the Formula:**
\[ B_1 = \frac{(4 \pi \times 10^{-7}) \times 8}{2 \pi \times 0.04} \]
Simplifying this:
\[ B_1 = \frac{(4 \times 8) \times 10^{-7}}{2 \times 0.04} \]
\[ B_1 = \frac{32 \times 10^{-7}}{0.08} \]
\[ B_1 = 4 \times 10^{-5} \, \text{T} \]

4. **Calculate the Resultant Magnetic Induction (B_net):**
The resultant magnetic induction (B_net) at the point is the vector sum of B1 and B2. Since B1 is directed towards the west (due to the right-hand rule) and B2 is directed towards the south, we can use the Pythagorean theorem:
\[ B_{\text{net}} = \sqrt{B_1^2 + B_2^2} \]
Substituting the values:
\[ B_{\text{net}} = \sqrt{(4 \times 10^{-5})^2 + (4 \times 10^{-5})^2} \]
\[ B_{\text{net}} = \sqrt{2 \times (4 \times 10^{-5})^2} \]
\[ B_{\text{net}} = 4 \sqrt{2} \times 10^{-5} \, \text{T} \]

5. **Final Result:**
The magnitude of the resultant magnetic induction at the specified point is:
\[ B_{\text{net}} = 4 \sqrt{2} \times 10^{-5} \, \text{T} \]