A long straight wire in the horizontal plane carries as curret of 75 A in north to south direction, magnitude and direction of field B at a point 3 m east of the wire is

Options

A. `4xx10^(-6)T`, vertical up
B. `5xx10^(-6)T`, vertical down
C. `5xx10^(-6)T`, vertical up
D. `4xx10^(-6)T`, vertical down

Explanation

To solve the problem of finding the magnitude and direction of the magnetic field \( B \) at a point 3 m east of a long straight wire carrying a current of 75 A in the north to south direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Current Direction**:
The wire carries a current of 75 A from north to south.

2. **Determine the Position of the Point**:
The point where we want to find the magnetic field is located 3 m east of the wire.

3. **Use the Right-Hand Thumb Rule**:
To find the direction of the magnetic field around a straight current-carrying wire, we can use the right-hand thumb rule. If you point your thumb in the direction of the current (south), your fingers will curl around the wire in the direction of the magnetic field lines.

4. **Determine the Magnetic Field Direction**:
At the point 3 m east of the wire, the fingers of your right hand will point upwards (out of the plane) when your thumb points south. Therefore, the direction of the magnetic field at this point is vertically upward.

5. **Calculate the Magnitude of the Magnetic Field**:
The formula for the magnetic field \( B \) around a long straight wire is given by:
\[ B = \frac{\mu_0 I}{2 \pi R} \]
where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) (permeability of free space),
- \( I = 75 \, \text{A} \) (current),
- \( R = 3 \, \text{m} \) (distance from the wire).

6. **Substituting the Values**:
\[ B = \frac{(4\pi \times 10^{-7}) \times 75}{2\pi \times 3} \]
Simplifying this:
\[ B = \frac{4 \times 75 \times 10^{-7}}{2 \times 3} = \frac{300 \times 10^{-7}}{6} = 50 \times 10^{-7} \, \text{T} \]
This can be rewritten as:
\[ B = 5 \times 10^{-6} \, \text{T} \]

7. **Final Result**:
The magnitude of the magnetic field \( B \) at the point 3 m east of the wire is \( 5 \times 10^{-6} \, \text{T} \) and the direction is vertically upward.