A horizontal overheadpowerline is at height of `4 m` from the ground and carries a current of `100 A` from east to west. The magnetic field directly below it on the ground is
`( nu_(0) = 4 pi xx 10^(-7) Tm A^(-1)`

Options

A. `2.5 xx10^(-7)` T northward
B. `5.0 xx10^(-7)` T southward
C. `5.0 xx10^(-7)` T northward
D. `2.5 xx10^(-7)` T southward

Answer

To find the magnetic field directly below a horizontal overhead power line carrying a current, we can use the formula for the magnetic field due to a long straight conductor:

\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2I}{R} \]

Where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space, given as \( 4\pi \times 10^{-7} \, \text{Tm/A} \),
- \( I \) is the current in amperes,
- \( R \) is the distance from the wire to the point where the magnetic field is being calculated.

### Step-by-step Solution:

1. **Identify the parameters**:
- Height of the power line from the ground, \( h = 4 \, \text{m} \).
- Current in the power line, \( I = 100 \, \text{A} \).
- Permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A} \).

2. **Calculate the distance \( R \)**:
- Since the magnetic field is being calculated directly below the power line, the distance \( R \) is equal to the height of the power line from the ground:
\[ R = h = 4 \, \text{m} \]

3. **Substitute the values into the formula**:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2I}{R} \]
Substituting the known values:
\[ B = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 100}{4} \]

4. **Simplify the expression**:
- The \( 4\pi \) terms cancel out:
\[ B = 10^{-7} \cdot \frac{200}{4} \]
\[ B = 10^{-7} \cdot 50 \]
\[ B = 5 \times 10^{-6} \, \text{T} \]

5. **Determine the direction of the magnetic field**:
- According to the right-hand rule, if the current flows from east to west, the magnetic field will be directed towards the south.

### Final Answer:
The magnetic field directly below the power line on the ground is \( 5 \times 10^{-6} \, \text{T} \) directed towards the south.