A horizontal overhead powerline 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 xx 10^(-7)` T southward
B. `5 xx 10^(-6) T` northward
C. `5 xx 10^(-6) T` southward
D. `2.5 xx 10^(-7)` northward

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 current-carrying wire. The formula is given by:

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

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A} \)),
- \( I \) is the current (in Amperes),
- \( r \) is the distance from the wire to the point where the magnetic field is being measured (in meters).

### Step 1: Identify the parameters
- The current \( I = 100 \, \text{A} \).
- The height of the power line from the ground \( h = 4 \, \text{m} \) (this is also the distance \( r \) from the wire to the point directly below it).

### Step 2: Substitute the values into the formula
Using the formula, we can substitute the values:

\[ B = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 100}{4} \]

### Step 3: Simplify the expression
The \( 4\pi \) in the numerator and denominator cancels out:

\[ B = 10^{-7} \cdot \frac{200}{4} \]

Calculating \( \frac{200}{4} \):

\[ \frac{200}{4} = 50 \]

So we have:

\[ B = 10^{-7} \cdot 50 = 5 \times 10^{-6} \, \text{T} \]

### Step 4: Determine the direction of the magnetic field
To find the direction of the magnetic field, we can use the right-hand thumb rule. Point your thumb in the direction of the current (from east to west), and your fingers will curl around the wire. The direction your fingers curl will indicate the direction of the magnetic field.

Since the current is flowing from east to west, the magnetic field directly below the wire will be directed southward.

### Final Answer
The magnetic field directly below the power line on the ground is:

\[ B = 5 \times 10^{-6} \, \text{T} \quad \text{(directed southward)} \]

---