To determine the distance from a power cable at which the magnetic field produced by the current is equal to the Earth's magnetic field, we can use the formula for the magnetic field around a long, straight current-carrying wire. This formula is derived from Ampère's law and is given by:
[tex]\[ B = \frac{\mu_0 I}{2 \pi r} \][/tex]
Where:
- \( B \) is the magnetic field
- \( \mu_0 \) is the permeability of free space (\(4 \pi \times 10^{-7} \, \text{T m/A}\))
- \( I \) is the current in the wire
- \( r \) is the distance from the wire
We are given the following values:
- Current, \( I = 100 \, \text{A} \)
- Earth's magnetic field, \( B = 5.00 \times 10^{-5} \, \text{T} \)
We need to solve for the distance \( r \). To do this, we rearrange the formula to solve for \( r \):
[tex]\[ r = \frac{\mu_0 I}{2 \pi B} \][/tex]
Substitute the given values into the equation:
[tex]\[ r = \frac{(4 \pi \times 10^{-7}) \times 100}{2 \pi \times 5.00 \times 10^{-5}} \][/tex]
We can simplify the expression step by step:
1. First, calculate the numerator:
[tex]\[ \mu_0 I = (4 \pi \times 10^{-7}) \times 100 = 4 \pi \times 10^{-5} \, \text{Tm/A} \][/tex]
2. Next, calculate the denominator:
[tex]\[ 2 \pi B = 2 \pi \times 5.00 \times 10^{-5} = 10 \pi \times 10^{-5} \, \text{T} \][/tex]
3. Divide the numerator by the denominator:
[tex]\[ r = \frac{4 \pi \times 10^{-5}}{10 \pi \times 10^{-5}} \][/tex]
[tex]\[ r = \frac{4}{10} = 0.4 \, \text{m} \][/tex]
Therefore, the distance from the wire at which the magnetic field is equal to that of the Earth is [tex]\( 0.4 \, \text{m} \)[/tex].
