Question

A surveyor is using a magnetic compass $\mathbf{6}\mathbf{.}\mathbf{1}\mathbf{}\mathbf{m}$ below a power line in which there is a steady current of $\mathbf{100}\mathbf{}\mathbf{A}$ .(a) What is the magnetic field at the site of the compass due to the power line? (b) Will this field interfere seriously with the compass reading? The horizontal component of Earth’s magnetic field at the site is role="math" localid="1663130453654" $\mathbf{20}\mathbf{mT}$ .

Short answer

a.$B=3.3\mu T$

b. It will affect the compass reading.

Step-by-step solution

Given Information

The radius is$r=6.1\text{m}$

The current is $i=100\text{A}$

Horizontal component of earth’s magnetic field at the site is $20\mu T.$

Determining the formulae

Consider equation of magnetic field due to current carrying wire at a point away from the wire at a distance r. Using this equation, calculate the magnetic field due to the wire and comparing it with the horizontal component of earth’s magnetic field, we can determine whether it will affect the compass reading or not.

Magnetic field$\left(\mathbf{B}\right)$ due to current$\left(\mathbf{i}\right)$ in the wire at any point at a distance r from the wire is: $\mathbf{B}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{i}}{\mathbf{2}\mathbf{\pi r}}$

Here, ${\mathbf{\mu }}_{\mathbf{0}}$ is the permeability of the free space.

(a) Calculating the magnetic field at the site of the compass due to the power line

Magnetic field $\left(B\right)$due to current $\left(i\right)$in the wire at a point and distance r from the wire is:

$B=\frac{{\mu }_{0}i}{2\pi r}$

$B=\frac{4\pi \times {10}^{-7}\times 100}{2\pi \times 6.1}=3.3\times {10}^{-6}T$

$B=3.3\mu T$

(b) Figuring out if this field will interfere seriously with the compass reading

Consider the horizontal component of earth’s magnetic field is$20\mu T$.The value we got is approximately one sixth of the magnetic field of earth.

Therefore, it will affect the compass reading.