Question

If the earth's magnetic field has a magnitude 3.4 × 10−5 T at the magnetic equator of the earth, what would be its value at the earth's geomagnetic poles?

Answer 1

Explanation:

Step 1:

Given data in the question  

The magnetic field on the equator,  $B=3.4 \times 10^{-5} T$

Let M be the magnetic moment of the magnetic dipole on Earth, and let R be the distance of the observation point from the middle of the magnetic dipole on Earth.

As the point on the magnetic equator is the equatorial position of the Earth's magnet, the equatorial origin of the magnetic field is

$\begin{aligned}B &=\frac{\mu_{0} M}{4 \pi R^{3}} \\\frac{\mu_{0} M}{4 \pi R^{3}} &=3.4 \times 10^{-5}\end{aligned}$

$\begin{aligned}M=& \frac{\left(3.4 \times 10^{-5} \times R^{3} \times 4 \pi\right)}{4 \pi \times 10^{-7}} \\M=& 3.4 \times 10^{2} R^{3}\end{aligned}$

Step 2:

Since the magnetic field on the geomagnetic poles of the Earth lies on the axial point of the magnetic dipole, the magnetic field at point B1 is given by

$\begin{aligned}\vec{B}_{1} &=\frac{\mu_{0} M}{4 \pi R^{3}} \\\frac{\mu_{0} M}{4 \pi R^{3}} &=3.4 \times 10^{-5}\end{aligned}$

$\begin{array}{c}{B_{1}=\frac{\left(3.4 \times 10^{-7} \times R^{3} \times 2 \times 10^{2}\right)}{R^{3}}} \\{B_{1}=6.8 \times 10^{-5} T}\end{array}$