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

Given:
Magnetic field at the magnetic equator, B = 3.4 × 10⁻⁵ T
Let M be the magnetic moment of Earth's magnetic dipole and R be the distance of the observation point from the centre of Earth's magnetic dipole.
As the point on the magnetic equator is on the equatorial position of Earth's magnet, the magnetic field at the equatorial point $\left(B\right)$ is given by
$$\begin{gathered} B=\frac{{\mu }_0}{4\mathrm{\pi }}\frac{M}{R^3} \\ \Rightarrow \frac{{\mu }_0}{4\mathrm{\pi }}\times \frac{M}{R^{\mathit{3}}}=3.4\times {10}^{-5} \\ \Rightarrow M=\frac{3.4\times {10}^{-5}\times {\mathrm{R}}^3\times 4\mathrm{\pi }}{4\mathrm{\pi }\times {10}^{-7}} \\ \Rightarrow M=3.4\times {10}^2{R}^3 \end{gathered}$$
As the magnetic field on Earth's geomagnetic poles lies on the axial point of the magnetic dipole, the magnetic field at the axial point $\left(B_1\right)$ is given by
$$\begin{gathered} {\overrightarrow{B}}_1=\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\times \frac{2M}{R^3} \\ \Rightarrow B_1=\frac{{10}^{-7}\times 2\times 3.4\times {10}^2\times R^3}{R^3} \\ \Rightarrow B_1=6.8\times {10}^{-5}\mathrm{T} \end{gathered}$$