Q55P - Fundamentals Of Physics

Question

Earth has a magnetic dipole moment of $\mathbf{\mu }\mathbf{=}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}\mathbf{J}\mathbf{/}\mathbf{T}$. (a) What current would have to be produced in a single turn of wire extending around Earth at its geomagnetic equator if we wished to set up such a dipole? Could such an arrangement be used to cancel out Earth’s magnetism (b) at points in space well above Earth’s surface or (c) on Earth’s surface?

Step-by-Step Solution

Answer

1. In a single turn of wire extending around Earth at its geometric equator, if we wished to set up such a dipole, the current would produce $\mathrm{I}=6.3\times {10}^8 \mathrm{A}$ .
2. Yes, because far away from the earth surface, both magnetic dipole moments are the same. If they are equal and opposite, then the net field will be zero.
3. No, because magnetic fields of both are not the same. Therefore, both the fields will not cancel out.

Step 1: Listing the given quantities

Magnetic dipole moment is $\mathrm{\mu }=8\times {10}^{22}\mathrm{J}/\mathrm{T}$.

Step 2: Understanding the concepts of magnetic dipole moment

We use the concept of magnetic dipole moment due to wire. Using the equation, we can find the current through the wire.

Formulae:

$\mathrm{\mu }=\mathrm{NIA}$

Step 3:(a)Calculations of the current would have to be produced in a single turn of wire extending around Earth at its geometric equator if we wished to set up such a dipole

Using the equation of magnetic moment,

$\mathrm{\mu }=\mathrm{NIA}$

Rearranging it for current,

$\mathrm{I}=\frac{\mathrm{\mu }}{\mathrm{NA}}$

Here area is $\mathrm{A}={\mathrm{\pi r}}^2$, and N is one.

We can write

$\begin{array}{c}\mathrm{I}=\frac{8\times {10}^{22}}{\left(1\right)\left(3.14\right)\times {(6.378\times {10}^6)}^2}\\ = \frac{8\times {10}^{22}}{1.2773\times {10}^{14}}\\ = 6.263\times {10}^8\\ =6.3\times {10}^8 \mathrm{A}\end{array}$

In a single turn of wire extending around Earth at its geometric equator, if we wished to set up such a dipole, the current would produce $\mathrm{I}=6.3\times {10}^8 \mathrm{A}$ .

Step 4: (b) Explanation for the such arrangement be used to cancel out Earth’s magnetism at points in space well above the Earth’s surface

Yes, because far away from the earth surface, both the magnetic dipole moments are the same. If they are equal and opposite, then the net field will be zero.

Step 5: (c) Explanation

No, because magnetic fields of both are not the same. Therefore, both the fields will not cancel out.