How Does Earth's Radius Influence Calculations in Magnetic Field Equations?

[Melqarthos]

Homework Statement

The Earth's magnetic field is essentially that of a magnetic dipole. If the field near the North Pole is about , what will it be (approximately) 1.4×104 above the surface at the North Pole?

Homework Equations

Apparently we need to use the Biot-Savart Law which I'm not sure how it even applies in this situation granted that Earth's magnetic field acts like a dipole.

The Attempt at a Solution

If anyone could give me a heads up on how to do this, that'd be great.

Melqarthos

 

[Delphi51]

Not the Biot-Savart Law. You just need the formula for the magnetic field around a magnetic dipole. Look it up in Wikipedia if your textbook doesn't have it handy.

 

[Melqarthos]

Well from what I understand we know that μ=NIA, which is the magnetic dipole moment of a coil and is considered a vector. I also know that:

torque= μ*B

This is all useful only when we're messing with currents. We can further our investigation by realizing that we can use μ and sub it into the equation of a magnetic field produced by a magnetic dipole (along the dipole axis):

B= [μ(permeability constant)/2pi]*[ μ/(R^2+x^2)^(3/2)]

Oh I see! Let me try to figure this one out.

 

[Melqarthos]

Not the Biot-Savart Law. You just need the formula for the magnetic field around a magnetic dipole. Look it up in Wikipedia if your textbook doesn't have it handy.

I tried the problem again. Check the picture to see my work. I used another value for distance from the North pole. Instead of a distance of 1.4*10^4 I used 1.3*10^4 km. The back of my book says the answer should be 3.6*10^-6 T, but that's not what I got.

Meqlarthos

 

[Melqarthos]

Not the Biot-Savart Law. You just need the formula for the magnetic field around a magnetic dipole. Look it up in Wikipedia if your textbook doesn't have it handy.

Okay I got it! But there's something I don't get. Why do we treat the radius of the Earth and the distance away from the magnetic dipole as 'one unified' distance, rather than two distances where the distance away from the Earth's surface should be x^2 as according to the magnetic dipole equation:

B= (μ0*μ)/(2*pi*(R^2+x^2)^(3/2)

and generally if x>>R, then:

B=(μ0*μ)/(2*pi*x^3)

I hope you can see where I'm coming from when I say why treat the Earth's radius and the distance away from the Earth's surface as one unified distance when the equations that give us the magnetic field don't tell us to do this.

Melqarthos