Magnetic Forces: Deriving Equation to Balance Rider Weight

[BOAS]

Hello,

i'm working through some prep required for my lab course and having some trouble understanding an equation. We're asked to show it's derivation and I think I've done this but the script itself is a bit ambigous about how to get there.

Homework Statement

The magnetic force of mutual repulsion between the wires is $F = \frac{\mu_{0} I^{2} L}{2 \pi r}$

Ideally this force should balance the weight of the rider, however there may also be a contribution from the Earth's magnetic field, which will add a force $B_{e} IL$

By replacing $I^{2}$ with $I_{1}I_{2}$ the effect of the external fields should cancel.

Homework Equations

The Attempt at a Solution

Depending on the direction of the current, the Earth's magnetic field is either contributing to or opposing the force due to the magnetic field of the wire, so

$F_{1} = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r} + B_{e} IL$

$F_{2} = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r} - B_{e} IL$

$F_{1} + F_{2} = \frac{\mu_{0} I_{1}I_{2} L + \mu_{0} I_{1}I_{2}L}{2 \pi r}$ (Earth's magnetic field cancels)

$2F = \frac{2(\mu_{0} I_{1}I_{2} L) }{2 \pi r}$

$F = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r}$ this is the equation I wanted to reach. Are my steps to get here ok? I don't really know how to put into words the importance of substituting I^2 for I1I2...

Since this force should balance the weight of the rider we can say;

$mg = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r}$ and rearrange to get

$I_{1}I_{2} = (\frac{g2 \pi r}{\mu_{0} L}m$ Which is what I want to plot a graph of)

Apologies if this is a bit of a vague post, but I don't have much practice of lab work and I don't know if my steps count as showing why the magnetic field of the Earth cancels due to the substitution of I^2 that I made. I do include a few more steps in the algebra in my book, showing explicitly the force due to the Earth's magnetic field cancelling etc.

Thanks,

BOAS

 

[BvU]

You're not getting much quick help here. Could it be the problem formulation is somewhat unclear to others ? They may know about forces between current carrying wires, but who is riding them ?
You are to do some experiment, measure something and plot the results in a smart way, right ?