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.

**1. Homework Statement**

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

Ideally this force should balance the weight of the rider, however there may also be a contribution from the earths magnetic field, which will add a force [itex]B_{e} IL[/itex]

By replacing [itex]I^{2}[/itex] with [itex]I_{1}I_{2}[/itex] the effect of the external fields should cancel.

**2. Homework Equations**

**3. 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

[itex]F_{1} = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r} + B_{e} IL[/itex]

[itex]F_{2} = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r} - B_{e} IL[/itex]

[itex] F_{1} + F_{2} = \frac{\mu_{0} I_{1}I_{2} L + \mu_{0} I_{1}I_{2}L}{2 \pi r}[/itex] (earths magnetic field cancels)

[itex] 2F = \frac{2(\mu_{0} I_{1}I_{2} L) }{2 \pi r}[/itex]

[itex] F = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r}[/itex] 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;

[itex] mg = \frac{\mu_{0} I_{1}I_{2} L}{2 \pi r}[/itex] and rearrange to get

[itex]I_{1}I_{2} = (\frac{g2 \pi r}{\mu_{0} L}m[/itex] 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 earths magnetic field cancelling etc.

Thanks,

BOAS