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Magnetic forces

  • Thread starter BOAS
  • Start date
544
19
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.

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
 

BvU

Science Advisor
Homework Helper
12,074
2,665
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 ?
 

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