- #1
BOAS
- 546
- 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.
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 Earth's 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.
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] (Earth's 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 Earth's magnetic field cancelling etc.
Thanks,
BOAS
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 [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 Earth's 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.
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
[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] (Earth's 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 Earth's magnetic field cancelling etc.
Thanks,
BOAS