# Magnetic Forces: Deriving Equation to Balance Rider Weight

• BOAS
In summary, the conversation is about a lab course and understanding an equation related to magnetic forces between wires. The equation includes a contribution from the Earth's magnetic field, which can be cancelled by replacing I^2 with I1I2. The steps to reach this conclusion are questioned, and there is uncertainty about the experiment and how to plot the results. There is also a mention of a lack of experience with lab work.
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.

## 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

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 ?

## 1. What is the equation for balancing rider weight using magnetic forces?

The equation for balancing rider weight using magnetic forces is Fm = mg, where Fm is the magnetic force, m is the mass of the rider, and g is the acceleration due to gravity.

## 2. How is the magnetic force calculated?

The magnetic force is calculated using the equation Fm = BIL, where B is the magnetic field strength, I is the current in the wire, and L is the length of the wire.

## 3. Why is it important to balance rider weight using magnetic forces?

Balancing rider weight using magnetic forces allows for a stable and controlled ride, preventing the rider from falling off or experiencing sudden shifts in weight distribution. It also reduces strain on the rider's muscles, making for a more comfortable and enjoyable experience.

## 4. What factors affect the strength of the magnetic force?

The strength of the magnetic force depends on the strength of the magnetic field, the current in the wire, and the length of the wire. Additionally, the distance between the rider and the magnetic field source can also affect the strength of the force.

## 5. Can the equation for balancing rider weight using magnetic forces be applied to other scenarios?

Yes, the equation can be applied to other scenarios involving the use of magnetic levitation, such as in maglev trains or hoverboards. However, the specific values for the variables may differ depending on the design and application of the magnetic force.

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