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## Homework Statement

I have attached a diagram. In case you can't view it, it shows an infintely long wire [itex] I_i = 5.00 [/itex] A on the positive y direction. 0.100 m to the right, there is a rectangular loop of dimensions 0.150 m x 0.450 m, the long side is parallel to the infinitely long wire to its left. The current is [itex] I_2 = 10.0 [/itex] A and also flowing up (in the position closest to the infinitely long wire).

a) Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire.

b) Find the force on the top, horizontal segment of the loop. Calculate this three ways: net force, average force, and the force on the midpoint.

## Homework Equations

Bio-Savaart equation.

## The Attempt at a Solution

I didn't have any trouble with (a). The force on the top and bottom horizontal segments just cancel, so I can disregard them. The rest is pretty simple.

Part (b) is giving me lots of trouble. I am not 100% sure I am calculating the magnetic force of the infinitely long wire on the horizontal part of the loop properly, and I really am not certain how to handle the vertical segments of the loop. These are not infinite, so there must be some error introduced in this case, but I have no idea how to approximate it. Would the errors just cancel, and the net force of these also be zero?

Here is what I have done so far, I hope this is ok:

For the net force:

[tex] B = \int_{0.100}^{0.250} \frac{(5.00)(10.0)(\mu_0)}{2\pi x}\,dx = 1.00268(\ln(x))_{0.100}^{0.250} = 9.19 \times 10^{-6} [/tex]

For the average force, I just used the mean value theorem, and multiplied the above by 1/(0.250-0.100) to get 6.12 x 10^{-15}.

To calculate the force on the midpoint, I first calculate the force from the vertical wire:

[tex] \frac{\mu_0(5.00)}{2\pi(0.100+\frac{0.150}{2})}\hat{i} = 5.73\times 10^{-6}\hat{i} [/tex]

Then the other force:

[tex] \frac{\mu_0(10.0)}{2\pi(0.100 + \frac{0.150}{2})}\hat{j} = 1.15 \times 10^{-5}\hat{j} [/tex]

Since these are perpendicular, I got the rest as

[tex] \sqrt{(5.73\times 10^{-6})^2 + (1.15 \times 10^{-5})^2} = 1.28 \times 10^{-5} [/tex]

Intuitively, I would have expected closer values with all these results, so I am surprised they are so far apart, and that makes me question my work.

Any advice gratefully accepted.

Thanks,

Brett

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