Finding the force between 2 finite length wires

[jimmyting]

[SOLVED] Finding the force between 2 finite length wires

Homework Statement

The two wires of length 2m are 3mm apart and carry a current of 10 A dc.
calculate the force between these wires.

Homework Equations

Well, I know that the force is found with $$F=IL\times B$$
Since we aren't given B, the equation for magnetic field is $$B= \frac{\mu\mi_{o}}{4\pi}\frac{IL\times\hat{r}}{r^{2}}$$

A possibly relating equation is the equation for an infinite long wire
$$B= \frac{\mu\mi_{o}I}{2\pi r}$$

The Attempt at a Solution

I plugged in what I knew for all the variables, and ended up with an answer of 4.44 repeating
I substituted 10 for I (both times since it is the same current for both wires)
r^ was 1 because the wires are // causing a perpendicular field
L was 2 because it is the length the current traveled in
r was .003 because it is the distance between which the force is being applied.

With those substitutions, what am I doing wrong?

 

[alphysicist]

Hi jimmyting,

You'll want to use the first and last equation here. You can actually combine them by inserting the value of B (the last equation) into the first equation for F. The result is a standard textbook equation for the force between two infinitely long, parallel wires:

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi d}$$

This is derived assuming infinitely long wires, but you can apply it here because the wires are so close compared to their length. What do you get for the force?

(Also, in your section 3 you said that r was 0.002, but the problem said they were 3 mm apart.)

 

[jimmyting]

Wow thank you for catching my error in the last part. It works out, thanks.
But if the segments were close in length comparably to the distance apart, would you need to apply the second equation?

 

[alphysicist]

Do you mean if the wires could not be considered infinitely long? Then the answer would be yes, you would need to use the differential form:

$$dB = \frac{\mu_0}{4\pi} \frac{I d\ell \times \hat r}{r^2}$$

and integrate to find the formula for B for that particular current configuration. (Of course, your textbook should have already done this for some common current configurations and you can just look up the formula for B for those cases.)