Force acting on conducting contour

[cdummie]

Homework Statement

We have contours C1 and C2 located in vacuum and there are constant currents in them, I1 and I2 respectively. Find the expression for magnetic force that acts on one very small element (dl) of the circle C2 and it's coming from one very small element of the contour C1. Sketch the contours with all the vectors that appear in the expression.

Homework Equations

F[/B]=IlxB

The Attempt at a Solution

This is how i tried to solve it.

First, the formula for magnetic force is F=IlxB but, since i have to choose very small elements of the contours it means that i am only interested in dl and not the whole length of the contour l so force coming form the one small element of C1 and it's acting on one small element of C2 should be F=I₂dl₂xB where I₂ is the current in C2, dl₂ is the small element of C2 and B is magnetic induction vector of one element of C1 in the point where dl₂ si located, which means that i should find B.Using Biot-Savart's law we have B=μ₀I₁dl₁/(4πr²)

where r is the distance from one element to another,

now, since axb=absinα , α-angle between a and b

then, the final solution is

F=μ₀I₂I₁dl₁dl₂/(4πr²)

Now, i need to know if this is correct, then i'll do the sketch. So, i'd appreciate any help here.

 

[rude man]

The Biot-Savart law gives B as a vector too, and in this case a differential one at that, so you should write
dB = μ₀I₁dl₁ x r/(4πr²)
with r being the unit vector pointing from dl₁ to dl₂.

Then you can write the element of force dF in terms of dB and dl₂.

 

[cdummie]

The Biot-Savart law gives B as a vector too, and in this case a differential one at that, so you should write
dB = μ₀I₁dl₁ x r/(4πr²)
with r being the unit vector pointing from dl₁ to dl₂.

Then you can write the element of force dF in terms of dB and dl₂.

Finally someone responded! :)

Ok, i know, B is a vector too, but if we look only at intensity, is my final solution in my previous post correct?

 

[rude man]

Finally someone responded! :)

Finally, you responded (5 days after I responded) :-)

Ok, i know, B is a vector too, but if we look only at intensity, is my final solution in my previous post correct?

Your final expression is correct only if the vector elements dl₁ and dl₂ have the same direction (or opposite direction). For example, if the two elements were at 90 degrees then dF = 0.

You didn't include a picture of the two circles so maybe that would resolve the issue. And in any case you force is also an element of force so needs to be written as dF or dF.

 

[cdummie]

Finally, you responded (5 days after I responded) :-)

Your final expression is correct only if the vector elements dl₁ and dl₂ have the same direction (or opposite direction). For example, if the two elements were at 90 degrees then dF = 0.

You didn't include a picture of the two circles so maybe that would resolve the issue. And in any case you force is also an element of force so needs to be written as dF or dF.

I understand what are you saying, it's because of the cross product. Thanks a lot for help!

 

[rude man]

I understand what are you saying, it's because of the cross product. Thanks a lot for help!

Right, and yer' welcome!