Magnetic field of semicurcular current

[woodysooner]

i just want to understand the integration process so just the math is what i want to understand. The problem is that I have a three dimensional field in which a semicircle lies in the x plane in which current travels. In the y plane there is a magnetic field (B) coming up out of the semicircle and it doesn't say how long (Question about that in a sec). The problem wanted to know what the cross product of these two is. So I took it at the axis and integrated with respect to sin theta because that what a cross product is. I(current) times B(magnetic field) times the sin of the angle between them. I took just half of the semicirlce and integrated sin from 0 to 90 degress and then multiplied my final answer by 2 because of symmetry.
Ok a few confusing things about this problem. I integrated from the axis in essence to you the symmtry to help me but the problem i see with this is that treating the magnetic field to be just at the origin but there are N amoun of field lines. So infinite lines that i must sum via integration, but how like i have thought about this all day and no matter how i integrate it it doesn't pan out because if i integrate from 0 90 for the bottom lines it doesn't help with other ones that aren't lying on the axis cause they have 360 degrees around them. I hope this is making sense a simple drawing well cover it very easily. Second problem was how we integrate with respect. As i was doing the problem i pictured this magnetic field to be short in lenth therefore sin would work but then i pictured it to be very long maybe even assumed to be infinite than i said sin would turn to cos and integrated with respect to cos theta but for both answers i got the same thing a negative answer which is fine but i just don't understand how to do the math it's complex to me. and to finish the problem after i integrated and was done i just multiplied the answer by N amount of lines. but i know that how you integrate one of the many N's is not the same as how you should integrate another N. It to me is like looking at a map of the United states and trying to take each mountain and some up the weight of the rock or material. You can't treat it as amorphic with smoothness each case of mountain is differnt each is a diffent height and you would have to integrate each one then sum up all that are in the US. It's almost as if you integrate and then have to integrate again but can't.
My thoughts are that if you have a semicircle and you have 5 random dots and you need to integrate them with respect to angle how do you treat them as a system or can you not because what you do to one will not help with the others(example if a dot lie on the very bottom of the semicircle all you could do is go pi around and you are done but go up just a lil to the next dot and it have 2pi around but a lot more up then down. I hope all this made sense and forget the physics this problem could be one of amillion situations you've seen not just cross products of magnetic current and force.

 

[Gza]

Wow. I hope the enter key, as well as the "." key are all working properly on your keyboard.

I was reading the post pretty well, until this statement (except for the x-plane y-plane stuff, the usual tradition is that it takes two axes to specify an orthogonal plane containing the origin):

Second problem was how we integrate with respect. As i was doing the problem i pictured this magnetic field to be short in lenth therefore sin would work but then i pictured it to be very long maybe even assumed to be infinite than i said sin would turn to cos and integrated with respect to cos theta but for both answers i got the same thing a negative answer which is fine but i just don't understand how to do the math it's complex to me.

The way you describe the situation makes the math seem complex to me too (and I've already finished an upper division series in E&M) I'm assuming you are using the integral version of the Biot-Savart law (excuse my spelling) for calculating the force on a current carrying wire, composed of differential current elements Idl. If that is the case, yes, it is just an integration of the cross product of the vector current element (Idl) and vector magnetic field (B). This gives you the total force on the current carrying wire from the B field. If it will help you, I can work out the problem in some detail to show you how it works.

 

[woodysooner]

gza your the man

ok yes using bio savart to the ILBsinx. Ok i will try to explain a lil better in the x direction you have a semicircle in which I current goes through and its a length of L so in essence its just piR in length and up in the y through the semicircle the B field. so taking the cross product leaves you with your force in the z. but what my problem is this. ok just doing ILBsinx to me would be alight if you had just a straight wire going in x and magetic field straight up in y, but the question was like there were visually millions of Bfield lines up in the y broken up in lil db's and what i assumed he was getting at is if you took the sin of the angle between each IL and db (going up higher and higher sin no longer holds, B reaches infinity and sin turns to cos) so if i wanted to inegrate ILBsinx as B approaches infinity then that would change it to ILBcosx then inegrated you get ILBsinx so i guess your back to square one. but do you see what I'm saying.
Next problem was the infinite B lines all i could figure was state the answer as IL(N)Bsinx and say n was number of field lines who knows? sorry for how i stated the question earlier i have thought of how it would work visually since then and i don't think i explained it correctly I am sure you will know how to do this cause this seems a lot easier to me but still not sure about it.

But on another note lol if you had any physics concept in which you needed to integrate something like say the B filed of even force in which you were doing it in respect to the angle but the angle for each field line was different like say the wire was hmm like slanted in the xy plane then what i explained yesterday is logical because each db has a differnt sinx and i am math man like i like physics concepts but i love the math and want it to work out and i thought maybe someone had a problem like this one so maybe you can give me a answer to both but if not no sweat I am just thankful you are helping.

 

[woodysooner]

another thing

this might not help at all but the b field doesn't go to negative its only in the positive dirction so the angle between IL and db starting with the first db is 90 then next db 90.1 all the way up till it approaches 180 with extreme infinity.

 

[Gza]

Alrite, now we're getting somewhere. Thanks for the clarification.

this might not help at all but the b field doesn't go to negative its only in the positive dirction so the angle between IL and db starting with the first db is 90 then next db 90.1 all the way up till it approaches 180 with extreme infinity.

The angles between IdL and B (not sure why you are referring to dB) are 90 degrees throughout the integration. This is of course true assuming the B field to be a uniform field.

 

[woodysooner]

making a lil more sense

I guess that makes sense what about the cross product and as B goes to infinity does that go to cos

 

[Gza]

I guess that makes sense what about the cross product and as B goes to infinity does that go to cos

Not too sure what you mean by "as B goes to infinity." The magnitude and direction of B are not changing in time. I would suggest that you use the proper cross product method (working out the determinant) instead of simply using B IdL sin() to get a better handle on what is happening in the problem. the sin() in the statement B IdL sin() is the sin() of the angle between B and IdL. It is a constant in this case with a value of 1. Make sure not to get that confused with the other angle you use to do the integration(if integrating in polar coordinates).

 

[woodysooner]

right

went talked to the prof about this problem and he said what you said about b to infinity he said if it was labeled B(x) then i would be right but B is just constant but what he also said was that the semicirlce of current IdL had to be done like rsin theta i hat plus rcostheta j hat cause the circle is in two dimensions. thanks for the help

 

[Gza]

Glad to be of service.