How Can We Calculate the Uniformity of a Magnetic Field in a Current Loop?

[Bleedingfingers]

Hi. I've been trying to calculate the uniformity of a magnetic field within a loop of current. I know that the magnetic field at a point r units way from a wire carrying current is given by mu not over 2pi times I over r. I thought that what I wanted to do was choose a point distance a from the center of the circular loop (but still inside it) and find a formula for the distance between that point and any point on the circle as a function of the angle that the point on the loop lies on. so I parameterized x and y as Rcos(t) and Rsin(t) respectively. Using the distance formula of pythagoras, and assuming that it doesn't change anything if I set the coordinates of the point I'm interested into (a,0), I get sqrt( (x-a)^2+(y)^2) which, carrying throught the operations gives me sqrt( R^2+a^2-Racos(t)). ing the integral of the inverse of this gives me a nasty looking integral which both mathematica and the integral tables I have referenced claim can only be solved using an eliptic integral which can only be evaluated between 0 and pi, wheras I need 0 to 2pi. Am I going about this in the proper manner? Any suggestions? After we get this, we will want to illustrate the uniformity of the magnetic field between two coils and show that the field between them is uniform, and find out exactly how uniform it is. We need to know this for the construction of a cyclotron. We want to be able to figure out how our electron/particle beam is going to fringe out when it reaches the edje of the magnetic field. Our goal is to build an exit port for the sucker (If we get around to it; probably won't happen till next quarter) so we want to be able to calculate where the particles are going to hit the side of the chamber so we can build the exit port there.

Thanks for the help on this.

 

[Gokul43201]

Why don't you start from the Biot-Savart Law ?

Still, for any point not at the center of the loop you'll likely have some messy integral to solve. For points near the center (distance from center << radius), you can use some suitable approximation to find the field. For any general point, and especially for points near the wire itself, it may just be best to do a numerical integration.