Magnetic field of a goffered circle

[bjnartowt]

Homework Statement

There is a constant current I = 10A in a conductor shaped as a “goffered” circle. Find the magnetic induction B at the center of the conductor. [The equation for the curve of the conductor, in polar coordinates, is

$${\textstyle{1 \over r}} = {\textstyle{1 \over a}} + b\cos (m\phi )$$

where m is an integer and a and b are constants.]

Homework Equations

$$\left| {\bf{B}} \right| \propto \int {\frac{{\sin \theta }}{{{r^2}}} \cdot d{\theta _{wire}}}$$

The Attempt at a Solution

Eventually, I wind up with,
$$\left| {\bf{B}} \right| = \int_0^{2\pi } {\frac{{{{(\alpha + b\cos (m\phi ))}^4}}}{{\sqrt {{{(bm\sin (m\phi ))}^2} + {{(\alpha + b\cos (m\phi ))}^2}} }}d\phi }$$

I put this integral into Maple, and it spits out a *Huge* number of what appears to be elliptic integrals at me. Can someone just name the integration technique I'd need in order to carry out this integral? I just need a lead, and I'll be off and running...

******UPDATE: Never mind...I figured it out. I got this slick cancellation when I realized I wasn't integrating with respect to d_phi, but rather d_l, which carries units of length*****

 

[mark726]

Hello,

Thank you for sharing your progress and solution with us. It seems like you have encountered some challenges while trying to solve this problem. I understand the importance of persistence and problem-solving skills in tackling complex scientific problems. In this case, it seems like you were able to identify the error in your integration and correct it, leading to a cancellation and ultimately arriving at the correct solution.

In terms of the integration technique needed for this problem, it appears that you may have used the substitution method to simplify the integral. This is a common technique used in solving integrals involving trigonometric functions. I am glad that you were able to figure it out and I hope this experience has helped you develop your problem-solving skills even further.

Best of luck with your future scientific endeavors! Keep up the good work.