The Bio-Savart Law applied to a square loop

[georgeh]

I made a typo. I meant, Biot- Savar Law( sorry ).1. Homework Statement
A conductor in the shape of a square Loop of edge length l= 0.400 m carries a current I = 10.0A. Calculate the magnitude and direction of the magnetic field at the center of the square.

Homework Equations

db = u_0/(4pi) * I ds x r_units vector /r^2

The Attempt at a Solution

I know that that I can take out the constants, u_0/4*pi and the I and integrate ds x r_unit vector /r^2
I know also that all four sides create a magnetic field B into the page. I just don't know how to state my r. I also know that
db = k* ds sin(thetha) /R^2
( with all the constatnts factored out, k = all the constants), but I am not sure how thetha varies, i am assuming r would be l/2 if we draw our axis at the point p. but i am not sure how thetha varies.. nor ds. any help would be appreciated.

 

[OlderDan]

RE: theta

Draw a triangle using one side of the square and the center of the square. What kind of triangle is this? How does your angle theta relate to this triangle?

If by r you mean the distance from a bit of the wire dl to the point where you are calculating the field at the center of the loop, it is a function of the angle between the wire and a line from the wire to the center of the loop. It may be L/2 somewhere, but certainly not everywhere.

 

[georgeh]

I get a 45-45-90 triangle. if i draw a perpendicular bisector from pt P to the wire, and draw an r at an angle thetha to the point P..

 

[OlderDan]

I get a 45-45-90 triangle. if i draw a perpendicular bisector from pt P to the wire, and draw an r at an angle thetha to the point P..

You also get a 45-45-90 if you draw from point P to two ends of one side. The lengths of those two lines are the intial and final values of r in the integral for one side. r gets shorter and then longer again as you move from one end of the wire to the other. The angle between dl and r starts at 45 degrees and increases as you move from end to end. How big does it get? Can you find a relationship between this angle, the distance from one end to a point on the wire and r from this point to the center of the loop? You might find it helpful to think about the relationship between the sine of an angle and the sine of the supplementary angle.