Magnetic field of a rectangular coil

[magnetized]

I got this formula for the magnetic field of a square-shaped coil: B = m0*2*sqrt(2)*I*N/(a*pi), where m0 is the permeability, and a is the side of the square... I can't really get why it is so, and my best tries resulted in approximation of the 'diameter' of the coil, for example by taking the diagonal of the square... thanks in advance!

 

[Meir Achuz]

That formula is for the field at the center of the square, although I don't get the factor of 2. It is derived by using the Biot-Savart law for a wire segment of length a four times.
The field of a wire segment of length a is found by integrating from -a/2 to +a/2, instead of infinity in the usual derivation for an infinite wire.

 

[magnetized]

hm, is there a way to avoid integrals? i found some similarities with the formula for a single wire segment, B = m0*I/(2*r*pi), as a single loop is just 4 of these put in shape of a square, and the orientation of the vector for every segment is the same if we look at the magnetic field in the center, but this turns out to be just four times the formula given above, and r = a/2, so i get B = 4*m0*I/(r*pi)

//edit
ok, i read the 'instead of infinity' now... i guess no hope for getting it right, i guess, as we haven't done integrals yet in school

 

[Meir Achuz]

You'd better hold off on square loops until integration.