Finding the magnetic field at center of a square loop

[lonewolf219]

My textbook says that at the center of a square conducting wire of length ω, the magnetic field is:

B=$\sqrt{2}$μ$_{0}$I/($\pi$R)

I am not sure how to calculate this...?

Because the Biot Savart law has a closed loop integral, we do not use piecewise addition of line integrals to find the magnetic field, as we would to find the magnetic force, is that correct? Do we treat the square loop as if it were a straight wire of total length 4ω?

 

[lonewolf219]

I think we can find the B field for one length of the square and then multiply by 4, because each side contributes the same magnitude. This is because each side is the same length, and each corner makes the same angle with respect to the test point, which would be a 45 degree angle...?

 

[Philip Wood]

You are right. R = (1/8) x perimeter of square.

 

[lonewolf219]

Thanks for the post, Philip Wood!

 

[sardar]

You are correct that the Biot Savart law is used to calculate the magnetic field at a point due to a closed loop of current. In the case of a square loop, we can consider it as a straight wire of total length 4ω, but we must also take into account the geometry of the loop. The formula given in your textbook is derived by considering the contribution of each side of the square loop to the magnetic field at the center.

To calculate the magnetic field at the center of a square loop, we can use the following formula: B = μ_0I/2R, where μ_0 is the permeability constant, I is the current flowing through the loop, and R is the distance from the center of the loop to the point where we want to find the magnetic field.

In the case of a square loop, the length ω is equal to half the side length of the square, so we can rewrite the formula as B = μ_0I/πω. This is the same formula given in your textbook, but with ω instead of R.

To better understand this formula, we can consider the magnetic field due to a straight wire of length 4ω at the center. The formula for this would be B = μ_0I/2ω. Notice that the length of the wire is now 4ω instead of πω. This is because the magnetic field at the center of a straight wire is directly proportional to the length of the wire, while for a square loop, it is inversely proportional to the length.

In summary, to find the magnetic field at the center of a square loop, we can treat it as a straight wire of length 4ω, but we must also take into account the geometry of the loop. The formula given in your textbook is derived by considering the contribution of each side of the loop to the magnetic field at the center.