Find an expression for the magnetic field at the center of a square loop of side a carrying current l.

The magnetic field at the center of a square loop can be found using Ampere's Law, which states that the magnetic field around a current carrying wire is given by B = μ0I/(2πr), where B is the magnetic field, μ0 is the permeability of free space, I is the current, and r is the distance from the wire.

Step 1: Consider one side of the square loop. The magnetic field at the center due to this side is given by B = μ0I/(2πr). But r is half the diagonal of the square, so r = a/√2, where a is the side of the square.

Step 2: Substitute r = a/√2 into the equation to get B = μ0I/(2πa/√2) = μ0√2I/(2πa).

Step 3: However, there are four sides to the square loop, and due to symmetry, the magnetic field at the center due to each side is the same. Therefore, the total magnetic field at the center is 4 times the magnetic field due to one side.

Step 4: Multiply the expression obtained in step 2 by 4 to get the total magnetic field at the center. B_total = 4μ0√2I/(2πa) = 2μ0√2I/πa.

So, the magnetic field at the center of a square loop of side a carrying current I is given by B = 2μ0√2I/πa.