In this post, we will find the formula of the magnetic field B at the center of a square loop of side a, carrying a current I.
Here is the diagram to understand the situation.
E is the center of the square ABCD. Current I is flowing through ABCD.
the distance between the center E and the midpoint of AB =r = Length of PE=a/2
Now, calculating the magnetic field at center E due to the current carrying AB side, we will use this formula,
=>BAB=[(μ0 2 I)/(4πa)](√2)
=>BAB=(μ0 2√2 I)/(4πa) ………….. (1)
Since all sides of the square create the same magnetic field at the center (the direction is also the same), so total magnetic field B will be
B=4×BAB=4×(μ0 2√2 I)/(4πa)
∴B=8√2[(μ0 I)/(4πa)] …………… (2)
Equation (2) can also be expressed as: B=2√2[(μ0 I)/(πa)]