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# Find the magnetic field B at the center of square loop of side a carrying a current I

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.

Solution:

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=(μ0I)(sinθ1+sinθ2)/(4πr)

BAB=(μ0I)(sin45o+sin45o)/(4πa/2)
=>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)]

See also  Calculate the magnetic field at point P located on the axis at a distance of 10 cm from the center of a circular wire of radius 5 cm carrying a current of 5 A.
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