A semicircular loop of radius a in free space carries a current I. Determine the magnetic flux density at the center of the loop.

the magnetic flux density at the center of the loop is μ₀I  / 4πα

Explanation:

Taking a look at the diagram;

we draw an imaginary curve to complete it as a circle.

we can now apply amperes law to write;

flux density B at centre;

∫B.dl = μ₀I

now since;

∫dl = 2πα

and field direction at centre is perpendicular to the screen, so it add up , and hence constant in magnitude.

so we can be taken out of the integral ,

B( 2πα ) = μ₀I

hence ;

B_circle = B = μ₀I  / 2πα

so if we remove the half part of this;

we get a semicircle, which is what we are looking for;

Now

B_semi = 1/2.B = 1/2 × μ₀I  / 2πα

B_semi = μ₀I  / 4πα

Therefore the magnetic flux density at the center of the loop is μ₀I  / 4πα