Answer: 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
