Question

A CURRENT I FLOWS IN AN INFINITELY LONG WIRE WITH CROSS SECTION IN THE FORM OF A SEMICIRCULAR RING OF RADIUS R . THE MAGNITUDE OF THE MAGNETIC INDUCTION ALONG ITS AXIS IS

Answer 1

Let semicircular ring of radius R as shown in figure ,
An elementary length dl cut for finding magnetic field .so, dl = Rdθ
so, elementary current along dl is di = i/πR × Rdθ = idθ/π
Now, you can see that elementary part of length is perpendicular upon dB
so, dB = $\bold{\frac{\mu_0di}{2\pi R}}$
$=\bold{\frac{\mu_0i}{2\pi^2R}}$

Now, magnetic field along axis B $=\bold{\int\limits^{\pi}_{0} {dB.sin\theta}\,d\theta}$ = 2dB
Now, put dB in here,
So, B = μ₀i/π²R