Q.Using the Biot-Savart law, derive the expression for the magnetic field at the centre of a circular current loop of radius $R$ carrying current $I$.

For a circular loop of radius $R$ carrying current $I$, every current element $I\,dl$ is at distance $R$ from the centre and perpendicular to the radius ($\theta=90^\circ$, $\sin\theta=1$), so $dB=(\mu_0/4\pi)(I\,dl/R^2)$, with every element's contribution pointing along the same axial direction at the centre. Integrating $dl$ around the full circumference $2\pi R$: …

Note every element is equidistant and perpendicular to the radius, so contributions add directly (no vector resolution …

• Forgetting the factor of $N$ for a multi-turn coil.
• Attempting vector resolution of the contributions when, by the loop's symmetry, they alr …