State Biot-Savart law. Use it to derive an expression for the magnetic field at the centre of a circular loop of radius R carrying a steady current I. Sketch the magnetic field lines for such a current-carrying loop.
Answer:B=2Rμ0I at the centre.
Solution
\textbf{Biot-Savart law:} dB=4πμ0r3Idℓ×r, magnitude 4πμ0r2Idℓsinθ. At the centre every element is at distance R with dℓ⊥r (θ=90∘), and all dB point the same way (along the axis), so
B=4πR2μ0I∮dℓ=4πR2μ0I(2πR)=2Rμ0I.
The field lines are concentric circles close to the wire, becoming straight and axial through the centre; the loop acts like a magnetic dipole with one face a north pole and the other a south pole.