Magnetic field produced along the axis of the current carrying circular coil:
Consider a current carrying circular loop of radius R and let I be the current flowing through the wire in the direction. The magnetic field at a point P on the axis of the circular coil at a distance z from its center of the coil O. It is computed by taking two diametrically opposite line elements of the coil each of length \(\vec {dl}\) at C and D. Let \(\hat r\) be the vector joining the current element (1 \(\vec {dl}\)) at C to the point p.
PC = PD = T =\(\sqrt{R^2 + Z^2}\) and angle ∠CPO = ∠DPO = θ
According to Biot-Savart’s law, the magnetic field at P due to the current element I \(\vec {dl}\) is
The magnitude of magnetic field due to current element l dl at C and D are equal because of equal distance from the coil. The magnetic field dB due to each current element I \(\vec {dl}\) is resolved into two components; dB sin θ along ydirection and dB cos θ along z-direction. Horizontal components of each current element cancels out while the vertical components (dB cos θ \(\hat k\) ) alone contribute to total magnetic field at the point P.
Current carrying circular loop using Biot -savart's law
If we integrate \(\vec {dl}\) around the loop, d \(\vec B\) sweeps out a cone, then the net magnetic field \(\vec B\) at point P is
Using Pythagorous theorem r2 = R2 + Z2 and integrating line element from 0 to 2πR, we get
Note that the magnetic field \(\vec B\) points along the direction from the point O to P. Suppose if the current flows in clockwise direction, then magnetic field points in the direction from the point P to O.
