Question

III. Extension to calculating the magnetic field on the axis of a solenoid . y! R A solenoid has a length of Land is comprised of N coaxial loops with radius R. All of the loops are connected so that they all carry the same current in the same direction. You want to determine the field at a point on the y axis. In the previous example you calculated the magnetic field ^ due to one loop. This is an extension to many loops. As you move from loop to loop what parameter(s) in your result for a single loop varies, y, R, I? How many loops are there in a distance along the solenoid dy? Write down the integral that results from adding contributions from loops along the y axis. If your variable is y is the integral (in terms of y) an easy one to integrate? If not, would it be easier to convert to an integral in terms of the angle between 8 and R)? If it's easier, write down the integral in terms of e. Perform the integration.

Answer 1

Answer: First we solve, the magnetic field at an axial point due to circular coil Carrying Current I 17 XOR "T Y from Biot-Savant law, magnetic field due to Small length element de" dB = Mo I de sino 47 ra de tr so o=90 dB = Mo I de 472 B = J de sine Cos e Component Cancel each other. Mo I B de Sino J 4 782 From triangle sino R 8 B = R de J Mo I 4T82 r=√ R² + x² 2TR B= Mo I R 4* (R²4 222, 3/2 de

So 2 ار Mo I R 소 2 (R²+ (2) 3/2 Now we see the solenoid case- axis गणि al we take small element, and see like this Toch in-02 01 02 let, in small area total turns are dy, so due to this magnetic field. dB = MoIR? dN 2 (R² + x 2, 3/2 Small distance da' in which these tums are dN= anda unit [-ve sign only for left side to origin, n turns per length x = R coto dx = -R Coseco de

→ Integral is easies in terms of O, AS above we calculated in above derivation Ba Mon los oat cos oz) 2