For the first case, ## a>> L ##. The first case does not work once ## L ## starts to get large enough to make a short solenoid. ## \\ ## Meanwhile, Ampere's law only works for the long solenoid geometry. Biot-Savart works for any geometry. Biot-Savart can readily be computed on-axis for the solenoid of medium length. Let me see if I can find the result in a google and give you a "link": https://notes.tyrocity.com/magnetic-field-along-axis-of-solenoid/ This "link" really needs a figure to show what the angles ## \Phi_1 ## and ## \Phi_2 ## are, but perhaps it is somewhat apparent. Here is a "link" with a diagram. See p.2. The angles are called ## \theta_1 ## and ## \theta_2 ## in this diagram. http://www.pas.rochester.edu/~dmw/phy217/Lectures/Lect_27b.pdf And see the formula at the bottom of p.6. This second "link" is using cgs units, so a couple conversion factors are necessary to get to the MKS result. ## \\ ## Editing: You can even use the formula ##B=\frac{\mu_o nI}{2}( \cos(\Phi_1)-\cos(\Phi_2)) ## to work the case with ## a>>L ##, and you do get the formula ## B=\frac{\mu_o NI}{2a} ## that you presented above. (You let ##n=\frac{N}{\Delta} ##, (with ## L=\Delta ##), and ## \Phi_1=\frac{\pi }{2}-\frac{\Delta}{2a} ##, and ## \Phi_2=\frac{\pi}{2} +\frac{\Delta}{2a} ##. In the limit ## \Delta \rightarrow 0 ##, you get the first formula above).al33 said:If we place many rings side by side, it looks just like a solenoid, right? And if we apply Ampere’s law on both cases, aren’t we supposed to get the same result? If not, how come? There must be some point that I haven’t figured out.
A magnetic field is a region in space around a magnet or electric current where a magnetic force can be observed. It is represented by lines of force that extend from the north pole of a magnet to the south pole.
Circular loops and solenoids are both structures that can be used to create magnetic fields. A circular loop is a single loop of wire, while a solenoid is a coil of wire with multiple loops. Both can produce a magnetic field when an electric current is passed through the wire.
The magnetic field of a circular loop is strongest at the center of the loop and weakens as you move away from it. In contrast, the magnetic field of a solenoid is strongest at the center of the coil and remains relatively constant along the length of the coil.
The direction of the magnetic field in a circular loop or solenoid is determined by the right-hand rule. If you point your thumb in the direction of the electric current, then your fingers will curl in the direction of the magnetic field.
The strength of the magnetic field in a circular loop or solenoid can be increased by increasing the number of loops in the solenoid, increasing the current passing through the wire, or using a material with higher magnetic permeability, such as iron, within the loop or solenoid.