Magnetic field of a solenoid using Ampere's Law -- not hand-waving!

In this video, we derive the magnetic field inside and outside a long solenoid starting with exactly one simple assumption: the magnetic field must approach zero in the large r limit.

🧠 Access full flipped physics courses with video lectures and examples at https://www.zakslabphysics.com/

This is the only assumption required to carefully prove that the magnetic field is zero outside the solenoid and B=mu_0*n*I inside the solenoid! In contrast, most lower division textbooks will assume that B=0 outside the solenoid and that B is uniform inside the solenoid. This hand-waving approach isn't necessary if you're willing to explore a few extra steps to find the magnetic field of a solenoid using Ampere's Law.

00:00 Introduction: given a solenoid of radius R carrying a current of I with linear turn density "n" turns per length, we use Ampere's law to derive the magnetic field inside and outside the solenoid. We only have to make one assumption to proceed with the calculation of field for a long solenoid: the magnetic field vanishes at a large distance from the symmetry axis!

01:12 Showing the magnetic field has no circumferential component. We draw a circumferential Amperian loop centered on the symmetry axis, and show that any circumferential component of the magnetic field must vanish when we apply Ampere's Law.

03:33 Showing the magnetic field has no radial component. We assume a radial component of magnetic field for the solenoid, but then we derive a contradiction: reversing the direction of current in the solenoid must reverse the direction of magnetic field. But reversing the direction of current is no different than picking up the solenoid and flipping it over! There's no way that picking a solenoid up and rotating it could cause an outward magnetic field component to reverse to inward, therefore the radial component of magnetic field must be zero.

04:56 Finding the direction of magnetic field inside and outside the solenoid. Now that we know the magnetic field is parallel to the solenoid axis, we show a cross section of the solenoid and use the right hand rule to infer the direction of magnetic field inside and outside the solenoid.

07:16 Choose an Amperian loop outside the solenoid to show that the magnetic field is zero outside the solenoid.

11:26 Choose an Amperian loop intersecting the solenoid to show that the magnetic field is mu_0*n*I within the solenoid.

#physics #magnetism #solenoid