Ampère's Law — Electromagnetism in 60 Seconds #Shorts

Added: 2026-05-26

[00:00:00]Push 5 amps of current up a copper wire.

[00:00:03]You can't see the magnetic field around it, but Ampere's law tells you exactly what's there. One line of math, no integral table required. The law in its integral form, the line integral of B around any closed loop equals mu zero times the current piercing through. Dot B to d a l equals mu zero zero I enclosed. Andre Marie Ampere, 1826. The magnetic analog of Gauss's law.

[00:00:30]Direction comes from the right-hand rule. Point your thumb along the current and your fingers curl in the direction of B. Around an upward current, the field circulates counterclockwise seen from above. The field forms closed circles around the wire. No beginning, no end. That's why there are no magnetic monopoles. Field lines don't terminate on charges. They loop back on themselves. Pick an Amperian loop, a circle of radius R centered on the wire.

[00:00:54]By symmetry, B has constant magnitude on the loop and points along d o. So, teach B t d l a collapses to B times the circumference, B to 2 pi r. Set that equal to mu zero I and solve. B equals mu zero I over 2 pi r. The field falls off as 1 over r, slower than gravity's inverse square. Double the distance, half the field. That single formula powers every magnetic compass and clamp on ammeter. Now coil the wire into a solenoid, n turns per meter. Choose a rectangular Amperian loop with one side inside, one outside. The integral collapses to B to L on the inside leg.

[00:01:32]The enclosed current is n to l i. Cancel L. B inside equals mu zero n i, uniform.

[00:01:39]Axial. Zero outside. But Ampere's original law breaks at a charging capacitor. Current flows in the leads, none in the gap, yet B exists in the gap. Maxwell, 1861 fixed it. Add epsilon zero d phi e d t, the displacement current. Changing electric flux acts like a current.

[00:01:59]In differential form, curl of B equals mu zero J plus mu zero epsilon zero d e by dt.

[00:02:07]Pair this with Faraday's law and you get the wave equation speed c equals one over root mu zero epsilon zero.

[00:02:14]Plug in the constants, it's the speed of light.

[00:02:17]Light is electromagnetic.

[00:02:19]One law from a humming transformer to the photon hitting your eye. Wire, solenoid, capacitor, light. Ampere's integral, Maxwell's correction, every electromagnet, every antenna, every wireless signal in the air.

[00:02:34]d f d d t l equals mu zero i.