Circulation, changing flux, and electromagnetic induction
Currents make magnetic fields, and changing magnetic fields make electric ones. These two laws are how electricity and magnetism feed each other.
Pick a loop position to enclose different wires. Only the wires inside the loop count toward the line integral of B; the ones outside drop out.
The dashed purple rectangle is the Amperian loop. Wires shown as circles with a dot (current out of screen) or cross (current into screen). Only enclosed currents contribute to the line integral.
Ampere's law says that the circulation of B around a closed loop equals μ₀ times the enclosed current. Faraday's law says that a changing magnetic flux induces an electric field: EMF = -dΦ/dt. Together, they describe how electric and magnetic fields create each other.
Draw an Amperian loop around current-carrying wires. Only the enclosed current contributes to the line integral of B. Wires outside the loop are irrelevant.
Key insight: Ampere's law is the magnetic analogue of Gauss's law. Just as enclosed charge determines electric flux, enclosed current determines magnetic circulation.
Change the magnet speed and watch the induced current in the coil. Faster motion means a faster change in flux, and a larger EMF.
A bar magnet oscillates near a conducting loop. Faster motion produces larger change in flux, resulting in greater induced EMF (Faraday's Law: EMF = -dΦ/dt).
A changing magnetic flux through a loop induces an EMF. Move a bar magnet toward a coil and watch the induced current flow. Faster motion means larger EMF.
Key insight: Faraday's law is the basis of electric generators, transformers, and wireless charging. It converts mechanical motion into electrical energy.
Switch between Magnet Approaching and Magnet Retreating. The induced current always pushes back: it repels an approaching magnet and pulls on a retreating one.
Lenz's Law: the induced current creates a magnetic field that opposes the change causing it. Approaching magnet is repelled; retreating magnet is attracted.
Lenz's law determines the direction of the induced current: it always opposes the change in flux. An approaching magnet is repelled; a retreating magnet is attracted. Nature resists changes in magnetic flux.
Key insight: Lenz's law is a consequence of energy conservation. If the induced current aided the change instead of opposing it, you could create energy from nothing.
Spot a mistake or have an idea?