Physics Speedrun - Electromagnetic Induction

Faraday's Law of Induction

An induced EMF is produced by a changing magnetic field

The electromotive force (EMF) induced in a circuit is equal to the changing rate of magnetic flux throught the circuit.

For a -loop coil:

Induction current:

Lenz's Law

An induced EMF is always in a direction that opposes the original change in flux that caused it.

Total induced charge is only determined by and , regardless the procedure.

Motional EMF

Conductor moves in magentic field

By Lorentz force:

indicates a non-electrostatic field.

Motional EMF:

The result can also be derived by Firaday's law.

A straight wire moving perpandicular to a uniform magnetic field

Motion on rails

Constant :

Constant EMF by source:

Vortex Electric Field

Vortex electric field

In changing magnetic field, forces on static charges is created by vortex electric field. Generalize the definition of :

The general form of Faraday's law:

where is the induced electric field. Differential form:

Induced EMF around uniformly changing cylindrical magnetic field


Self Inductance

Self inductance

Magnetic flux current

Self inductance of the coil , unit: Henry

A coil with significant is an inductor. EMF induced in an inductor:

  1. shows the electromagnetic inertia of a coil
  2. depends on geometry and ferromagnetics
  3. Inductors in AC curcuit have the effect of reactance / impedance

Inductance of a long solenoid

where denotes the volume.

Inductance of coaxial cable

Inductance of a rectangular toroid

Mutual Inductance

Two coils near each other

The change of current in coil 1 causes EMF in coil 2. Total flux in coil 2 created by current :

is the mutual inductance of coil 2 with respect to coil 1, depending on geometry and ferromagnetics. The change of current in coil 2 also causes EMF in coil 1:

It can be proved that

Mutual inductance , unit: Henry

To minimize mutal inductance
Two ideal coupling coils


Total inductance when connecting 2 and 3 (Same direction):

Magnetic Energy Storage

Energy stored in an inductor:

A long solenoid

Energy per unit volume / Energy density:

LC Circuit

A simple LC circuit

In a SHM:

Displacement Current

the discontinuity of current in a charging capacitor

Follow the path , the Ampere's law shows that the magnet field produced by should satisfy

However, under the pervious defination of steady current model, the current enclosed by equals to zero, indicating that

The result is contradictionary since there is no actual current passing through the capacitor:

An extra term is required in Ampere's law to handle the changing . We call normal current in wire the conduction current:

Electric field between plates:

We define the displacement current to reveal changing between the plates:

Ampere's law in a general form:

And continuity of total current:

Maxwell's Equation

Electric field:

Electrostatic Field Induced (Vortex) Electric Field Total Electric Field

Magnetic field:

Conduction Current Displacement Current Total Current

Maxwell's equations:

Maxwell's equations in differential form:

Maxwell's equations for steady magnetic field and electrostatic field:

Maxwell's equations with magnetic charges (magnetic monopoles):

Electromagnetic Waves


Maxwell's equations in free space without charges or conduction currents:

Wave Equations

3D wave equation of electromagnetic wave:

1D (plane) wave equation:


Compared with standard wave equation:

The speed of EM wave:

EM ware is transverse wave in phase

Particular solution of wave equation:

, and are always in phase:


Total enegry stored per unit volume:

With :

Energy transport through EM wave

Energy transports per unit time per unit area:

The direction of energy transporting: