# Physics Speedrun - Electromagnetic Induction

## Faraday's Law of Induction

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

For a

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

By Lorentz force:

Motional EMF:

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

Constant

Constant EMF by source:

### 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

## Inductor

### Self Inductance

Magnetic flux

Self inductance of the coil

A coil with significant

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

where

### Mutual Inductance

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

**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

Generally,

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

### Magnetic Energy Storage

Energy stored in an inductor:

Energy per unit volume / Energy density:

### LC Circuit

In a SHM:

## Displacement Current

Follow the path

However, under the pervious defination of steady current model, the *current* enclosed by

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 **conduction current**:

Electric field between plates:

We define the **displacement current** to reveal changing

**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

## 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:

### Properties

Compared with standard wave equation:

The speed of EM wave:

Particular solution of wave equation:

### Energy

Total enegry stored per unit volume:

Energy transports per unit time per unit area:

The direction of energy transporting: