Electromagnetic Propagation

Electromagnetic Propagation Model

The Main Idea

What is the ideal model for the propagation of electromagnetic waves? Does it look like a sheet of electrons? Maybe it can be described as a spring? A mountain? No! The following explains why the best model is a traveling slab of perpendicular waves that oscillate back and forth.

A Mathematical Model

Gauss's Law: [math]\displaystyle{ \sum \phi_{electric} = \frac{q_{inside}}{\epsilon_0} }[/math] [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

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Simple

Introduction

Waves may be a familiar concept. We can imagine waves as the familiar up-and-down movement of crests and troughs. The concept of a wave is also applicable to the physics of radiation. According to Maxwell’s equations, a time varying electric field produces a magnetic field, just as a time varying electric field produces a magnetic field. We can try to picture what such a time varying field would “look” like. Let’s propose a model for it.

We can call this single traveling box of field a pulse. There are no charges inside or outside of it. It is simply a traveling box of electric field. It generates a perpendicular magnetic field. Both electric field and magnetic field are perpendicular to the direction that the fields are moving. We can call this a “pulse” of electromagnetic field.

In order to be an acceptable model, it must satisfy all four of the equations of Maxwell. Applying Guass’s law for electricity and magnetism we see that the model works out.

Mathematical Principles

Gauss's Law

Faraday's Law

Applying Faraday’s law involves realizing that the pulse in moving through space, and, for an imaginary path, the flux is changing through that path. We can pick a path h high and w wide, and allow part of the path to be inside the box of pulse and some of the path to be outside the pulse, initially. A short time, Δt, later the pulse has moved, encompassing more of the path. The change in the flux is the magnitude of the magnetic field times the velocity of the particle times the height of the path. The change in flux is equal to the emf around the path. The electric field times the height is equal to the emf. So Eh = Bvh. Simplifying that solution yields that the electric field is equal to the velocity times the magnetic field.

Ampere-Maxwell Law

This law states that the magnetic field times the path length is equal to mu naught times the quantity of the sum of currents inside the path plus the change in electric flux times epsilon naught. The change in electric flux is equal to Evh. The magnetic field integral is Bh. Equating the two yields that mu naught, epsilon naughtEvh is equal to Bh. Simplifying yields that B = mu naught epislon naught vE. So using our model of a box-like particle and applying Maxwell’s equations to it, we’ve found that E = vB and B = mu epsilongVE. Also the electric flux and magnetic flux equal zero. Solving for v we find that v = 3e8 m/s. This is the speed of light.

Difficult

Connectedness

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History

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Edited by Chiagoziem Obi