Electromagnetic Propagation: Difference between revisions
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===Ampere-Maxwell Law=== | ===Ampere-Maxwell Law=== | ||
This law states that the magnetic field times the path length is equal to <math>\mu_0 \sum I_{enclosed}+ | This law states that the magnetic field times the path length is equal to <math>\mu_0 \sum I_{enclosed}+ \frac{d\phi_{electric}}{dt}\mu_0\epsilon_0</math> The change in electric flux is equal to Evh. The magnetic field integral is Bh. Equating the two yields that <math>\mu_0\epsilon_0Evh = Bh </math> Simplifying yields that <math> B = \mu_0\epsilon_0vE</math> | ||
So using our model of a box-like particle and applying Ampere-Maxwell’s equations to it, we’ve found that E = vB and | So using our model of a box-like particle and applying Ampere-Maxwell’s equations to it, we’ve found that <math> E = vB </math> and <math> B = \mu_0\epsilon_0vE </math> 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=== | ===Difficult=== | ||
Revision as of 21:49, 2 December 2015
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: Maxwell's Equations
Gauss's Law for Electricity and Magnetism
[math]\displaystyle{ \sum \phi_{electric} = \frac{q_{inside}}{\epsilon_0} = \int \vec{E} \bullet d\vec{A} }[/math] [math]\displaystyle{ \sum \phi_{magnetic} = 0 = \int \vec{B} \bullet d\vec{A} }[/math]
Faraday's Law
[math]\displaystyle{ \int \vec{E} \bullet d\vec{l} = -\frac{d\phi_{magnetic}}{dt} }[/math]
Ampere-Maxwell Law
[math]\displaystyle{ \int \vec{B} \bullet d\vec{l} = \mu_0\epsilon_0\frac{d\phi_{electric}}{dt} + \mu_0I_{enclosed} }[/math]
An Interactive Computational Model
Follow this link to find an interesting little animation.
http://www.walter-fendt.de/ph14e/emwave.htm
Proposed Model
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.
Does the Model Fit the Math?
Gauss's Law
Lets use the slab as a Gauss surface. Calculate the flux through the surface. Applying Gauss' Law shows us that since there is equal amounts of magnetic and electric flux going into and out of the slab, there is no net charge or magnetic monopole through the slab.
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 [math]\displaystyle{ \mu_0 \sum I_{enclosed}+ \frac{d\phi_{electric}}{dt}\mu_0\epsilon_0 }[/math] The change in electric flux is equal to Evh. The magnetic field integral is Bh. Equating the two yields that [math]\displaystyle{ \mu_0\epsilon_0Evh = Bh }[/math] Simplifying yields that [math]\displaystyle{ B = \mu_0\epsilon_0vE }[/math] So using our model of a box-like particle and applying Ampere-Maxwell’s equations to it, we’ve found that [math]\displaystyle{ E = vB }[/math] and [math]\displaystyle{ B = \mu_0\epsilon_0vE }[/math] 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
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Edited by Chiagoziem Obi