Introducing Basic Electrodynamics

Probably one of the more important areas of physics is electrodynamics, not only because of its innate applications (as to electro-magnetic waves and phenomena) but its use in other related disciplines such as plasma physics, astrophysics and solar physics. We already saw (very briefly and generically) for example, how electrodynamics figures into the Earth's magnetosphere and its magnetic field, e.g.

http://brane-space.blogspot.com/2011/03/look-at-earths-magnetic-dipole-field.html

So let's begin by looking at some basic field derivations using the diagram shown, based on a rectangular wire loop placed in a crossed (E, B) field in a rectangular coordinate system (x,y, z). The field intensities E, B are shown in the directions indicated.

We begin by the general definition for the magnetic flux, φ:

φ = B (A)

where B is the magnetic induction and A the area of the region through which the flux passes.


Now, from the diagram, since the area of the wire loop is: A = l dx

Then the magnetic flux through it is:

φ= B ldx

and the rate of change of magnetic flux (from Faraday's law) is:

dφ/dt = d/dt {B ldx} = dB/dt (ldx)

Further, taking the integral around the loops we find:

INT(0) E*ds = -dφ/dt

->

(@E/@x) dx l = -l dx @B/@t

where @ denote partials.

Thence: @E/@x = -@B/@t

Similarly,

INT(0) B*ds = l dx ( -@B/@x)

And the electric flux through the rectangle is expressed:

@φ(E)/@t = ldx (@E/@t)


More completely, with the appropriate constants of proportionality put in (u_o, e_o):

(where u_o is the magnetic permeability = 4 π x 10^-7 H/m

and e_o is the electric permittivity of free space, viz.

e_o = 8.85 x 10^-12 F/m)


- (@B/@x) dxl = u_o e_o ldx (@E/@t)


or finally,

(@B/@x) = -u_o e_o (@E/@t)

and if one takes the derivative of: @E/@x = -@B/@t

and combines with the preceding equation, one obtains the Maxwell wave equations:


1) (@^2E/@x^2) = u_o e_o (@^2E/@t^2)

2) (@^2B/@x^2) = -u_o e_o (@^2B/@t^2)

Note that in all these cases so far we are looking at plane wave solutions of the forms:

E= E(max) cos (kx - wt)

B = B(max) cos (kx - wt)

where the 'max' indices refer to maximum amplitude values of the field components, E, B, and w = 2 πf, where f is the frequency.

Using the above, interested readers ought to be able to satisfy the Maxwell wave equations given!

Next: The Energy carried by Electro-magnetic waves.


Readers are invited to do the differentiation and substitution and verify this for themselves!