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Maxwell's Equations

Integral form:
Ε = ∫E.dl = ∂φ_{B}/∂t where φ_{B} = BA (Faraday)
∫B.dl = μ_{0}I + μ_{0}ε_{0}∂φ_{E}/∂t where φ_{E} = EA (Ampere  Maxwell)
The displacement current correction was added to
Ampere's law by Maxwell to explain the presence of
a magnetic field between capacitor plates even
though no current is threading through the path.
∫E.dA = Q/ε_{0} (Gauss for E)
∫B.dA = 0 (Gauss for B, no monopoles)
Differential form:
∇ x E = ∂B/∂t ... 1.
∇ x B = μ_{0}J + μ_{0}ε_{0}∂E/∂t ... 2.
∇.E = ρ/ε_{0}
∇.B = 0
The differential and integral formulations of the
equations are mathematically equivalent, by the
divergence theorem in the case of Gauss's law and
Gauss's law for magnetism, and by Stokes theorem
in the case of Faraday's law and Ampere's law.
Proof:
Gauss's Law

∫∫E.dS = q/ε_{0}
^{s}
q = ∫∫∫ρdV
^{v}
∫∫E.dS = (1/ε_{0})∫∫∫ρdV
^{s} ^{v}
Divergence Theorem:
∫∫∫divEdV = ∫∫E.dS
^{v} ^{S}
= (1/ε_{0})∫∫∫ρdV
^{v}
∫∫∫(divE  ρ/ε)dV = 0
^{v}
divE  ρ/ε = 0
Faraday's Law

∫∫B.dS = ρ/ε_{0}
^{s}
If we replace E with B and set ρ = 0 (since there
is no such thing as magenetic charge (monopole)), we
get:
divB = 0
φ = ∫∫B.dS
^{s}
V = dφ/dt and V = ∮E.dr (E = V/r)
^{c}
∮E.dr = dφ/dt = d/dt{∫∫B.dS}
^{c} ^{s}
= ∫∫(∂B/∂t).dS)
^{s}
Now apply Stokes Theorem, ∮E.dr = ∫∫curlE.dS, to get:
^{c} ^{s}
∫∫{curlE + ∂B/∂t).dS = 0
^{s}
So that:
curlE + ∂B/∂t = 0
Ampere's Law

∮B.dr = μ_{0}I
^{c}
Now, I = ∫∫j.dS
^{s}
Therefore,
∮B.dr = ∫∫μ_{0}j.dS
^{c} ^{s}
Now apply Stokes Theorem to get:
∫∫(curlB  μ_{0}j).dS = 0
^{s}
So that:
curlB = μ_{0}j
with Maxwell's correction we get:
curlB = μ_{0}(j + ε_{0}∂E/∂t)
Both the differential and integral formulations
are useful. The integral formulation can often
be used to simply and directly calculate fields
from symmetric distributions of charges and
currents. On the other hand, the differential
formulation is a more natural starting point for
calculating the fields in more complicated (less
symmetric) situations.
Wave Equation:
From 1. ∇ x (∇ x E) =  ∂(∇ x B)/∂t
Sub 2. ∇ x (∇ x E) = μ_{0}ε_{0}∂^{2}E/∂t^{2}
Use identity ∇ x (∇xE) = ∇^{2}E + ∇(∇.E)
ρ = J = 0 (free space vacuum ) so,
∇.E = 0 => ∇^{2}E = μ_{0}ε_{0}∂^{2}E/∂t^{2}
One solution is E = E_{0}sin{2π(x  vt)/λ}
Substitute into above => v^{2} = 1/μ_{0}ε_{0} = c^{2}
∇^{2}E = (1/c^{2})∂^{2}E/∂t^{2}
Similarly, we can also do
From 2. ∇ x (∇ x B) = ∇ x {μ_{0}ε_{0}∂E/∂t}
∇(∇.B)  ∇^{2}B = μ_{0}ε_{0}∂(∇ x E)/∂t
but ∇.B = 0 so
∇^{2}B = μ_{0}ε_{0}∂(∂B/∂t)/∂t
∇^{2}B = μ_{0}ε_{0}∂^{2}B/∂t^{2}
This has a solution similar to E:
B = B_{0}sin{2π(x  vt)/λ}
Light is an self sustaining em wave! Changing E
field => changing B field => changing E field. The
B field is perpendicular to the E field and both
have the same phase (i.e. zero phase difference).
From 1 again. ∇ x E = ∂B/∂t
Therefore,
∂E_{0}sin{2π(x  vt)/λ}/∂x = ∂B_{0}sin{2π(x  vt)/λ}/∂t
E_{0}(2π/λ)cos{2π(x  vt)/λ} = B_{0}(2πv/λ)cos{2π(x  vt)/λ}
E_{0} = B_{0}v
v = c = E_{0}/B_{0}
Thus, the ratio of the electric to magnetic
fields in an electromagnetic wave in free
space is always equal to the speed of light.
Energy density of E field: U_{E} = ε_{0}E^{2}/2 J/m^{3}
This is derived from the energy stored in a capacitor.
Energy density of B field: U_{B} = B^{2}/2μ_{o} J/m^{3}
This is derived from the energy stored in a solenoid.
U_{Total} = ε_{0}E^{2}/2 + B^{2}/2μ_{o}
Now B = E/c so
U_{B} = E^{2}/2μ_{0}c^{2}
Now, c^{2} = 1/ε_{0}μ_{0} so
U_{B} = E^{2}ε_{0}μ_{0}/2μ_{0}
= E^{2}/2ε_{0}
Thus, we can write U_{Total} in the following equivalent
forms:
U_{Total} = ε_{0}E^{2}
≡ B^{2}/μ_{o} ... substituting E = cB in the U_{E} equation.
≡ cε_{0}EB
The intensity can be found by taking the energy
density (energy per unit volume) at a point in
space and multiplying it by the velocity at which
the energy is moving. The resulting vector has
the units of power divided by area.
I_{0} = cU_{0} = cε_{0}E_{0}^{2}/2 + cB_{0}^{2}/2μ_{o}
= cε_{0}E_{0}^{2}
≡ cB_{0}^{2}/μ_{o}
≡ c^{2}ε_{0}E_{0}B_{0}
Now,
c^{2} = 1/(μ_{0}ε_{0})
Therefore,
I_{0} = {1/(μ_{0}ε_{0})}{ε_{0}E_{0}B_{0}}
= (1/μ_{0})EB
This is the POYNTING VECTOR. This more formally
defined as:

S = (1/μ_{0})ExB
= (1/μ_{0})EBsinθ
= (1/μ_{0})EB since E and B are orthogonal
The Poynting vector represents the rate of energy
transport per unit area (energy flux) in W per meter^{2}).
The modulus of S, S is equal to the intensity, I.
All electromagnetic waves (radio, light, Xrays, etc.)
obey the inversesquare law thus the intensity of an
electromagnetic wave is proportional to the inverse of
the square of the distance from a point source.
E_{RMS} = E_{0}/√2 and B_{RMS} = B_{0}/√2
I_{Average} = cε_{0}E_{RMS}^{2}
= cB_{RMS}^{2}/μ_{o}
Photons

In the quantum description, the electromagnetic field
is an observable property of photons. Photons can be
thought of as mini E/M wave segments that consists of
an oscillating electric field component, E, and an
oscillating magnetic field component, B. The electric
and magnetic fields are orthogonal (perpendicular) to
each other, and they are orthogonal to the direction
of propogation of the photon. The E and B fields flip
direction as the photon travels. The number of
oscillations that occur in one second is the frequency,
f. The superposition of a sufficiently large number
of photons has the characteristics of a continuous
electromagnetic wave. The energy of the photon is given
by E = hf and a wavelength is equal to c/f. There is a
correspondence between the energy of a photon stream
and the Poynting vector in the classical approach.