Redshift Academy

Wolfram Alpha:

Last modified: January 26, 2018
```Maxwell's Equations
-------------------
Integral form:

Ε = ∫E.dl = -∂φB/∂t   where φB = BA (Faraday)

∫B.dl = μ0I + μ0ε0∂φE/∂t  where φE = 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/ε0 (Gauss for E)

∫B.dA = 0 (Gauss for B, no monopoles)

Differential form:

∇ x E = -∂B/∂t    ... 1.

∇ x B = μ0J + μ0ε0∂E/∂t  ... 2.

∇.E = ρ/ε0

∇.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/ε0
s
q = ∫∫∫ρdV
v
∫∫E.dS = (1/ε0)∫∫∫ρdV
s               v
Divergence Theorem:

∫∫∫divEdV = ∫∫E.dS
v          S
= (1/ε0)∫∫∫ρdV
v
∫∫∫(divE - ρ/ε)dV = 0
v
divE - ρ/ε = 0

Faraday's Law
-------------

∫∫B.dS = ρ/ε0
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 = μ0I
c
Now, I = ∫∫j.dS
s

Therefore,

∮B.dr = ∫∫μ0j.dS
c        s

Now apply Stokes Theorem to get:

∫∫(curlB - μ0j).dS = 0
s

So that:

curlB = μ0j

with Maxwell's correction we get:

curlB = μ0(j + ε0∂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) = -μ0ε0∂2E/∂t2

Use identity ∇ x (∇xE) = -∇2E + ∇(∇.E)

ρ = J = 0 (free space vacuum ) so,

∇.E = 0 => ∇2E = μ0ε0∂2E/∂t2

One solution is E = E0sin{2π(x - vt)/λ}

Substitute into above => v2 = 1/μ0ε0 = c2

∇2E = (1/c2)∂2E/∂t2

Similarly, we can also do

From 2. ∇ x (∇ x B) = ∇ x {μ0ε0∂E/∂t}

∇(∇.B) - ∇2B = μ0ε0∂(∇ x E)/∂t

but ∇.B = 0 so

∇2B =  μ0ε0∂(-∂B/∂t)/∂t

∇2B =  μ0ε0∂2B/∂t2

This has a solution similar to E:

B = B0sin{2π(x - vt)/λ}

Light is an self sustaining e-m 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,

∂E0sin{2π(x - vt)/λ}/∂x = -∂B0sin{2π(x - vt)/λ}/∂t

E0(2π/λ)cos{2π(x - vt)/λ} = B0(2πv/λ)cos{2π(x - vt)/λ}

E0 = B0v

v = c = E0/B0

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: UE = ε0E2/2 J/m3

This is derived from the energy stored in a capacitor.

Energy density of B field: UB = B2/2μo J/m3

This is derived from the energy stored in a solenoid.

UTotal = ε0E2/2 + B2/2μo

Now B = E/c so

UB = E2/2μ0c2

Now, c2 = 1/ε0μ0 so

UB = E2ε0μ0/2μ0

=  E2/2ε0

Thus, we can write UTotal in the following equivalent
forms:

UTotal = ε0E2

≡ B2/μo ... substituting E = cB in the UE equation.

≡ cε0EB

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.

I0 = cU0 = cε0E02/2 + cB02/2μo

= cε0E02

≡ cB02/μo

≡ c2ε0E0B0

Now,

c2 = 1/(μ0ε0)

Therefore,

I0 = {1/(μ0ε0)}{ε0E0B0}

= (1/μ0)EB

This is the POYNTING VECTOR.  This more formally
defined as:
-
S = (1/μ0)ExB

= (1/μ0)EBsinθ

= (1/μ0)EB since E and B are orthogonal

The Poynting vector represents the rate of energy
transport per unit area (energy flux) in W per meter2).
The modulus of S, |S| is equal to the intensity, I.

All electromagnetic waves (radio, light, X-rays, etc.)
obey the inverse-square law thus the intensity of an
electromagnetic wave is proportional to the inverse of
the square of the distance from a point source.

ERMS = E0/√2 and BRMS = B0/√2

IAverage = cε0ERMS2

= cBRMS2/μo

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.
```