Physics 249 Lecture 2, Sep 7th 2012
Reading: Today Chapter 3, Monday Chapter 4
HW: Posted on web site later today.
1) The discreet nature of electric charge.
Matter can have electric charge. For instance, an object can be charged up and a wire can
carry a flow of electric charge.
Michael Faraday discovered that the same quantity of electricity could decompose
Avagadro’s number of molecules into monovalent ions. It didn’t matter what molecule
as long as the ionic bonds in the atoms were of the type that involved monovalent ions,
ions with one valence electron. This meant that there was Avagadro’s number of
discrete objects(?) in the electricity that when injected in the system broke up the bond. I
= qNA. In this case we now understand that the electron fills the orbit in one of the
molecules that was previously filled by the atoms sharing an electron in an ionic bond.
With a separate estimate of Avagodro’s number it was even possible to measure the
charge of the object. However, again the object in question had not yet been observed.
Observation of the electron.
When there are electrical discharges in a gas cathode rays are emitted. This was true of
many materials. JJ Thompson measured the charge over mass ratio of these particles.
The measurement required an understanding of Newtons law’s and the laws of electricity
and magnetism
Force of a magnetic field. FB = qvB
Force to bend a particle into an orbit radius r. F = mac = mv2/r
Equating the two such that the magnetic field is the force: qvB = mv2/r, q/m = v/RB
Though you still need to measure v which isn’t easy to do for fast particles. This best can
be done by balancing electric and magnetic forces: qvB = qE, v = E/B.
From this you could measure the mass of the particles if it were the same in Faraday’s
and Thompson’s experiments. However, that had to be proved.
Millkan’s oil drop experiment. By charging individual oil drops of known mass you
could levitate them in an electric field qE=mag. Indeed the charge was the same.
Charge 1.601x10-19 C
Mass 9.109x10-31 kg (2000 times less than the lightest atom, the hydrogen atom).
2) Discrete nature of radiation
Matter can absorb and emit radiation. Absorbed radiation will increase the (oscillatory)
kinetic energy and Temperature of the matter and emission from the oscillating
(accelerating) charges in the matter will lower the kinetic energy and Temperature of the
matter. The emission process is known as thermal emission. When the rate of absorption
equals the rate of emission the matter is in thermal equilibrium. A perfect absorber and
emitter is called an ideal black body. The spectrum of this thermal radiation and total
power of a black body can be analyzed.
The total power can be found experimentally or derived classically as proportional to
εσT4 , where the proportionality constant, ε, for an ideal black body is 1 and σ is the
Stefan–Boltzmann constant. However neither the shape of the distribution or the most
probably value of the distribution which varies inversely with temperature was predicted
by classical physics. Further if you attempt to derive the energy distribution and integrate
it to find the total energy or power the classical approach fails.
The classical and quantum derivations approximate an ideal black body as a cavity with a
small hole where the light can shine in or out. This system absorbs almost all light
incident on the hole as an ideal blackbody does, since the light can’t bounce directly out.
The only difference is that the amount of energy in an out is reduced by the size of the
hole relative to the area of the cavity. If you don’t think this approximation seems
reasonable the comparison could be done with an actual cavity with the same result.
The classical approach involves finding the distribution of the number of modes of
oscillation of the classical electromagnetic field in the cavity per unit wavelength or the
number density, multiplying it by the average energy per mode to get the energy of the
waves and then integrating over wavelengths to find the total energy and power.
The number density of wavelengths depends on the geometry of the cavity and what
standing modes are allowed and can be calculated as:
𝑛(𝜆) = 8𝜋𝜆−4
The average energy is found by understanding oscillator energy distribution or the
fraction of oscillators with energy E per unit energy at a given temperature T and then
integrating the energy E times the fraction to find the average. This is based on the idea
that energy absorbed by the system is causing harmonic oscillation in the elements of the
system raising their energy and when they emit light the frequency of that light is the
same as the frequency of the oscillation and then has energy E. This is true of light
produced by accelerated (oscillating) charges.
First for the distribution where f is fraction of oscillators with energy E and when
integrated will add up to one.
𝑓(𝐸) = 𝐴𝑒 −𝐸/𝑘𝑇
This is taken from the Maxwell-Boltzmann distribution for harmonic oscillators in a
system with Temperature T and is derived using statistical mechanics and k is the
Boltzmann constant.
Then we find and average energy:
∞
∞
𝐸̅ = ∫ 𝐸𝑓(𝐸)𝑑𝐸 = ∫ 𝐸𝐴𝑒 −𝐸/𝑘𝑇 𝑑𝐸 = 𝑘𝑇
0
0
The energy density is:
𝑢(𝜆) = 8𝜋𝑘𝑇𝜆−4
Which both has an infinite density (and peak wavelength) at small wavelengths and
integrates to infinity.