Electricity + Magnetism + Optics=303

In the course of this course, we shall see that the historically separate fields of Electricity, Magnetism, and Optics can be seen as different facets of one, fundamentally electric force and theory.

Today we'll try to put this electric (or "electro-magnetic") force into context, comparing it to some of the other forces and pointing out the regimes in which different forces are more or less important.

Rumble of the forces

We know about four different forces...

    Gravity / Electric / Strong / Weak

Coulomb's law

Much of the first half of the course might be summarized by Coulomb's law which specifies the electric force between two "point" charges: $$ \myv F_{\text el} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{\rr ^2}\uv \rr$$

We'll use SI units -- meters, kilograms, seconds, Newtons -- in which charge is in Coulombs.

The permittivity of free space is $\epsilon_0= 8.85\times 10^{-12}C^2/(Nm^2)$.

$q_1$ and $q_2$ are the charges of two "point" charges at positions $\myv r$ and $\myv r'$, where their relative separation is $\myv \rr \equiv \myv r-\myv r'$.

The charge of an electron is ~$-1.6\times 10^{-19}$ C.

Calculate the magnitude of the electric force between two electrons separated by a distance of 1 m.

Compared to gravity

Newtonian gravity also has a $1/\rr^2$ dependence on separation between two point masses: $$\myv F_{\text grav}=G\frac{m_1 m_2}{\rr^2} \uv\rr.$$ The mass of an electron is $9.1 \times 10^{-31}$ kg, and $G=6.67\times10^{-11} N\cdot m^2/kg^2$.

Calculate the magnitude of the gravitational force between two electrons which are 1 m apart from each other. And then compare to the magnitude of the electric force between them.

$$\begineq F_e&=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{\rr^2}\\ &= \frac{(-1.6\times 10^{-19} C)^2}{4\pi \left[8.85\times 10^{-12}C^2/(Nm^2)\right] 1m^2}=2.3 \times 10^{-28}N\endeq$$

$$\begineq F_g &= G\frac{m_1m_2}{\rr^2}\\ &=6.67\times10^{-11}N m^2 / k g^2 \frac{(9.1 \times 10^{-31} k g)^2} {1m^2}=5.5\times10^{-71} N\endeq$$

$\frac{F_e}{F_g} \approx 10^{42}$ : Looks like the electric force is much stronger than gravity.

Who cares about gravity?

How is it that we are even dimly aware of gravity if the electric force is so much stronger?

Any amount of positive charge attracts negative charge to itself (and vice versa), until the amount of positive and negative charge in any macroscopic body is very nearly equal $\Rightarrow$ the body is 'electrically neutral'.


Gravitational forces alone do an excellent job of explaining Earth's orbit and period. Therefore the Earth and Sun must be ~electrically neutral.


But our Sun is so hot that it is a plasma: A mixture of particles in which electrons are not localized to any particular nucleus. So, to understand circulation of this plasma, solar flares, we *do* need to take into account the EM interactions.

E compared to nuclear forces

The nucleus

At the heart of an atom is the positively charged nucleus surrounded by a cloud of electrons.

Is there anything about this picture that makes you uncomfortable?

Like the fact that those protons aren't flying away from each other?

The strong force

The "strong" force is...
  • stronger than the electric force.
  • stronger than gravity.
  • attractive between any combination of protons and neutrons.

If you're so strong...

If the strong force is so strong, why don't *all* the protons and neutrons in the world universe stick together?

Despite the attraction between nucleons that is stronger than their E-M repulsion, ${}_{83}^{209}$Bi is the largest stable nucleus. Every nucleus with an atomic number greater than 83 is unstable -- that is, (radioactive) -- that is, they all fall apart spontaneously!!.

The Yukawa potential

In his 1935 paper, Hideki Yukawa proposed this general form for the field potential that applies to all four of the forces: $$V_\text{Yukawa}=-g^2\frac {e^{-\alpha m_B r}}{r}.$$ where $g^2$ is the amplitude of the potential (which is assumed to be attractive). For electric interactions, $g^2=q/(4\pi \epsilon_0)$. $m_B$ is the mass of the boson particle that "mediates" the interaction. $\alpha$ is a constant, such that the range of the force is $1/(\alpha m_B)$, and $r$ is the radial distance from a source (charge) of the field.

  • The electric field is mediated by photons which are massless, so $$V_\text{Yukawa}\to V_e \propto \lim_{m_B\to 0}\frac{e^{-\alpha m_B r}}{r}=\frac 1r.$$ Because the photon, the vector boson of the E-M field, has no mass(*).
  • $V_s \propto e^{-\alpha m_\text{meson} r}$ -- vector bosons are mesons with non-zero mass:
    • Strong force mediated by gluons
    • Weak force (responsible for $\beta$ decay in the nucleus) mediated by W and Z bosons.

    (*)We say that $c$ is the "speed of light". But more generally, $c$ could be called "the speed of massless particles in our universe".

    • Electric force mediated by photons
    • Gravitational force mediated by hypothetical "gravitons".
    • Do gravitons have mass? 2017 collision of neutron stars shows nearly simultaneous arrival of light / gravity waves at Earth. (See equation (1).) So, we think gravitons are also massless particles, just like light.
    • Strong > E at small distances, but range is $\approx 1/\beta$.
    • E > Strong at "large" distances (compared to $1/\beta$).

    The strong force (as well as the "weak" force) is short-range. One piece of evidence for this is the limited size of the periodic table.

  • The Electric force turns out to be the most important for these parts of every-day experience:

    • Chemistry is all changes in the configurations of electrons.
    • The contact forces that arise when we push on objects are ultimately electrical.

    Charge

    Electrical charge....

    • comes in two flavors, which we call + and - (plus and minus).
    • appears to be precisely conserved in the universe: both globally and locally.
    • Charge is quantized. The charge of any macroscopic body is only ever $Q= n e$: a positive (or negative) multiple of $e$, the charge on the electron.
    • Why is the charge of the electron--a pointlike "lepton"--precisely the same magnitude (though opposite sign) of a compound "baryon" like the proton (made up of three quarks)?

    Unification of theories

    The following three subjects were considered separate realms of physics before ~1800:

    Magnetism

    Ancient Indians, Greeks and Chinese (at least) were aware of the singular properties of lodestone (magnetized magnetite): of how it attracted other lodestones, as well as iron.

    By the 11th century, the Chinese were using suspended "floating fish": magnetized needles or pieces of iron floating on water as to keep track of direction at night or with low visibility. The Ming dynasty diagram shows the names for different compass headings.

    Electricity


    Electricus is Greek for 'amber-like'.

    ~600 BC: Thales of Miletus observed that amber rubbed with cat's fur could attract light objects. (He believed this was induced 'magnetite-ism'.)

    Other electrical phenomena: static electricity generators, the two varieties of charge, current as a flow of charge, batteries, conductors and insulators.

    Optics

    Newton argued that light consists of particles. Thomas Young showed ~1800 with his slit experiments, that light acts as a wave.

    Optical phenomena included the reading stones invented by Abbas Ibn Firnas in the 9th century, spectacles invented in the 13th century, Galileo's telescope, the splitting of the sun's light via a prism.

    At right, detail from Death of the Virgin by the Master of Heiligenkreuz, early 15th century.

    Unification

    • 1820. Oersted notices that an electric current causes a magnetic needle to change direction.
    • 1831. Faraday finds that a moving magnet can cause an electric current to flow.
    • 1864. James Clerk Maxwell combined electric and magnetic equations into one theory of "electromagnetism", that also implied traveling waves in the electric and magnetic fields.

    These waves travel with velocity $c$, and turned out to be light.

    20th century

    Are the E-M fields just an accounting device for keeping track of the force that one bunch of charge exerts on another bunch?

    We say the interference of E-M waves is what accounts for...

    The interference pattern of a double-slit apparatus--detected as individual particle (photon) impulses.

    QED sez: Waves of the E-M field are the probability amplitudes associated with the Q-M field quanta: photons.

    In summary...

    • Maxwell's theory unifies electricity, magnetism, optics.
    • The theory was already relativistic before relativity.
    • The theory gave the same results as quantum field theories before quantum mechanics: The photon is the massless quanta of the electromagnetic field.

    Image credits

    Ming dynasty compass diagram, Matthew Sullivan, via Earthsky.org, NASA