A brief review of field theory concepts

• Reading:
  • ``Gauge Fields, Knots and Gravity'' by John Baez  & Javier P. Muniain , World Scientific Publishing Co. Pty Ltd, 1994, ISBN 9810220340
  • Only the first 158 pages: in fact much less$\ldots$, we'll review the most basic stuff only.
  • Material covered by the lecture should be sufficient.
• An example of field equations: 
  
      

  $$\boldsymbol{\nabla}\cdot\boldsymbol{E} = \frac{1}{\epsilon_0}\rho\qquad \hbox{Coulomb Law}$$ (3.1)

  $$\boldsymbol{\nabla}\cdot\boldsymbol{B} = 0\qquad \hbox{No magnetic monopoles}$$ (3.2)

  $$\frac{\partial\boldsymbol{E}}{\partial t} = c^2\boldsymbol{\nabla}\times\boldsymbol{B} - \frac{1}{\epsilon_0}\boldsymbol{j}\qquad \hbox{Amp\\lq ere-Maxwell Law}$$ (3.3)

  $$\frac{\partial\boldsymbol{B}}{\partial t} = - \boldsymbol{\nabla}\times\boldsymbol{E}\qquad \hbox{Faraday Law}$$ (3.4)

  
  

• The operators $\boldsymbol{\nabla}$ are vector   differentiation operators, which, in Cartesian coordinates only, look as follows  
  

  $$\boldsymbol{\nabla} \equiv \left[ \frac{\partial}{\partial ... ...\partial}{\partial y}, \frac{\partial}{\partial z} \right]$$ (3.5)

  
  

• t is time
• E is the electric field
• B is the magnetic field
• $\rho$ is the charge density
• j is the current density
• c and $\epsilon_0$ are constants, which depend on the choice of a system of units.

• It is easy to see that c is the   speed of light. Take $\partial/\partial t$ of equation (3.4), and assume vacuum, i.e., $\rho = 0$ and j = :
  
  $$\begin{eqnarray*}\frac{\partial}{\partial t}\frac{\partial}{\partial t}\boldsymb... ... - \nabla^2\boldsymbol{B} \right) = c^2\nabla^2\boldsymbol{B} \end{eqnarray*}$$
  
  hence
  

  $$\nabla^2\boldsymbol{B} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \boldsymbol{B} = \boldsymbol{0}$$ (3.6)

  
  
  Here $\nabla^2$ is the Laplace   operator, which, in Cartesian coordinates only, is given by:
  

  $$\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$ (3.7)

  
  

• Observe that $\partial t^2$ is multiplied by c² in the   wave equation. This is necessary, so that both differential operators would divide B by the same thing, i.e., $\hbox{m}^2$.
• Emacs Calc has   numerous functions for operations on units, which
  • simplify units
  • convert units
  • remove units
  • autorange units (i.e., prefixes such as ``k'' or ``M'' are automatically added so that the numbers don't get ridiculously large or small)
  • let you define your own units.
  • example:

    '150 km
    '3 hr
    /
    1:  50 km / hr
    uc
    New units: mi/hr
    1:  31.0685596119 mi / hr
    uc
    New units: m/s
    13.8888888889 m / s
    '172 cm
    uc
    New units: ft+in
    1:  5 ft + 7.71653543307 in
    '69 kg
    uc
    New units: lb
    1:  152.118960908 lb
    

  • If you don't know how to input a particular unit, view calc-units.el in

    /afs/ovpit.indiana.edu/common/gnu/share/emacs/site-lisp/calc-2.02f
    

• Emacs Calc also knows about fundamental physics constants, e.g., the Planck   constant, Gravitational   constant, Avogadro   constant and many others:

  '1 Grav
  uc
  New units: N
  1:  6.67259e-17 N m^2 / g^2
  

• Similarly   we can derive the wave equation for E by taking the time derivative of the Ampère-Maxwell Law (equation (3.3))
  
  $$\begin{eqnarray*}\frac{\partial}{\partial t}\frac{\partial}{\partial t} \boldsym... ...) - \nabla^2\boldsymbol{E} \right) = c^2\nabla^2\boldsymbol{E} \end{eqnarray*}$$
  
  hence
  

  $$\nabla^2\boldsymbol{E} - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \boldsymbol{E} = \boldsymbol{0}$$ (3.8)

  
  

• In order to get these wave equations for B and E we have assumed $\rho = 0$ and j = too. In presence of charges and currents things get complicated.
  • waves in conductors  propagate only above the   plasma frequency, i.e., longer waves reflect, shorter waves pass through; cf. colour of gold, reflection of AM waves from the ionosphere;
  • waves in semiconductors   propagate only if photon energy is less than the bandgap (this is a quantum effect that cannot be explained using Maxwell equations);
  • waves in   dielectrics (glass, water) propagate but with a different phase velocity, and usually with some dispersion, i.e., different wavelengths have different phase velocities.
  • nonlinear effects   can sometimes act in the opposite direction to dispersion and form   the so called solitons. These are of increasing importance in physics of optical fibres and laser beams.