LOG#204. Mechanics and light.

Gravitational or electrical forces are inverse squared laws:

$F_N=G_N\dfrac{Mm}{R^2}$

$F_C=K_C\dfrac{Qq}{R^2}$

Strikingly similar, they are both also conservative forces. For gravity:

$U_g=-G_N\dfrac{Mm}{R}$

and

$U_e=K_C\dfrac{Qq}{R}$

for the electrical force. Defining the potentials $V_g=U_g/m$ , and $V_e=U_e/q$ , you get the gravitational and electrical potentials

$V_g(r)=-G_N\dfrac{M}{R}$

$V_e(r)=K_C\dfrac{Q}{R}$

Conservative fields are defined from these potentials

$E=-\nabla V_e$

$g=-\nabla V_g$

In general, for any field $\Psi$ , if conservative, $\Psi=-\nabla V$ . The gravitational field reads, from newtonian gravity (module a sign)

$g=\dfrac{F}{m}=G_N\dfrac{M}{R^2}$

and you would get $E=-K_CQ/R^2$ in the coulombian field case! Focusing on the gravitational case (a similar field could be done with the electrical field)…The momentum

$p=mv=m\dfrac{dx}{dt}$

is conserved under any vertical (radial) gravitational field. Imagine you do a traslation

$x'=x-\alpha$

The momentum in the $x$ component reads $p_x=m\dfrac{dx}{dt}=m\dfrac{dx'}{dt}$ ! Note the momentum in the $y$ or vertical component would not be conserved due to $F_g$ ! Thus, symmetry is important. Imagine a spring holding from the upper horizontal surface. Then

$x(t)=A\sin(\omega t)$

where $A=constant$ and

$\dot{x}=A\omega \cos (\omega t)$

with $m\omega^2=k$ , then

$m\ddot{x}=-kx$

and

$E=T+U=\dfrac{m}{2}\dot{x}^2+\dfrac{k}{2}x^2=A^2\dfrac{k}{2}=\dfrac{m\omega^2 A^2}{2}=constant$

Since the spring force is conservative, $F=-kx=-kd(x^2/2)/dx$ , the total energy is conserved. Note the symmetry that says $E$ does not depend on the time and it is constant!

Going 3D is important here. We will use components to avoid vector arrows for convenience. Newton’s second law is

$\sum_i F_i=ma_i$

$v_i=\dfrac{d}{dt}x_i$

$a_i=\dfrac{d}{dt}v_i=\dot{v}_i=\dfrac{d^2}{dt^2}x_i=\ddot{x}_i$

$F_i=\dfrac{d}{dt}p_i$

If $F_i=0$ , then $p_i=constant$ !

Kinetic energy for non-relativistic particles read

$T=\dfrac{1}{2}mv^2=\dfrac{1}{2}m\dot{q_i}^2=\dfrac{p_i^2}{2m}$

If $p_i$ is conserved, then the kinetic energy is also conserved. This is valid for the free particle. In the case of conservative forces, the potential energy reads

$a_i=\dfrac{d^2}{dt^2}x_i=\dfrac{f_i}{m}$

and it yields a uniform motion with solution

$v_i(t)=v_{0i}+\dfrac{f_i}{m}t$

$x_i(t)=x_{0i}+v_{0i}t+\dfrac{f_i}{2m}t^2$

and

$f(x_i-x_{0i})=\dfrac{1}{2}m(v^2_i-v_{0i})$

The first term is precisely:

$W(0\rightarrow f)=\Delta T=\dfrac{1}{2}\Delta v^2$

And thus,

$W(i\rightarrow f)=-\Delta E_p=-\Delta U$

with

$U=-F\cdot \Delta x$

$f_i=-\dfrac{dU}{dx^i}$

i. e., $f=-\nabla U$ , Q.E.D. for any conservative force. $E=T+U$ holds for conservative forces with certain properties in the potential energy (depending on coordinates in a homogeneus way). For the harmonic oscillator:

$a=\ddot{x}$

$\ddot{x}+\omega^2x=0$

and the solution

$x(t)=A\sin (\omega t)+B\cos (\omega t)$

with $x(0)=x_0$ at $t=t_0$ . $t=t_0=0$ in general, so

$x(t)=x_0\cos (\omega t)$

or any other sinusoidal waveform as well. The velocity

$v(t)=\dot{x}=-x_0 \omega \sin (\omega t)$

$a(t)=-x_0\omega^2\cos^2(\omega t)$

and then

$E=\dfrac{1}{2}v(t)^2+\dfrac{1}{2}x(t)^2=\dfrac{1}{2}mx_0^2\omega^2=constant$

as before!

Light can NOT be described with classical NEWTONIAN mechanics. It took several decades an roughly speaking several centuries to code electromagnetic laws into a single set of equations. Maxwell wrote the synthesis of our current electromagnetic knowledge of light:

Gauss law for the electric field: $\nabla \cdot E=div E=\rho/\varepsilon_0=4\pi K_C\rho$ . Equivalently, $\phi=\oint_S E\cdot dS=4\pi K_CQ=Q/\varepsilon_0$ . For point particles, this law provides $E_i=K_CQx_i/r_i^3=Qu_i/4\pi\varepsilon_0 r_i^2$ . Moreover, $F_i=qE_i$ , and the gauge field $E_i=-\nabla_i\varphi$ . $\varphi=V=Q/4\pi\varepsilon_0 r_i$ is the potential.
Faraday’s law: $\nabla\times E=-\partial_t B$ , or equivalently $\oint_\Gamma E\cdot dl=-\partial_t\oint_C B\cdot dS$ .
Gauss law for the magnetic field (no magnetic monopoles in standard electromagnetism): $\nabla\cdot B=0$ , or $\oint_SB\cdot dS=\phi_B=0$ .
Ampere’s law: $\nabla\times B=j/\varepsilon_0c^2$ . This original Ampere’s law does not conserve electric charge, so Maxwell added an extra term, the displacement current, yielding
$\nabla\times B=j/\varepsilon_0c^2+(\partial_t E)/c^2$

.

The combination of the 4 equations above produces wave-like equations for $E, B$ :

$\dfrac{1}{c^2}\partial_t^2 E_i-\nabla^2 E_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) E_i=0$

$\dfrac{1}{c^2}\partial_t^2 B_i-\nabla^2 B_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) B_i$

Plane wave solutions are allowed, with $\sin(\omega t-k_i x^i)$ or generally

$E_i\sim c B_i\sim e^{\omega t-k_i x^i}$

Wave speed is given by

$\dfrac{1}{c^2}=\varepsilon_0\mu_0$

so Maxwell cleverly pointed out that light should be an electromagnetic wave! Furthermore, $E\perp B\perp v$ in general. Light can also be polarized. Polarization or fluctuations in the directions of $(E, B)$ is due to the transverse character of the electromagnetic waves. Maxwell’s equations unify $E,B$ into a single framework. All the electromagnetic phenomena from a common dynamics. Special relativity allows to condense Maxwell equations into $\partial_\mu F^{\mu\nu}=j^\nu$ and $\varepsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0$ . Clifford algebra simplify these equations into a single $\partial F=j$ . Differential forms also allows for such a simplification. Maxwell equations have a new invariance beyond galilean invariance: Lorentz invariance. Essentially, Maxwell equations imply Special Relativity.