## The Electrodynamics of Comet Dust

It was believed that the principle force acting on the dust in a comet’s trail was radiation pressure. In 1982, Professor M.K. Wallis and Professor M.H.A. Hassan (later Dean of the University of Khartoum and President of the Network of Academies of Science in Africa) showed that in a certain regime electromagnetic forces dominated those from radiation pressure. This had very practical consequences as certain spacecraft (such as Giotto and Vega) had shields that did not account for this fact.

1. A particle in a uniform magnetic field will naturally go in a circle with a characteristic radius and frequency (known equivalently as the cyclotron radius, Larmor radius, gyroradius, etc.). Given that $\omega=10^{-23}\frac{\omega_p}{a^2}$ where the initial factor is due to a plasma physics calculation, $.01 \mu m\leq a\leq .1 \mu m$ is the radius of our piece of comet dust, and $\omega_p$ is the cyclotron radius of the proton in a $10^{-4}$ Gauss magnetic field. Calculate the cyclotron radius of the proton and give the order of magnitude of the expected radius and period of circular motion using the fact that a typical speed of a grain caught in a solar wind is $\sim 10^5ms^{-1}$. Compare this to the 270m width of Hailey’s comet and the time it takes the comet to be in an entirely new place $\sim$.1s.
2. The long length and timescale of cyclotron motion compared to the comet means that linear motion is a good approximation in our case; show this by calculating the dominant contribution to  $m\dot{v}=q(E+v\times B)$, the acceleration tangential to the comet.
3. The smaller grains are also subject to radiation acceleration $a_r\sim g_1*(\frac{a}{.1\mu m})^s$. Calculate $v_{tot}$ assuming the grain undergoes cylotron motion with radius $8*10^7m$ (this radius is due to effects on the charge of the grain that we have elided for simplicity).
4. The stream of particles generated by these effects will be confined to a characteristic radius, calculate $L=\frac{v_{tot}^2}{|a_{tot}|}$ to see how far one must be from the comet to avoid this high-speed stream of charged particles.

References:

Electrodynamics of submicron dust in the cometary coma.
Wallis, M. K. ; Hassan, M. H. A.
Astronomy and Astrophysics (ISSN 0004-6361), vol. 121, no. 1, May 1983, p. 10-14.

## Noether’s Theorem

1. Write the kinetic energy for a particle in projectile motion
2. Write the potential energy for a particle in projectile motion
3. We can now construct the Lagrangian $L=T-V$. Use the Euler-Lagrange Equations ($\frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}=\frac{\partial L}{\partial x_i}$) to find two conserved momenta and the equation of motion for free-fall.
4. Can you see under what general condition momentum in a given direction will be conserved by studying the Euler-Lagrange Equations?

The fact that every differentiable symmetry of a Lagrangian yields a conserved quantity is known as Noether’s Theorem and was first proved by Emmy Noether in 1915. One of the founders of Abstract Algebra and a member of the Göttingen school of Mathematics, she fled to the United States in 1933 after being expelled from her position by the Nazi party. A selection of biographies of Emmy Noether can be found here.

## The Wu Experiment

For a long time it was believed that parity symmetry (the interchange of left and right) held in every interaction. In 1956, the Chinese-American physicist Chien-Shiung Wu showed that it is in fact violated, specifically in the interaction $^{60}_ {27}Co\rightarrow^{60}_ {28}Ni+e^-+\bar{\nu}_e+2\gamma$. T.D Lee and C.N. Yang had suggested to her that pseudo scalar quantities such as  $\langle\sigma \cdot p_e\rangle$, where $\sigma$ – is the nuclear spin and $p_e$ is the electron momentum – might actually not be invariant under parity conservation. No physicist had ever measured such a quantity, so C. S. Wu quickly devised a novel experiment to do so. In this problem we will follow a calculation of Wu to investigate why parity conservation implies that the expectation value of the chosen pseudo-scalar is 0

1. The parity operator acts on a wave function as follows $P\psi(x)=\psi(-x)$. Show for an odd or even wavefunction viz. (odd: $\psi(-x)=-\psi(x)$, even: $\psi(-x)=\psi(x)$) that the probability the wave function is between $-a$ and $a$ is invariant under parity.
2. Show that parity invariance implies that $\langle\sigma \cdot p_e\rangle=0$

References:

The discovery of the parity violation in weak interactions and its recent developments
Chien-Shiung Wu
Published in: Lect.Notes Phys. 746 (2008), 43-70

Experimental Test of Parity Conservation in Beta Decay
C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson
Phys. Rev. 105, 1413 – Published 15 February 1957