# Presentation: 505522

## Maxwell Equations

**Maxwell Equations**
**and Special Relativity in Accelerators**

**David Robin**

## Need to know how particles will move in the presence of electric and magnetic fields. Present a basic review of classical physics*
Equations of Motion
Calculations of the Fields
Special Relativity
Give a couple examples

* Will ignore quantum mechanical effects for now

- Equations of Motion
- Calculations of the Fields
- Special Relativity
- Give a couple examples
** Will ignore quantum mechanical effects for now*

## Feynman Lectures by R. Feynman, R. Leighton, and M Sands
Introduction to Electrodynamics by D. Griffith
Spacetime Physics by E. Taylor and J. Archibald
Particle Accelerator Physics, Basic Principles and Linear Beam Dynamics by H. Wiedemann

- Introduction to Electrodynamics by D. Griffith
- Spacetime Physics by E. Taylor and J. Archibald
- Particle Accelerator Physics, Basic Principles and Linear Beam Dynamics by H. Wiedemann

## Newton’s Law’s of Motion

- Lorentz Force Equation – Force on a charged particle traveling with velocity, *v*, in the presence of an electric, *E*, or magnetic, *B*, field

- To determine the particle motion one needs to know the electric and magnetic fields – Maxwell’s Equations

## I. Divergence Theorem 

- II. Curl Theorem

## Curl Theorem (Stoke’s Theorem)

- Divergence Theorem

- Volume Integral

- Surface Integral (Flux)

- Line Integral (Circuitation)

- or

- (***F*** is conservative if curl ***F*** is zero)

## I. Gauss’ Law 

- (Flux of *E* through a closed surface) = (Charge inside/0)

- II. (No Name)
- (Flux of *B* through a closed surface) = 0

- permittivity of free space

## III. Faraday’s Law

- (Line Integral of *E* around a loop) =
- 					-d/dt(Flux of B through the loop)

- IV. (Ampere’s Law modified by Maxwell)
- (Integral of B around a loop) = (Current through the loop)/0
- 				+d/dt(Flux of E through the loop)

- permeability of free space

## In free space combining Gauss’ Law and Faraday’s Law

- In free space combining Ampere’s Law and the last Maxwell Equations

- Equation of a wave in three dimensions is

## Slide 10

## Perhaps Light is an electromagnetic wave

- Equation of a wave in three dimensions is

- Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves traveling at the speed of light
**Perhaps Light is an electromagnetic wave**
**“****We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena” - Maxwell**

## Slide 12

## The principle of relativity. The laws of physics apply in all inertial reference systems.

** ****The universal Speed of light.** The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source.

## where v is the velocity of the particle and 

- where *v* is the velocity of the particle and
*c* is the velocity of light

- where *m* is the rest mass of the particle

## Rest Energy, Eo : Eo = mc2

- Rest Energy, *E**o** : E**o** = mc**2*
- Total Energy, *E : E = m****c**2*
- Momentum, *p : p = m****v*

## Two particles have equal rest mass m0.

**Laboratory Frame (LF):** one particle at rest, total energy is E.

**Centre of Mass Frame (CMF):** Velocities are equal and opposite, total energy is Ecm.

- The quantity		is invariant.
- In the **CMF****,**** **we have
- In general
- In the **LF****,**** **we have		       and
- And finally

## Two inertial frames moving with respect to each other with velocity,v

- Two inertial frames moving with respect to each other with velocity,*v*

## Two celebrated consequences of the transformation are 

- Two celebrated consequences of the transformation are
- Time dilation and Lorentz contraction
**Time dilation**. A clock in the primed frame located at *x = vt* will show a time dilation, *t’ = 1/***
**Lorentz contraction**. An object in the primed frame with length *L’* along the *x’* axis and is at rest in the primed frame will be of length *L = L’/*** in the unprimed frame

## Slide 19

## Wish to thank Y. Papaphilippou and N.Catalan-Lasheras for sharing the tranparencies that they used in the USPAS, Cornell University, Ithaca, NY 20th June – 1st July 2005

**Wish to thank Y. Papaphilippou and N.Catalan-Lasheras for sharing the tranparencies that they used in the USPAS, Cornell University, Ithaca, NY 20th June – 1st July 2005**

## Problem 1.

**Problem 1.**
**Protons are accelerated to a kinetic energy of 200 MeV at the end of the Fermilab Alvarez linear accelerator. Calculate their total energy, their momentum and their velocity in units of the velocity of light.**
**Problem 2.**
**A charge pion has a rest energy of 139.568 MeV and a mean life time of ******** = 26.029 nsec in its rest frame. What are the pion life times, if accelerated to a kinetic energy of 20 MeV? And 100 MeV? A pion beam decays exponentially like e****-********/t****. At what distance from the source will the pion beam intensity have fallen to 50%, if the kinetic energy is 20 MeV? Or 100 MeV?**
**Problem 3.**
**A positron beam accelerated to 50 GeV in the linac hits a fixed hydrogen target. What is the available energy from a collision with a target electron assumed to be at rest? Compare this available energy with that obtained in a linear collider where electrons and positrons from two similar linacs collide head on at the same energy.**
**Rest energy of an electron = 0.511 MeV**
**Rest energy of a proton = 936 MeV**

## Show that a function satisfying f(x,t)=f(x-vt) automatically satisfies the wave equation.

**Show that a function satisfying f(x,t)=f(x-vt) automatically satisfies the wave equation.**
**A muon has a rest mass of 105.7MeV and a lifetime at rest of 2.2e-6 s. **
**Consider a muon traveling at 0.9c with respect to the lab frame. What is its lifetime? How far does the muon travel? How does this compare to the distance it would travel if there were no time dilation?**
**Consider a muon accelerated to 1GeV. **
**	****What is its velocity? How long does it live?**
**For a non-relativistic charge moving in the z direction, calculate the general particle trajectory when subjected to a field B****x****=B****z****=0, and B****y****=sin(2********z/********) for 0<z<z****0****, and B****y****=0 elsewhere.**