Speed and energy of particles – relativistic calculation
Question 20S: Short Answer		

In these questions you will make calculation about charged particles accelerated by different potential differences, at energies where relativistic effects are important.

At any speed up to close to the speed of light the total energy is given by:

where

The kinetic energy given to a particle with charge q accelerated through a potential difference V is:

Relationships to use

At rest, with v = 0, g = 1 and the energy is rest energy Erest = mc2. 

The relativistic factor g can be found from the ratio of total energy to rest energy:

The kinetic energy is the difference between total energy and rest energy, Ekinetic = Etotal  – Erest , which can be written as

Data to use
Charge on proton = 1.6 ´ 10–19 C	Mass of proton = 1.7 ´ 10–27 kg, equivalent to rest energy nearly 1 GeV
Speed of light = 3.0 ´ 108 m s–1
Advice: work with energy units
For making relativistic calculations it is best to express quantities in energy units. The electron volt (eV) is often convenient.
Kinetic energy and g factor
1.	The table below shows the energies achieved by various early particle accelerators. Taking Erest = 1 GeV for a proton, add to the table giving the relativistic factor g. Use g = Etotal / Erest with Etotal = Ekinetic + Erest.

Accelerator	Date	Particle accelerated	Kinetic energy	g factor	Ratio v / c	
Cockcroft and Walton	1932	Proton	770 keV			
Van de Graaff	1932	Proton	1.5 MeV			
Cyclotron	1937	Proton	3 MeV			
Synchrocyclotron	1947	Proton	500 MeV			
Synchrotron	1954	Proton	6 GeV			

2.	Add to the table the ratio v / c for the protons in each accelerator. Use

					so that

3.	By 1960, an accelerator giving protons a kinetic energy of 30 GeV had started operating at Brookhaven, USA. Explain why you can at once write down the value 30 as an approximation to the g factor for these protons.

4.	As a rule of thumb, relativistic effects only need to be taken into account for speeds exceeding 20% of the speed of light. For which of these accelerators are relativistic effects certainly important?