Question

You wish to build a linear accelerator that will accelerate electrons to ?=89 % of the speed of light. You want adjacent drift tubes to be at +5000V and −5000V potentials. To work this problem, you will need the relativistic formula for kinetic energy. (?=??2/2 doesn't work for really fast particles.) Kinetic energy is ?=?0(?−1) where ?0 is the rest energy of an object. The rest energy of an electron is 0.511 MeV. [Remember that if we want to convert eV to SI units, all we have to do is put ?=1.60×10−19 C into the equation; however, for this problem, it's easiest just to leave everything in eV.] How many drift tubes do you need? Hints: The energy an electron gains is ??=??? in each gap. Just find the total kinetic energy by using the formula. Remember that you need one more drift tube than there are gaps. Range of answers: 45 to 115

hint: β= 90.7 %, correct answer is 72

β= 86.9, correct answer is 54

Answer 1

Let us first calculate the relativistic kinetic energy of the electron travelling at a speed of 0.89c(i.e.89% of c)

Where m_{​​​​​​0} =mass of electron= 9.10×10^-31 kg.

V=velocity of electron=0.89c

c=speed of light.

We already know the rest mass energy of electron =0.511Mev

So the kinetic energy of electron moving at 0.89c is 0.01Mev.

So the number of gaps(n) required to accelerate the electron to 0.89c can be calculated as follows:

Therefore the number of drift tubes required=62+1=63.

Now let us calculate how much energy is gained by the electron in each gap.

The potential difference in the gap between the two drift tubes is 10000V and the charge on the electron is 1.6×10^-19 C. So we can calculate the energy gained in each gap ∆K as,