Question

An electron is accelerated within a particle accelerator using a 100 MV electric potential. The 100 MeV electron moves along an evacuated tube of length 4 m, fixed to the laboratory frame.

What length of the tube would be measured by an observer moving with the electron?

Answer 1

The length of the tube is 3.92 m.

Step-by-step explanation:

Given that,

Electric potential = 100 MV

Length = 4 m

Energy = 100 MeV

We need to calculate the value of
`\gamma`

Using formula of relativistic energy

`E=m_(0)c^2(\frac{1}{\sqrt{1-(v^2)/(c^2)}}-1)`

Put the value into the formula

`1.6*10^(-15)= 9.1*`10^(-31)*9*10^(16)(\frac{1}{\sqrt{1-(v^2)/(c^2)}}-1)`

`(\frac{1}{\sqrt{1-(v^2)/(c^2)}}-1)=(1.6*10^(-15))/(9.1*10^(-31)*9*10^(16))`

Here,
`\gamma-1=(\frac{1}{\sqrt{1-(v^2)/(c^2)}}-1)`

`\gamma-1=0.01953`

`\gamma=0.01953+1`

`\gamma=1.01953`

We need to calculate the length

Using formula of length

`L'=(L)/(\gamma)`

Put the value into the formula

`L'=(4)/(1.01953)`

`L'=3.92\ m`

Hence, The length of the tube is 3.92 m.