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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 :

Answer:

The length of the tube is 3.92 m.

Explanation:

Given that,

Electric potential = 100 MV

Length = 4 m

Energy = 100 MeV

We need to calculate the value of [tex]\gamma[/tex]

Using formula of relativistic energy

[tex]E=m_{0}c^2(\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1)[/tex]

Put the value into the formula

[tex]1.6\times10^{-15}= 9.1\times`10^{-31}\times9\times10^{16}(\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1)[/tex]

[tex](\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1)=\dfrac{1.6\times10^{-15}}{9.1\times10^{-31}\times9\times10^{16}}[/tex]

Here, [tex]\gamma-1=(\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1)[/tex]

[tex]\gamma-1=0.01953[/tex]

[tex]\gamma=0.01953+1[/tex]

[tex]\gamma=1.01953[/tex]

We need to calculate the length

Using formula of length

[tex]L'=\dfrac{L}{\gamma}[/tex]

Put the value into the formula

[tex]L'=\dfrac{4}{1.01953}[/tex]

[tex]L'=3.92\ m[/tex]

Hence, The length of the tube is 3.92 m.

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