Question

An electron moving to the right at$0.90c$collides with a positron moving to the left at$0.90c$. The two particles annihilate and produce two gamma-ray photons. What is the wavelength of the photons?

Short answer

The wavelength of the photons is$1.023pm$.

Step-by-step solution

Step 1:Given Information 

We need to find the wavelength of the photons.

Simplify

Consider:

$$\begin{gathered} v=0.90c \\ {m}_e=9.0\times {10}^{-31}kg \\ c=3\times {10}^{8}m/s \\ hc=1240eV \\ {m}_e{c}^2=0.511eV \end{gathered}$$

using formula :

$E=\gamma m{c}^2$ ...(1)

we get relation for $\lambda$, from equation (1)

role="math" localid="1649961398708" $$\begin{gathered} E=\gamma m{c}^2 \\ \frac{hc}{\lambda }=\gamma m{c}^2 \\ hc=\lambda \gamma m{c}^2 \\ \lambda =\frac{hc}{\gamma m{c}^2} \\ \lambda =\frac{hc}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}m{c}^2} \\ \lambda =\frac{1200}{\sqrt{1-0.81}\times 0.511} \\ \lambda =\frac{1200}{0.43589\times 0.511} \\ \lambda =1.023pm \end{gathered}$$