In problems involving electromagenetism it is often convenlent and informative to express answers in terms of a constant, `alpha`, which is a combination of the Coulomb constant, `k_(e) = 1//4piepsi_(0)`, the charge of the electron, e, and `h = h//2`, h being Planck's constant. For instant, the lowest energy that a hydrogen atom can have is given by `E = 1//2 alpha^(2) mc^(2)`, wherer m is the mass of the electron and c is the speed of light. Which of the following is the correct expression for `alpha`?
(HINt : non-relativistic kinetic energy is `1//2 mv^(2)`, where v is speed.)

Options

A. `(k_(e)e^(2))/(hc)`
B. `(h)/(k_(e)e^(2)c)`
C. `(k_(e)e^(2)h)/(c)`
D. `(k_(e)e^(2)pi)/(hc)`

Answer

D

Explanation

`E = (z^(2))/(n^(2)) (me^(4))/(8epsi_(0)h^(2))`
For hydrogen `E = (me^(4))/(8epsi_(0)^(2)h^(2))`
`((16pi^(2)K_(e)^(2))/(8h^(2))) me^(4) rArr ((pi^(2)K_(e)^(2))/(h^(2))e^(4)) m/2`
`E=((piK_(e)e^(2))/(hc)) (1/2mc^(2)) = 1/2 alpha^(2)mc^(2)`