Q$3(20)$: Assume a hydrogen$-$like $($single electron$)$ helium ion. It is currently frozen in time $($no movement, including by the electron$).$

a$)$ How far from the center of the helium nucleus is the center of mass of the combined system? Draw a helpful model and label it with the center of mass, helium nucleus, and the electron.

b$)$ If the helium nucleus rotated around the center of mass while the electron stayed frozen in its place, how much greater energy of attraction would there be between the nucleus and electron when they were closest together than when they are farthest apart? $($Hint: think about re$-$drawing the model and think through it$.)$

c$)$ What wavelength of light is equal to this difference from part b$?$

d$)$ Is the center of mass within the helium nucleus? Why$/$why not is this the case?