How Do You Calculate the Fine Structure Energy Difference in Hydrogen?

[helpcometk]

Homework Statement

Calculate the numerical value of the energy difference between the 2P _3/2and the 2P _1/2 states in Hydrogen (the fine structure).

Homework Equations

possible relevant equations are in the attachement

The Attempt at a Solution

It is quite unclear weather the relativistic (sexcond)or ,the equation which takes into account the magnetic field corrections (the first equation )of the proton around the electron has to be used.it seems that we should use the first equation.
The graph on the attachement tells us which values i have to substitute.
I would like someone who specialises in atomic physics ,to tell me if I am thinking correctly in order to solve the problem.im very uncertain about if this answer is correct.

 

[NateD]

I=1/2 J=3/2 and J'=1/2ΔE=(-e^4/(8*pi*epsilon_0)^2*m_e*c^2*Z^4*α^3)/(n_1^3*(j+1/2)^2-n_2^3*(j'+1/2)^2)ΔE=(-e^4/(8*pi*epsilon_0)^2*m_e*c^2*1^4*1/137^3)/(2^3*(3/2+1/2)^2-2^3*(1/2+1/2)^2)ΔE=(-e^4/(8*pi*epsilon_0)^2*m_e*c^2*1^4*1/137^3)/(2^3*2^2-2^3*1^2)ΔE=(-e^4/(8*pi*epsilon_0)^2*m_e*c^2*1^4*1/137^3)/(8-4)ΔE=(-e^4/(8*pi*epsilon_0)^2*m_e*c^2*1^4*1/137^3)/4ΔE=(-9.109383*10^-31*(5.63*10^-34)^2*2.998*10^8*1^4*1/137^3)/4ΔE=-5.17*10^-24 Joules