Assume: nucleus g = 1 and I = 1/2
Apply field: B_0 = 2T
What is photon energy for photons that will be absorbed if no hyperfine interaction?
First calculate distance between 2 hyperfine levels. This is the energy difference between the m_J = -1/2 and 1/2 level. We have E = -g_J*mu_B*B_0*m_J so the difference is delta E = -g_J*mu_B*B_0*(-1/2-1/2) = g_J*mu_B*B_0 = 11.56 eV. Using E = h*f, we find a frequency of f = 2.82*10^15 Hz.
Now turn on the hyperfine interaction with B_hf = 10T. We use E_hf = m * mu_B * B_hf. Then we find delta E new = delta E – mu_B * B_hf = 11.56 – 5.78*10^-4 eV. We find f = 2.8*10^15 Hz.
I am not sure why the question talks about m_J = -3/2 levels, while the nuclear spin is I=1/2.