 Solution

For an ultrarelativistic electron of Lorentz factor $\gamma =E/(m_e{c}^2)=3000/0.511=5870.84$, the synchrotron critical frequency gives ${\epsilon }_c=\hslash {\omega }_c=3\hslash c{\gamma }^3/(2\rho )$. Thus, using the supplied constants,The angle-integrated spectral power per unit photon energy is proportional to $x{\int }_x^{\infty }{K}_{5/3}(u)du$, with $x=\epsilon /{\epsilon }_c$. It peaks at $x\simeq 0.286$, soThis is a power spectrum per linear energy interval; power per logarithmic interval and photon-count spectra have different peaks. The magnetic rigidity condition is $p=|e|B\rho$, with $pc\simeq E$, givingThe critical energy sets a broad spectral scale, not a maximum allowed photon energy.^[ref]