= Solution
For an ultrarelativistic electron of <Lorentz factor> $\gamma=E/(m_ec^2)=3000/0.511=5870.84$, the <synchrotron critical frequency> gives $\varepsilon_c=\hbar\omega_c=3\hbar c\gamma^3/(2\rho)$. Thus, using the supplied constants,
$$
\boxed{\rho=\frac{3\hbar c\gamma^3}{2\varepsilon_c}=7.47\,\mathrm m.}
$$
The angle-integrated spectral power per unit photon energy is proportional to $x\int_x^\infty K_{5/3}(u)du$, with $x=\varepsilon/\varepsilon_c$. It peaks at $x\simeq0.286$, so
$$
\boxed{\varepsilon_{\rm peak}\simeq2.29\,\mathrm{keV}.}
$$
This 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 $pc\simeq E$, giving
$$
\boxed{B\simeq\frac{E}{|e|c\rho}=1.34\,\mathrm T.}
$$
The critical energy sets a broad spectral scale, not a maximum allowed photon energy.<Synchrotron Radiation in the Geant4 Physics Reference Manual by Geant4 Collaboration>[critical energy and angle-integrated synchrotron spectrum]{ref}
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