Curvature radiation refers to the light emitted by charged particles moving along curved magnetic field lines. To sustain this emission, there has to be an electric field component parallel to the magnetic field lines, such that the electrostatic work replenishes the radiated energy. The magnitude of the electric field is bounded by the Schwinger limit

$ E_s^2 \approx \frac{m_e^3 c^4}{r_e \hbar^2} $

where $ m_e $ is the mass of the electron, $ c $ is the speed of light, $ r_e $ is the classical electron radius and $ \hbar $ is the reduced Planck constant. The size of the radiating region has to be comparable to the curvature radius $ r $ and so the maximum luminosity is

$ L \approx E_s^2 r^2 c $

Another way to interpret this result is as follows. The force on a single electron is bounded by $ E_s q $ where $ q $ is the elementary charge. The power exerted on a single relativistic electron is bounded by $ E_s q c $. The electric field due to the radiating electrons cannot exceed the Schwinger limit, so the upper bound on the electron number density $ n $ is $ q n r < E_s $ and the maximum luminosity is given by $ E_s q c n r^3 \approx E_s^2 r^2 c $