Problem 4.216

A non-relativistic charged particle moves in a transverse uniform magnetic field with induction \(B\). Find the time dependence of the particle's kinetic energy diminishing due to radiation. How soon will its kinetic energy decrease e-fold? Calculate this time interval for the case (a) of an electron, (b) of a proton.

Reveal Answer

\begin{aligned} &\text { 4.216. } T=T_{0} \mathrm{e}^{-\alpha t}, \text { where } \alpha=1 / 3 e^{4} B^{2} / \pi \varepsilon_{0} c^{3} m^{3} . \text { After } t_{0}=\frac{1}{\alpha}=\\ &=\left\{\begin{array}{ll} 2.5 \mathrm{~s} & \text { for the electron, } \\ 1.6 \cdot 10^{10} \mathrm{~s}=0.5 \cdot 10^{3} & \text { years for the proton. } \end{array}\right. \end{aligned}

Electromagnetic Waves. Radiation