A charged particle moving in a uniform magnetic field penetrates a layer of lead and thereby loses one half of its kinetic energy. How does the radius of curvature of its path change?

Answer

(a) Nonrelativistic particle `R_(1)//R_(2)=sqrt(K_(1)//K_(2))`
(b) Relativistic particle `R_(1)//R_(2)=sqrt((K_(1)(2epsi_(0)+K_(1))/(K_(2) (2epsi_(0)+K_(2))`

Explanation

Since the charge of the particle and the magnetic field induction remain unchanged, the radii of the tracks of the particles are proportional to their momenta: `R_1//R_(2)= p_1/p_2.` The relation to the kinetic energy depends on the nature of motion.
(a) Nonrelativistic particles. The momenta of the particle are proportional to the square roots of their kinetic energies, and therefore also proportional to the radii of their tracks.
(b) Relativistic particles. In this case the dependence of the momentum on the kinetic energy of the particle is more complex: `p=1/c sqrt(K(2epis_(0)+K))`
Hence we obtain the desired ratio of the radii of the tracks.