A beam comprising protons and electrons, travelling at the same...

The radius of the circular path of a charged particle in a perpendicular magnetic field is given by $R = \dfrac{mv}{qB}$.

For a thin region of magnetic field of width $d$, the angle of deviation $\theta$ is given by $\sin\theta = \dfrac{d}{R} = \dfrac{qBd}{mv}$.

Since protons and electrons have different masses ($m_p \gg m_e$) but the same magnitude of charge and same speed, their radii of curvature will be different ($R_p > R_e$).

Therefore, their angles of deviation will be unequal ($\theta_p < \theta_e$).

Additionally, due to their opposite charges, they will experience magnetic forces in opposite directions. Thus, they will deviate by unequal angles and thereby separate.