Magnetism and Electrons — Question 5

Question 5

A neutral atomic beam separates according to moment projection

Neutral atoms of mass M travel at speed v through a magnet of length ell, then drift distance D. Each atom has moment projection $\mu_z=\pm\mu_B$. On the beam axis $\partial B_z/\partial z=G$ is constant. Assume small deflections and fixed projections.

1. Find the vertical displacement acquired inside the magnet.

2. Include the field-free drift and find the separation of the two spots on the screen.

Question 5 - Solution

A neutral atomic beam separates according to moment projection

(a) Method and calculation

$$F_z=\mu_zG,\quad a_z=\pm\mu_BG/M,\quad t_m=\ell/v.$$$$\boxed{z_m=\pm\frac{\mu_BG\ell^2}{2Mv^2}}.$$

(b) Interpretation and further calculation

$$v_{z,\mathrm{exit}}=\pm\frac{\mu_BG\ell}{Mv},\quad z_{\mathrm{screen}}=\pm\frac{\mu_BG\ell}{Mv^2}(D+\ell/2).$$$$\boxed{\Delta z=\frac{2\mu_B|G|\ell}{Mv^2}(D+\ell/2)}.$$

Physical check

Neutral atoms avoid the charge-force complication of a free electron beam; the gradient model applies locally on the beam axis.