10.1.13. a. A vacuum device consists of a coaxial cylinder of radius $R$ and a wire placed in a longitudinal magnetic field of induction $B$. When the wire is heated, electrons with kinetic energy $K$ are emitted from its surface; in this case, a current $I$ flows in the external circuit between the cylinder and the wire. Draw the dependence of $I$ on $B$. Find the values of $B$ at which the current in vacuum is zero.
b. The figure shows two dependencies of $I$ on $B$ at different residual gas pressures $P_1$ and $P_2$. Which pressure is higher?
Solution
a. We neglect the transverse dimension of the wire. We will assume that electrons are emitted from the wire radially and their velocities $v$ are strictly perpendicular to the field lines. The Lorentz force acts on the electron, twisting it into a circle. According to Newton's second law: $$eBv = \frac{m_ev^2}{r} \implies r = \frac{m_ev}{eB} \quad (1)$$
From the definition of kinetic energy: $$\frac{1}{2}m_ev^2 = K \implies v = \sqrt{\frac{2K}{m_e}} \quad (2)$$
Substitute (2) into (1): $$r = \frac{m_e}{eB}\sqrt{\frac{2K}{m_e}} = \frac{\sqrt{2m_eK}}{eB}$$
We see that the larger $B$ is, the smaller $r$ is. The condition for the electron to reach the cylinder wall of radius $R$: $$2r \ge R$$ $$2 \cdot \frac{\sqrt{2m_eK}}{eB} \ge R$$
Hence the boundary value of the magnetic induction: $$B_0 = \frac{2\sqrt{2m_eK}}{eR}$$
At $B \le B_0$ in vacuum, all electrons will reach the cylinder and some constant current $I_0$ will flow. At $B > B_0$, the electrons will not reach the cylinder wall and the current will be absent. The graph of the dependence $I(B)$ in vacuum will be in the form of a step.
b. In the presence of residual gas, electrons can collide with gas molecules. Such collisions change the directions of velocities — this gives the electron a chance to reach the cylinder wall even if initially $2r < R$. Therefore, at a higher pressure, more collisions occur and the current is maintained at larger values of $B$. From the graph in the problem statement, it is seen that the boundary value of the magnetic induction: $$B_{0_2} > B_{0_1}$$
