– Lerenta force:
\[
\boldsymbol{r}=\boldsymbol{q} \mathbf{R}+\mathbf{q}|\mathbf{\mathrm { B }}| \text {. }
\]
– Motion equation of a relativistie partiele:
– Period of revolutios of a charged particle is a unilorm magetie field:
\[
r=\frac{2 \mathrm{am}}{\sqrt{B}} \text {. }
\]
where $m$ is the relativiatic mass of the particle, $m=m d \sqrt{-(6 e r}$.
– Betatron condition, that is the condition for as electron to move along a circular orbit in a betatron:
\[
B_{4}=\frac{1}{2}(B) \text {, }
\]
where $B_{\text {, }}$ is the mapetic induction ot an erbit’s point, $(B)$ is the mean value of the induction inside the orbit.
3.372. At the moment $t=0$ an electron leaves one plate of a parallel-plate capacitor with a negligible velocity. An accelerating voltage, varying as $V=a t$, where $a=100 \mathrm{~V} / \mathrm{s}$, is applied between the plates. The separation between the plates is $t=5.0 \mathrm{~cm}$. What is the velocity of the electron at the mement it reaches the opposite plate?
3.373. A proton accelerated by a potential difference $V$ gets into the uniform electric field of a parallel-plate capacitor whose plates extend over a length $l$ in the motion direction. The field strength varies with time as $E=a t$, where $a$ is a constant. Assuming the proton to be non-relativistic, fiad the aagle between the motion directions of the proton before and after its flight through the capactior; the proton gets in the field at the moment $t=0$. The edge effects are to be neglected.
3.374. A particle with specific charge q/m moves rectilinearly due to an electrie fleld $E=E_{0}-a x$, where $a$ is a positive constant, $x$ is the distance from the point where the particle was initially at rest. Find:
(a) the distance covered by the particle till the moment it came te a standstill;
(b) the acceleration of the particle at that moment.
3.375. An electron starts moving in a uniform electric field of strength $E=10 \mathrm{kV} / \mathrm{cm}$. Ifow soon after the start will the kinetic energy of the electron become equal to its rest energy?
3.376. Determine the acceleration of a relativistie electron moving along a uniform electric field of strength $E$ at the moment when its kinetic energy becomes equal to $T$.
3.377. At the moment $t=0$ a relativistie proton flies with a velocity $\mathbf{v}_{0}$ inte the region where there is a uniform transverse electric field of strength $\mathrm{E}$, with $\mathrm{v}_{4} \perp \mathrm{E}$. Find the time dependence of

3.386. A non-relativistic charged particle flies through the electric field of a cylindrical capacitor and gets into a un iform transverse magnetic field with induction $B$ (Fig. 3.100). In the capacitor the particle moves along the are of a circle, in the magnetic field, along a semi-eircle of radius $r$. The potential diflerence applied to the capacitor is equal to $\boldsymbol{V}$, the radii of the electrodes are equal to $a$ and $b$, with $a<b$. Find the velocity of the particle and its specife charge $q / m$.
ท. $\mathbf{2 . 1 0 0}$.
Fie. a.tot.
3.387. Uniform electric and magnetic fields with strength $E$ and induction $B$ respectively are directed along the $y$ axis (Fig. 3.101). A particle with specific charge $q / m$ leaves the origin $O$ in the direction of the $x$ axis with an initial non-relativiatic velocity $v_{6}$. Find:
(a) the coordinate $y_{n}$ of the particle when it crosses the $y$ axis for the $n$th time;
(b) the angle a between the particle’s velocity vecter and the $y$ axis at that moment.
3.388. A narrow beam of identical ions with specific charge $\mathrm{g} / \mathrm{m}$, possessing different velocities, enters the region of space, where there are uniform parallel electric and magnetic fields with strength $E$ and induction $B$, at the point $O$ (see Fig. 3.101). The beam direction coincides with the $x$ axis at the point $O$. A plane screen oriented at right angles to the $x$ axis is located at adistance $l$ from the point $O$. Find the equation of the trace that the ions leave on the screen. Demonstrate that at $z<i$ it is the equation of a parabola.
3.389. A non-relativistic proton beam passes without deviation through the region of space where there are uniform transverse mutually perpendicular electrie and magnetic fields with $E=120 \mathrm{kV} / \mathrm{m}$ and $B=50 \mathrm{mT}$. Then the beam strikes a grounded target. Find the force with which the beam acts on the target if the beam current is equal to $I=0.80 \mathrm{~mA}$.
3.390. Non-relativistic protons move rectilinearly in the region of space where there are uniform mutually perpendicular electric and magnetic fields with $E=4.0 \mathrm{kV} / \mathrm{m}$ and $B=50 \mathrm{~m}$. The trajectory of the protons lies in the plane $x z$ (Fig. 3.102) and forms an angle $9-30^{\circ}$ with the $x$ axis. Find the pitch of the helical trajectory along which the protons will move after the electrie field is switched off.
162
162
3.391. A beam of non-relativistic charged particles moves without deviation through the region of space $A$ (Fig. 3.103) where there are transverse mutually perpendlcular electric and magnetic fields with
Fis. 3.102.
Fe. 3.103.
strength $E$ and induction $B$. When the magnetic field is switched off, the trace of the beam on the screen $S$ shifts by $\Delta x$. Knowing the distances and $b$, find the specifie charge $q / m$ of the particles.
3.392. A particle with specific charge $\mathrm{q} / \mathrm{m}$ moves in the region of space where there are uniform mutually perpendicular electric and magnetic fields with strength $\mathbf{E}$ and induction B (Fig. 3.104). At the moment $t=0$ the particle was located at the point $O$ and had sero velocity. For the non-relativistie case find:
(a) the law of motion $x(t)$ and $y(t)$ of the partiele; the shape of the trajectory;
(b) the length of the segment of the trajectory between two nearest points at which the velocity of the particle turns into zero:
(c) the mean value of the particle’s veloc-
ทie. 3.104. ity vecter projection on the $x$ axis (the drift velocity).
3.393. A system consists of a long cylindrical anode of radius a and a coaxial cylindrical cathode of radius $b(b<a)$. A flament located along the axis of the system carries a heating current $I$ producing a magnetic field in the surrounding space. Find the least potential difference between the cathode and anode at which the thermal electrons leaving the cathode without initial velocity start reaching the anode.
3.394. Magnetron is a device consisting of a flament of radius a and a coaxial cylindrical anode of radius $b$ which are located in a uniform magnetic feld parallel to the flament. An accelerating potential difference $V$ is applied between the filament and the anode. Find the value of magnetie induction at which the electrons leaving the filament with tero velocity reach the anode.
3.395. A charged particle with specific charge $\mathrm{q} / \mathrm{m}$ starts moving in the region of space where there are uniform mutually perpendicular electric and magnetic fields. The magnetic field is constant and has an induction $B$ while the strength of the electric field varies with time as $E=E_{0}$ cos ot, where $\omega=q B / m$. For the non-relativistic case find the law of motion $x(f)$ and $y(t)$ of the particle if at the moment $t=0$ it was located at the point $O$ (see Fig. 3.104). What is the approximate shape of the trajectery of the particle?
3.396. The cyclotron’s oscillator frequency is equal to $\mathrm{v}=10 \mathrm{MHz}$. Find the effective accelerating voltage applied across the dees of that cyclotron if the distance between the neighbouring trajectories of protons is not less than $\Delta r=1.0 \mathrm{~cm}$, with the trajectory radius being equal to $r=0.5 \mathrm{~m}$.
3.397. Protons are accelerated in a cyclotron so that the maximum eurvature radies of their trajectory is equal to $r=50 \mathrm{~cm}$. Find:
(a) the kinetic energy of the protons when the acceleration is completed if the magnetic induction in the cyclotron is $B=1.0 \mathrm{~T}$;
(b) the minimum frequency of the cyclotron’s oscillator at which the kinetic energy of the protons amouats to $T=20 \mathrm{MeV}$ by the end of acceleration.
3.398. Singly charged ions $\mathrm{He}^{*}$ are accelerated in a cyelotron so that their maximum orbital radius is $r=60 \mathrm{~cm}$. The frequency of a cyclotron’s oscillator is equal to $\mathrm{v}=10.0 \mathrm{MHz}$, the effective accelerating voltage across the dees is $V=50 \mathrm{kV}$. Neglecting the gap between the dees, find:
(a) the total time of acceleration of the ion;
(b) the appreximate distance covered by the ion in the process of its acceleration.
3.399. Since the period of revolution of electrons in a uniform magnetic field rapidly increases with the growth of energy, a cyclotron is unsuitable for their acceleration. This drawback is rectified in a mierotron (Fig. 3.105) in which a change $\Delta T$ in the period of revolution of as electron is made multiple with the period of accelerating field $T_{4}$. How many times has as electron to cross the accelerating gap of a microtron to acquire an energy $W=$ $=4.6 \mathrm{MeV}$ if $\Delta T=T_{\text {, }}$. the magnetic induction is equal to $B=107 \mathrm{mT}$, and the freguency of accelerating field to

Fie. 3.105 . $\mathrm{v}-3000 \mathrm{MHz}$ ?
3.400. The ill effects associated with the variation of the period of revolution of the particle in a cyclotron due to the increase of its energy are eliminated by slow monitoring (modulating) the frequency of accelerating field. According to what law $\theta(f)$ should this frequen$c y$ be monitored if the magnetic induction is equal to $B$ and the particle acquires an energy $\Delta \boldsymbol{W}$ per revolution? The charge of the particle is $q$ and its mass is $m$.
3.401. A particle with specifie charge $\mathrm{g} / \mathrm{m}$ is located inside a round solenoid at a distance $r$ from its axis. With the current switched into