# Problems in General Physics → Electrodynamics → Motion of Charged Particles in Electric and Magnetic Fields

 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 = at, where a = 100 V/s, is applied between the plates. The separation between the plates is l = 5.0 cm. What is the velocity of the electron at the moment it reaches the opposite plate? Free solution >> 3.374. A particle with specific charge q/m moves rectilinearly due to an electric field E = E0 - ax, 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 to a standstill; (b) the acceleration of the particle at that moment. Free solution >> 3.376. Determine the acceleration of a relativistic electron moving along a uniform electric field of strength E at the moment when its kinetic energy becomes equal to T. Free solution >> 3.377. At the moment t = 0 a relativistic proton flies with a velocity v0 into the region where there is a uniform transverse electric field of strength E, with v0 ⊥ E. Find the time dependence of (a) the angle θ between the proton's velocity vector v and the initial direction of its motion; (b) the projection vx of the vector v on the initial direction of motion. Free solution >> 3.378. A proton accelerated by a potential difference V = 500 kV flies through a uniform transverse magnetic field with induction B = 0.51 T. The field occupies a region of space d = 10 cm in thickness (Fig. 3.99). Find the angle α through which the proton deviates from the initial direction of its motion. Free solution >> 3.379. A charged particle moves along a circle of radius r = 100 mm in a uniform magnetic field with induction B = 10.0 mT. Find its velocity and period of revolution if that particle is (a) a non-relativistic proton; (b) a relativistic electron. Free solution >> 3.380. A relativistic particle with charge q and rest mass m0 moves along a circle of radius r in a uniform magnetic field of induction B. Find: (a) the modulus of the particle's momentum vector; (b) the kinetic energy of the particle; (c) the acceleration of the particle. Free solution >> 3.382. An electron accelerated by a potential difference V = 1.0 kV moves in a uniform magnetic field at an angle α = 30° to the vector B whose modulus is B = 29 mT. Find the pitch of the helical trajectory of the electron. Free solution >> 3.383. A slightly divergent beam of non-relativistic charged particles accelerated by a potential difference V propagates from a point A along the axis of a straight solenoid. The beam is brought into focus at a distance l from the point A at two successive values of magnetic induction B1 and B2. Find the specific charge q/m of the particles. Free solution >> 3.384. A non-relativistic electron originates at a point A lying on the axis of a straight solenoid and moves with velocity v at an angle α to the axis. The magnetic induction of the field is equal to B. Find the distance r from the axis to the point on the screen into which the electron strikes. The screen is oriented at right angles to the axis and is located at a distance l from the point A. Free solution >> 3.385. From the surface of a round wire of radius a carrying a direct current I an electron escapes with a velocity v0 perpendicular to the surface. Find what will be the maximum distance of the electron from the axis of the wire before it turns back due to the action of the magnetic field generated by the current. Free solution >> 3.386. A non-relativistic charged particle flies through the electric field of a cylindrical capacitor and gets into a uniform transverse magnetic field with induction B (Fig. 3.100). In the capacitor the particle moves along the arc of a circle, in the magnetic field, along a semi-circle of radius r. The potential difference applied to the capacitor is equal to V, the radii of the electrodes are equal to a and b, with a < b. Find the velocity of the particle and its specific charge q/m. Free solution >> 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-relativistic velocity v0. Find: (a) the coordinate yn of the particle when it crosses the y axis for the nth time; (b) the angle α between the particle's velocity vector and the y axis at that moment. Free solution >> 3.389. A non-relativistic proton beam passes without deviation through the region of space where there are uniform transverse mutually perpendicular electric and magnetic fields with E = 120 kV/m and B = 50 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 mA. Free solution >> 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 kV/m and B = 50 mT. The trajectory of the protons lies in the plane xz (Fig. 3.102) and forms an angle φ = 30° with the x axis. Find the pitch of the helical trajectory along which the protons will move after the electric field is switched off. Free solution >> 3.393. A system consists of a long cylindrical anode of radius a and a coaxial cylindrical cathode of radius b (b < a). A filament 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. Free solution >> 3.396. The cyclotron's oscillator frequency is equal to ν = 10 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 Δr = 1.0 cm, with the trajectory radius being equal to r = 0.5 m. Free solution >> 3.397. Protons are accelerated in a cyclotron so that the maximum curvature radius of their trajectory is equal to r = 50 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 T; (b) the minimum frequency of the cyclotron's oscillator at which the kinetic energy of the protons amounts to T = 20 MeV by the end of acceleration. Free solution >> 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 microtron (Fig. 3.105) in which a change ΔT in the period of revolution of an electron is made multiple with the period of accelerating field T0. How many times has an electron to cross the accelerating gap of a microtron to acquire an energy W = 4.6 MeV if ΔT = T0, the magnetic induction is equal to B = 107 mT, and the frequency of accelerating field to ν = 3000 MHz? Free solution >> 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 ω(t) should this frequency be monitored if the magnetic induction is equal to B and the particle acquires an energy ΔW per revolution? The charge of the particle is q and its mass is m. Free solution >> 3.401. A particle with specific charge q/m is located inside a round solenoid at a distance r from its axis. With the current switched into the winding, the magnetic induction of the field generated by the solenoid amounts to B. Find the velocity of the particle and the curvature radius of its trajectory, assuming that during the increase of current flowing in the solenoid the particle shifts by a negligible distance. Free solution >>