Particle charge in a magnetic field
1. Explain the behavior of a charged particle in a magnetic field!
Solution:
If a particle with an electric charge (electron, proton, ion, …) enters a magnetic field perpendicularly to the induction lines, the magnetic field acts on the particle with a magnetic force
Fₘ = B·Q·v.
Under the influence of this force, the particle will move in the field with velocity v, whereby
it acquires a centrifugal force
F_OD = (m·v²) / r
If the forces are in equilibrium, the trajectory of the particle will be a circle with radius
r = (m·v) / (B·Q)
r = radius of the circular trajectory,
v = speed of the particle,
B = magnetic induction,
Q = electric charge of the particle.
It holds that:
If the particle moves in both an electric and a magnetic field, the Lorentz force applies:
F = Q·(E + v·B·sinα)
2. Derive the formula for the radius of the circular trajectory of a charged particle in a homogeneous magnetic field!
Solution:
3. With what velocity did a proton (mp = 1.673·10–27kg, Qp = 1.602·10-19C) move in a magnetic field (B = 1T) if its trajectory was a circle with radius r = 60cm? With what frequency did the proton orbit the circle?
Solution:
Analysis:
mp = 1.673·10–27kg, Qp = 1.602·10-19C, B = 1T, r = 60cm = 0.6m, v = ?, f = ?
The velocity of the proton in the magnetic field was 5.75·107m·s-1, its frequency f = 15.26 MHz.
4. What kinetic energy does a proton (mp = 1.673·10-27kg, Qp = 1.602·10-19C) have when moving in a circle of radius r = 10cm in a homogeneous magnetic field with B = 0.1T, perpendicular to the induction lines?
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5. An electron (me = 9.1·10–31kg, Qe = 1.602·10–19C) after being accelerated in an electric field with U = 100V enters a homogeneous magnetic field with B = 10mT perpendicular to the induction lines. Along what trajectory will the electron move?
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6. What magnitude must the magnetic induction of a homogeneous magnetic field have so that an electron moves in it in a circle with radius r = 40cm at velocity v = 3.8·107m·s-1? What will be the kinetic energy of the electron?
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7. With what velocity would a proton (mp = 1.673·10–27kg, Qp = 1.602·10–19C) have to move in the Earth's magnetic field (B = 5·10–5T – tables) perpendicular to the induction lines so that the magnetic force equals the gravitational force acting on it?
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8. Into a homogeneous magnetic field with B = 2·10–5T in a vacuum entered a proton (mp = 1.673·10–27kg, Qp = 1.602·10-19C) in a direction perpendicular to the induction lines. What will be the frequency of its circular motion?
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9. Two electrons moving with the same velocities entered two homogeneous magnetic fields (B1 = 0.1T, B2 = 0.2T) and began moving in circular trajectories. Determine the ratio of their orbital periods T1 : T2.
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10. An electron started moving from rest and after passing through a potential difference of 220V entered perpendicularly into a homogeneous magnetic field with induction 5mT. In the magnetic field the electron moved in a circular trajectory with radius 1cm. Determine the mass of the electron.
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11.A proton moves in a homogeneous magnetic field. The magnetic induction of the field is B = 15 mT, the mass of the proton is mp = 1.67·10–27 kg, its charge is Qp = 1.602·10–19 C. Determine the radius of its circular path r, if the proton’s speed is v = 2·106 m·s-1.
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12.A particle with charge Q = 1.602·10-19 C entered a homogeneous magnetic field with magnetic induction B = 10-2 T. In this magnetic field it moves along a circle with radius r = 0.45·10-2 m. Determine the magnitude of the particle’s momentum.
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13.A proton moves at speed v = 1·106 m·s-1 in a homogeneous magnetic field with magnetic induction B = 1 T.
- What force acts on the proton (mp = 1.673·10–27 kg, QP = 1.602·10–19 C)
- Along what trajectory will the proton move?
14.An electron, after being accelerated in an electric field with U = 100 V, enters a homogeneous magnetic field with magnetic induction B = 10 mT perpendicular to the field lines. Along what trajectory will it move?
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15.What must be the voltage between the anode and cathode in a glass tube so that the luminous circular trace marking the trajectory of electrons has radius r = 5·10-2 m? The magnetic induction in the tube is B = 8.2·10-4 T.
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16.An α particle moves along a circular path with radius r = 4.5 cm in a homogeneous magnetic field with magnetic induction B = 1.2 T. Calculate:
- a.) its speed
- b.) its period of motion
17.A beam of protons moves in a vacuum tube such that their speed is v = 0.1c. The electrostatic force is Fe = 3·10–13 N.
- a.) What is the value of the ratio E/B if the net force acting on the proton is zero?
- b.) What is the magnitude of the magnetic induction B?
18.Between the anode and cathode of an electron gun in a vacuum cathode tube, electrons travel at speed v = 9.19·106 m·s-1. The distance between the plates is d = 1 cm with voltage U = 10 V. Calculate the magnitude of the magnetic induction B of a homogeneous magnetic field for which the electrons will not be deflected from the axis of the tube.
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19.A proton and an α particle (mHe = 4·mp, QHe = 2·Qp) enter a homogeneous magnetic field perpendicular to the field lines. Compare the radii of the circular trajectories of the particles, if the particles have the same speed.
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20.An electron entered a homogeneous magnetic field with B = 10 mT perpendicular to the induction lines, with kinetic energy Ek = 30 keV. Determine the radius r of its circular trajectory.
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