☰ Contents · Physics

The Lorentz force

Lessons 6 · 1 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 11, 1st edition. Niso Poligraf Publishing House, Tashkent, 2018
6

Motion of a charged particle in a uniform magnetic field. The Lorentz force

Textbook: pp. 17–23
GoalCalculate the Lorentz force F = |q|·υ·B·sinα, the radius R = mυ/(|q|B) and the period T = 2πm/(|q|B) of a charged particle in a field, and know the principle of the mass spectrometer.
New words
Lorentz force · Lorens kuchiradius of the circular path · aylanish radiusiperiod of revolution · aylanish davrimass spectrometer · mass-spektrometr
Explanation

A charged particle moving in a magnetic field is acted on by the Lorentz force F = |q|·υ·B·sinα, where α is the angle between the velocity and B. Its direction for a positive charge is found with the left-hand rule (it is reversed for a negative charge). If the particle moves along the field lines (α = 0° or 180°) the force is zero; if it enters perpendicular to them (α = 90°) the force is greatest. Because the Lorentz force is perpendicular to the velocity it does no work: the speed and kinetic energy do not change, only the direction does. A particle entering perpendicularly moves uniformly along a circle of radius R = mυ/(|q|B) (the Lorentz force plays the role of the centripetal force), and the period T = 2πm/(|q|B) does not depend on the speed. This principle is used in the mass spectrometer to separate particles by mass: among particles of the same speed and charge the heavier one traces the larger circle.

Worked examples
A proton (m = 1.67·10⁻²⁷ kg, q = 1.6·10⁻¹⁹ C) enters a field B = 0.1 T perpendicularly at υ = 2·10⁶ m/s. F = qυB = 1.6·10⁻¹⁹·2·10⁶·0.1 = 3.2·10⁻¹⁴ N; R = mυ/(qB) = 1.67·10⁻²⁷·2·10⁶/(1.6·10⁻¹⁹·0.1) ≈ 0.21 m.
An electron (m = 9.1·10⁻³¹ kg) circles in a field B = 1 mT. T = 2πm/(eB) = 2·3.14·9.1·10⁻³¹/(1.6·10⁻¹⁹·10⁻³) ≈ 3.6·10⁻⁸ s ≈ 36 ns; if its speed doubles the period stays the same, while the radius doubles.
Class activity

In your notebook draw the paths of a positive and a negative particle in a field directed into the page (which way does each circle turn?). Explain the same rule using the electron beam in an old cathode-ray tube. Keep strong magnets away from electronic devices and cards.

Practice
1
Is there a Lorentz force on a particle moving along the field lines?
2
q = 3.2·10⁻¹⁹ C, υ = 5·10⁵ m/s, B = 0.4 T, α = 90°. What is the force in units of 10⁻¹⁵ N?
3
An electron enters a field B = 1 mT perpendicularly at υ = 1.76·10⁷ m/s (e/m = 1.76·10¹¹ C/kg). What is the radius in cm?
4
Why does the Lorentz force change only the direction of a particle's velocity, not its speed?