Section 5.3: Motion of Charged Particles

The motion of charged particles in electric and magnetic fields is fundamental to particle physics and applications like cyclotrons, mass spectrometers, and cathode ray tubes.

Uniform Magnetic Field (perpendicular velocity)

Charged particle moves in a circle due to Lorentz force:

Radius: \( r = \frac{mv}{qB} \)
Angular speed: \( \omega = \frac{qB}{m} \)

Period of revolution: \( T = \frac{2\pi m}{qB} \)

Uniform Electric Field

Particle accelerates along field lines:

Acceleration: \( a = \frac{qE}{m} \)

Velocity changes linearly in direction of field, motion is straight-line.

Motion of charged particle placeholder

Example 1

An electron moves with velocity 2×10⁶ m/s perpendicular to 0.01 T magnetic field. Determine the radius and period of its circular path.

Radius: \( r = \frac{mv}{qB} = \frac{9.11\times10^{-31} \cdot 2\times10^6}{1.6\times10^{-19} \cdot 0.01} \approx 1.14\times10^{-3} m \)

Period: \( T = \frac{2\pi m}{qB} = \frac{2\pi \cdot 9.11\times10^{-31}}{1.6\times10^{-19} \cdot 0.01} \approx 3.58\times10^{-9} s \)

Example 2

A proton accelerates from rest in a uniform electric field of 500 N/C. Find acceleration if m = 1.67×10⁻²⁷ kg.

Acceleration: \( a = \frac{qE}{m} = \frac{1.6\times10^{-19} \cdot 500}{1.67\times10^{-27}} \approx 4.79\times10^{10}\ m/s^2 \)

Practice Problems

  1. A particle of mass 2×10⁻³ kg and charge 4 μC moves at 10 m/s perpendicular to a 0.02 T magnetic field. Find radius of motion.
  2. Electron enters a 5000 N/C uniform electric field from rest. Find speed after 1 μs.
  3. Proton moves at 2×10⁵ m/s at 45° to magnetic field 0.01 T. Compute Lorentz force and circular motion radius component.
  4. Describe qualitatively the trajectory of a charged particle entering uniform E and B fields perpendicular to each other.
  5. A cyclotron accelerates protons to 1×10⁷ m/s. If B = 1 T, compute the orbit radius.
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