Section 4.6: Motion of Charges

Charged particles experience forces when placed in electric fields, causing them to accelerate according to Newton’s second law. Motion can be linear or curved depending on the field.

Equations of Motion:
  • Force on a charge: \( \vec{F} = q \vec{E} \)
  • Acceleration: \( \vec{a} = \frac{\vec{F}}{m} = \frac{q \vec{E}}{m} \)
  • Velocity after time \(t\): \( \vec{v} = \vec{v_0} + \vec{a} t \)
  • Displacement: \( \vec{s} = \vec{v_0} t + \frac{1}{2} \vec{a} t^2 \)
  • In uniform fields, motion is analogous to projectile motion.

Example 1

A proton (mass \(1.67 \times 10^{-27} \text{kg}\), charge \(1.6 \times 10^{-19} \text{C}\)) is placed in a uniform electric field of 500 N/C. Find its acceleration.

\( a = \frac{F}{m} = \frac{qE}{m} = \frac{1.6 \times 10^{-19} * 500}{1.67 \times 10^{-27}} \approx 4.79 \times 10^{10} \text{ m/s²} \)

Example 2

An electron enters a uniform electric field of 200 N/C with an initial horizontal velocity of \(2 \times 10^6\) m/s. Find its vertical displacement after 1 μs.

\( a = \frac{qE}{m} = \frac{1.6 \times 10^{-19} * 200}{9.11 \times 10^{-31}} \approx 3.51 \times 10^{13} \text{ m/s²} \)

\( y = \frac{1}{2} a t^2 = 0.5 * 3.51 \times 10^{13} * (1 \times 10^{-6})^2 \approx 17.55 \text{ m} \)

Practice Problems

  1. Calculate the acceleration of a 2 C charge with mass 0.5 kg in a 100 N/C field.
  2. An electron moves in a 300 N/C uniform field. Find the force on it.
  3. Determine displacement after 2 μs for a proton in a 400 N/C field starting from rest.
  4. Compare accelerations of proton and electron in the same electric field.
  5. A charge moves with initial velocity in a uniform field. Determine final velocity after t seconds.
  6. Explain why motion in a uniform electric field is parabolic.
  7. Calculate time to reach a speed of 1e5 m/s for an electron in a 500 N/C field.
  8. Determine the trajectory equation for a charge starting from origin with horizontal velocity \(v_0\).
  9. Explain effect of mass on motion of different charged particles in same field.
  10. A proton moves in an electric field of 1000 N/C. Find its kinetic energy after moving 0.01 m.
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