D3-4. Charged Particles in Electric Fields

1. Deflection in a Uniform Electric Field

The Concept

  • A charged particle in an electric field will experience a constant electric force $F = qE$.
  • If moving perpendicularly through a uniform electric field (e.g. entering the gap between two charged parallel plates), it travels in a parabolic trajectory (projectile motion).

Rules of Deflection:

  • Positive charges are deflected towards the negative plate.
  • Negative charges are deflected towards the positive plate.
  • Uncharged particles (e.g., neutrons) experience zero force and travel perfectly straight.

Factors Affecting Deflection Curve:

  • Charge: Greater magnitude of charge $\rightarrow$ Greater force $\rightarrow$ Tighter curve/more deflection.
  • Mass: Greater mass $\rightarrow$ Harder to accelerate ($a = F/m$) $\rightarrow$ Less deflection.
  • Speed: Faster particle $\rightarrow$ Less time spent in the field $\rightarrow$ Less total deflection.

2. Examples

Example 1

Problem: A single proton travelling with a constant horizontal velocity enters a uniform electric field between two parallel charged plates. Describe the path taken by a Boron nucleus (Atomic number = 5, Mass number = 11) that enters the electric field at the same point and with the same velocity as the proton.

Solution:

Step 1 (Compare Charges): Boron has 5 protons, meaning its charge is $5\times$ greater than the proton. The electrical force $F$ pulling it will be $5\times$ greater ($F = qE$).
Step 2 (Compare Masses): The Boron nucleus has 11 nucleons, meaning its mass $m$ is roughly $11\times$ greater than the proton.
Conclusion: The acceleration is $a = qE/m$. Since the mass factor ($11\times$) is much greater than the force factor ($5\times$), its vertical acceleration will be roughly half that of the proton. Because it accelerates slower vertically, the Boron nucleus will be less deflected and hit the plate further down the line.