Overview
The lecture demonstrates how to apply Coulomb's Law to a multi-charge system, emphasizing symmetry and geometry to simplify force and acceleration calculations.
Problem Setup
- Five fixed protons are arranged in a wedge shape at 30° angles, each 1 micron apart.
- A sixth, movable proton is placed between charges four and five.
- The goal is to find the acceleration of the movable proton when released.
Using Symmetry
- Rotating the system by 15° aligns it so the net force on the movable proton will have only an x-component.
- Forces from charges four and five cancel out due to symmetry and equal distance from the movable proton.
- Only charges one, two, and three contribute to the net force calculation.
Displacement Vectors
- The distance from each relevant charge to the movable proton is D (1 micron).
- R20 = D[cos(15°) x̂ + sin(15°) ŷ]; R30 is a mirror image with -sin(15°) in ŷ.
- R10 = 2D cos(15°) in the x direction.
Applying Coulomb's Law
- Coulomb’s force: F = k q₁ q₂ / r²; use the appropriate r for each contributing charge.
- For charge 1: r = 2D cos(15°); direction is x̂.
- For charges 2 and 3: r = D; directions are along R20 and R30 unit vectors.
Combining Forces
- The y-components of the forces from charges two and three cancel each other.
- The net x-component is: F_net = (k e² / D²) × [1/4 cos²(15°) + 2cos(15°)].
- The geometry of the arrangement determines the expression in brackets.
- Calculating the numerical factor yields approximately 2.2.
Final Calculations
- Substituting constants: k = 8.99 × 10⁹ N·m²/C², e = 1.62 × 10⁻¹⁹ C, D = 1 × 10⁻⁶ m.
- The calculated force: F ≈ 5.08 × 10⁻¹⁶ N.
- The mass of a proton is 1.67 × 10⁻²⁷ kg.
- Acceleration: a = F/m ≈ 3 × 10¹¹ m/s².
Important Notes
- The calculated acceleration is instantaneous; it changes as the proton moves.
- Once the proton moves, forces from charges four and five will also affect the net force in the x direction.
- Acceleration is not constant in this arrangement.
Key Terms & Definitions
- Coulomb’s Law — The force between two electric charges is F = k q₁ q₂ / r², where k is Coulomb's constant.
- Symmetry — Using geometric properties to simplify physics problems by canceling out equal and opposite contributions.
- Micron (μm) — One millionth of a meter, or 1 × 10⁻⁶ m.
Action Items / Next Steps
- Review trigonometric calculations for displacement vectors.
- Practice applying symmetry to simplify force calculations in multi-charge arrangements.
- Prepare for further problems involving non-constant acceleration in electric fields.