Transcripts of 21.8

Overview of Electric Fields and Parallel Plates

• The presentation covers the electric field produced by charged plates, specifically in Section 21.8 of Module 21.

• Recap of previous material: The electric field due to a large conducting plate with surface charge density sigma (σ).

Electric Field from a Single Conducting Plate

• Electric field (E) at a distance from a conducting plate with surface charge density σ:

  • E = σ / (2 * ε₀)

  • This electric field is constant regardless of distance from the plate, assuming the distance is small compared to the plate dimensions.

• Positive charges cause the electric field to move away from them, while negative charges attract electric field lines.

Configuration of Parallel Plates

• Introduction of a second parallel plate with negative charge of magnitude -q.

• Both plates create electric fields:

  • E from positive plate away from it.

  • E from negative plate heading towards it.

• The total electric field in the middle of the two plates is:

  • E_total = E_positive + E_negative = (σ / (2 * ε₀)) + (σ / (2 * ε₀)) = σ / ε₀

• This results in a uniform electric field between the plates, while outside the plates, the fields cancel or are greatly diminished, effectively making E ≈ 0.

Charging the Parallel Plates

• To charge the plates, a battery can be connected:

  • Positive terminal connects to the top plate, making it positively charged.

  • Negative terminal connects to the bottom plate, making it negatively charged.

• Charge cannot flow through the gap (approx. 32,000 volts per inch needed to create a breakdown); hence, the assumption of an electric field generated in a vacuum or air.

Force on Charged Particles in an Electric Field

• When a charge q is placed in the electric field, it experiences a force (F) given by:

  • F = E * q

• This force causes the charge to accelerate:

  • a = F / m = (E * q) / m

• Relationship between acceleration and the electric field direction:

  • Positive charge accelerates parallel to the electric field.

  • Negative charge accelerates anti-parallel.

Equations of Motion in Electric Fields

• For a charge experiencing uniform acceleration in the electric field, kinematics equations can be used:

  • v_final = v_initial + a * t

  • s = v_initial * t + (1/2) * a * t²

• Substituting acceleration (a) yields:

  • Distance traveled, y, for a charge in an electric field:

    • y = (1/2) * (q/m) * E * t²

Work-Energy Theorem in Electrostatics

• The work done on the charge by the electric field results in a change in kinetic energy:

  • Work = F * d

• The electric force does work on the charge, leading to:

  • ΔKE = q * E * d

• This represents the work-energy theorem applied to electrostatics and is consistent with classical mechanics.

Deflection of Charges in Electric Fields

• Example of a charge injected horizontally into a uniform electric field:

  • Initial velocity in the x-direction with zero velocity in the y-direction.

  • Force causes an upward acceleration, deflecting the trajectory of the charge in a parabolic path similar to projectile motion.

• Equations derived for horizontal and vertical movements involve:

  • The time of flight based on horizontal travel (constant x velocity).

  • Vertical deflection related to time spent in the electric field and resulting acceleration from the force due to the electric field.

• Final deflection equation:

  • y_deflection = (1/2) * (q/m) * E * (x_initial / v⁰_x)² * x²

  • Results in parabolic motion characteristics, consistent with previous physics learning.

Conclusion

• The discussion integrates fundamental physics concepts from earlier courses to explain motion within a constant electric field.

• Students are encouraged to revisit the associated textbook sections for clarity and deeper understanding as they advance to Module 23.