Lecture 6

Electric Wind and Ion Propulsion

• Electric Wind: Uses the sharp point effect for propulsion.

• Deep Space 1 Mission: Utilized ion propulsion to reach asteroid (9969 Braille).

  • Produced low thrust (0.1 N) and low acceleration but sustained for long duration (Δv = 15,000 km/h).

  • Highly efficient, consuming less than 159 pounds of fuel over 16,000 hours.

  • Allows for smaller and lower-cost launch vehicles due to reduced fuel requirements.

Previous Concepts Recap

• Charge Density

• Electric Field of a Charge Distribution

• Electric Field of a Uniformly Charged Rod

  • Near the finite rod (r << L): Formula from far-close approximation valid for infinite rod as well.

  • General formula provided with constants.

Current Goals

• Determine Fields of Uniformly Charged:

  • Ring

  • Disk

  • Infinite Plane

  • Two Infinite Planes

Electric Field Characteristics

• Point Charge vs Line Charge vs Plane Charge:

  • Electric field varies inversely with the distance for point charge (E ∝ 1/r) and inversely squared for line charge (E ∝ 1/r²).

  • Electric field generated by uniformly charged ring along its axis.

Finding Electric Fields Along a Ring

• Goal: E along the z-axis of a ring up to radius 'a'.

• Point Charge Contribution:

  • Use formula: E = (1/4πε₀) * (q / |r|²) * r̂.

  • E = (1/4πε₀) * q * (a² + z²)^{-3/2} * r̂.

Integration Approach

• Consider electric field contribution from infinitesimal charges on the ring.

• Electric fields contribute symmetrically; only the z-component accumulates.

• Field Expression for a Ring:

  • Total electric field along axis derived from integration of point charges.

Calculating Electric Field Along a Disk

• Initial Goal: Find total electric field along the z-axis with a disk radius R.

• Set up integration to account for uniformly distributed charge across the disk area.

• Solution approach includes evaluation of z-component of electric field due to disk's symmetry.

Final Observations

• When observing from a distance z >> R, the field of a uniformly charged disk achieves characteristics similar to a point charge.

• Infinite Plane Analysis:

  • E field derived from uniform surface charge densityσ leads to constant fields.

  • Superpositions considered for two infinite planes resulting in zero field.

Summary of Topics to Understand

• Objectives for week:

  • Understand electric fields for a ring, disk, infinite plane, and configurations with two infinite planes.

• Review symmetry considerations and methods for deriving electric fields from charge distributions.