Electric Fields: Disks, Infinite Planes, and Gauss's Law

Announcements

  • Homework Set 3: Due this coming Sunday by 10:00 PM.
  • Homework Set 4: Will be issued, likely with more conceptually and mathematically sophisticated problems, but not a large quantity.
  • Topic 1 Retakes: Information on signing up for retakes will be provided once Topic 1 scores are returned.

Review of Previous Concepts

Finite Rod Problem

  • Last session, the calculation for the finite wire/line of charge was completed, specifically calculating the y-component of the electric field at a target location after previously determining the x-component.

Charged Ring Problem

  • The electric field of a thin, uniformly charged ring was explored, focusing on the field along its axis of symmetry (e.g., the z-axis if the ring is in the xy-plane).
  • Result (with Correction): The instructor noted a crucial error in the displayed formula, which should have a power of three-halves in the denominator.
    • The electric field along the z-axis for a ring of charge Q and radius R at a distance z from its center is: Ez = \frac{kc Q z}{(z^2 + R^2)^{3/2}}\hat{k}. Where \hat{k} is the unit vector in the z-direction.
  • Dimensional Analysis: This correction (from two-halves to three-halves) is essential for the units to be consistent.
    • Electric field units must be newtons per coulomb (N/C), which are equivalent to k_c \times \text{Charge} / \text{Distance}^2.
    • If the numerator has a distance term (like z), the denominator must be effectively cubic (distance^3) to result in 1/\text{distance}^2 overall. The (z^2 + R^2)^{3/2} achieves this.

Electric Field of a Uniformly Charged Disk

Problem Setup

  • Objective: Calculate the electric field of a uniformly charged disk at a point along its axis of symmetry.
  • Difficulty: Calculating the exact field anywhere is very difficult, so the focus remains on the central axis.
  • Methodology: The disk is conceived as an infinite number of concentric, thin rings. The known electric field formula for a single ring will be used, and then integrated over all these rings.
  • Surface Charge Density: Since the charge is spread over an area, we use surface charge density, denoted by the Greek letter sigma (\sigma).
    • \sigma = Q / (\pi R^2), where Q is the total charge and R is the total radius of the disk.
  • Analogy: This method is similar to previous calculations where objects were broken down into point particles, but here, the