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: $E<em>z = \frac{k</em>c Q z}{(z^2 + R^2)^{3/2}}\hat{k}$. 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