Electromagnetics Theory Lecture: Electric Field on the Axis of a Uniformly Charged Ring

Introduction

• Professor Itesh Dolakia presents the topic.
• Focus: Electric field on the axis of a uniformly charged ring.

Overview of the Ring

• Charged Ring: A ring with total charge Q.
• Radius: Denoted as a.
• Line Charge Density:
  • Formula: ( , \lambda = \frac{Q}{2\pi a} ) (total charge divided by the length of the ring).

Electric Field on the Axis

• Axis Definition: Defined as the x-axis extending from the center of the ring.
• Point of Interest: Distance x from the center of the ring.
• Differential Charge Element:
  • Consider a small charge element dq, contributing to the electric field at a point on the axis.

Components of Electric Field

• Distance Calculation:
  • Distance from dq to the point on the axis: ( r = \sqrt{x^2 + a^2} ).
• Electric Field Contribution:
  • Electric field due to dq: ( dE = \frac{K , dq}{r^2} ).
• Angle Analysis:
  • Angles formed by the electric field components are considered (angles theta).
• Component Breakdown:
  • Cosine Component: ( dE \cos(\theta) ) contributes to the net electric field.
  • Sine Component: Opposite directions cancel out.

Calculation of Electric Field

• Cosine of Angle:
  • From triangle: ( \cos(\theta) = \frac{x}{r} ).
• Net Electric Field Calculation:
  • ( dE_{net} = \frac{K , x , dq}{(x^2 + a^2)^{3/2}} ) directed along ax._

Total Electric Field

• Integration: To find the total electric field:
  • Total electric field:
    [ E = \int dE_{net} = \frac{K , x , Q}{(x^2 + a^2)^{3/2}} \text{ in the direction of } ax. ]
• Alternative Expression:
  • Using line charge density:
    [ E = \frac{K , x , \lambda 2\pi a}{(x^2 + a^2)^{3/2}} \text{ with } \lambda = \frac{Q}{2\pi a}. ]_

Key Takeaways

• Remember both formulas for calculating the electric field due to a uniformly charged ring.
• The formulas will be useful for future calculations, including those related to a uniformly charged disc.

Next Lecture

• Topic: Calculation of electric field due to a uniformly charged disc.
• Encouragement for questions in the comment box.