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Electric Field from Line Charges

Sep 3, 2025

Overview

This lecture covers how to calculate the electric field at a specific point due to two finite lines of charge, breaking the problem into vertical and horizontal charge segments and deriving general solutions using integration.

Problem Setup

  • Two line charges of length L, each with linear charge density λ, are considered.
  • Point P is located at (L, L/2) relative to the origin.
  • The goal is to find the electric field at point P due to both segments.

Vertical Segment Calculation

  • The vertical charge runs from y = –L/2 to y = +L/2 along the y-axis.
  • A small charge element: dQ = λ dy.
  • Each dQ creates an electric field: dE = k dQ / r² in the r̂ direction.
  • The field is split into x and y components using trigonometry.
  • Eₓ and Eᵧ are found by integrating over the segment.
  • General results for the field from a vertical segment at horizontal distance L:
    • Eₓ = [2 ln(√5)/5] (k λ / L)
    • Eᵧ = 0 (for this configuration)

Horizontal Segment Calculation

  • The horizontal charge runs along the x-axis from x = –L to x = 0.
  • A small charge element: dQ = λ dx.
  • dE = k dQ / r² with r changing along x.
  • Field components are resolved similarly as before but with axes swapped.
  • General solutions are derived for Eₓ and Eᵧ, analogous to the vertical case but interchanging x/y and L/a.

Combining Results

  • The vertical segment contributes only to the x-component of the field at the point.
  • The horizontal segment contributes to both x and y components.
  • Total field at point P is the vector sum of these contributions:
    • E_total = [2 k λ / L] x̂ + [2 ln(5)/5] ŷ.

Key Terms & Definitions

  • Linear charge density (λ) — Charge per unit length along a line.
  • Electric field (E) — Force per unit charge, here due to distributed charges.
  • k — Coulomb’s constant.
  • r̂ — Unit vector pointing from the charge element to the field point.

Action Items / Next Steps

  • Review the general expressions for the electric field due to finite and infinite line charges.
  • Practice deriving the field at different positions for both vertical and horizontal segments.
  • Check the textbook’s integral tables for similar problems.

Full transcript