Overview
This lecture explains how to calculate the electric field created by a continuous line of charges using calculus, focusing on converting discrete charge sums into integrals and deriving electric field components.
Discrete to Continuous Charge Distribution
- Calculating the field from many point charges individually is complex due to messy vector addition.
- Instead, treat the line of charges as a continuous charge distribution.
- Divide the bar into small segments of width (dx), each containing charge (dq).
Linear Charge Density and Setup
- Total charge (Q) spread over length (L) gives linear charge density (\lambda = Q/L).
- The charge in each segment is (dq = \lambda , dx).
- The electric field from each segment: (k , dq / r^2) in the (\hat{r}) direction.
- To find the total field, integrate across the length of the line.
Integral Formulation
- The field at point (P) is an integral: (E = \int_{x_1}^{x_2} k \lambda / r^2 , \hat{r} , dx).
- The field has both (x) and (y) components; integration must be done for each.
Component Decomposition and Trigonometry
- Express (r) and (\hat{r}) in terms of (x, y), and trigonometric functions.
- (r = \sqrt{x^2 + y^2}); (\sin \theta = x/r), (\cos \theta = y/r).
- (E_x = k \lambda \int x / r^3 , dx), (E_y = k \lambda \int y / r^3 , dx).
Variable Substitution
- Use the triangle: (\tan \theta = x/y \Rightarrow x = y \tan \theta).
- Rewrite (dx = y \sec^2 \theta , d\theta); substitute this into the integrals.
Simplified Integrals and Results
- After substitution, (E_x = (k \lambda / y) [\cos \theta_2 - \cos \theta_1]).
- (E_y = (k \lambda / y) [\sin \theta_2 - \sin \theta_1]).
- (\theta_1, \theta_2) are the angles at the ends of the charge line relative to point (P).
Special Cases and Results
- At the center above the line, (E_x = 0) due to symmetry.
- For a very long line, (E_y = 2k\lambda / y); the field points directly away and decreases as (1/y).
Key Terms & Definitions
- Linear Charge Density ((\lambda)) — Charge per unit length on a line of charge.
- (\mathbf{k}) — Coulomb's constant ((1/4\pi\epsilon_0)).
- (\hat{r}) — Unit vector pointing from the source to the field point.
- (dq) — Infinitesimal charge segment: (dq = \lambda dx).
- Trigonometric Substitution — Using triangle relations to express variables for integration.
Action Items / Next Steps
- Review calculus integration techniques, especially trigonometric substitution.
- Read the textbook example for this problem for alternative solution methods.