L27.1 Separation of variables - Cartesian coordinates - Example 3.4 P-I
#electrodynamics #griffiths #sayphysics
00:00 - Introduction to Example 3.4
00:06 - Example 3.4 Problem Statement
00:20 - Setup of the Two-Dimensional Problem with Grounded Metal Plates
00:31 - Description of Metal Plates at y = 0 and y = a
00:39 - Metal Strips at x = ±b and Potential v₀
01:19 - Objective: Find the Potential Inside the Rectangular System
01:28 - Laplace Equation for Two-Dimensional Potential
02:13 - Boundary Conditions for the Problem
02:32 - First Boundary Condition: v = 0 at y = 0
02:47 - Second Boundary Condition: v = 0 at y = a
02:51 - Third Boundary Condition: v = v₀ at x = ±b
03:37 - Comparison to Example 3.3 and Explanation of Differences
04:01 - Solving the Equations with Separation of Variables
04:32 - Using Old Solutions for This Setup
05:08 - Potential Equation in Terms of Exponentials and Trigonometric Functions
05:51 - Symmetry of the Enclosure and Implications on Potential
06:33 - Derivation of the General Solution
07:07 - Relationship Between Constants A and B
07:44 - Simplification to Two Hyperbolic Cosine Terms
08:58 - Final Expression for the Potential v(x, y)
09:51 - Applying Boundary Conditions to Final Equation
10:17 - Boundary Condition at x = ±b to Solve for Constants
10:39 - Final Step to Solve for Potential v₀
Example 3.4
Two infinitely long grounded metal plates, again at y = 0 and y = a, are connected at x = ±b by metal strips maintained at a constant potential Vo, as shown in Fig. 3.20 (a thin layer of insulation at each comer prevents them from shorting out). Find the potential inside the resulting rectangular pipe.
In this video, we solve Example 3.4 from the renowned textbook 'Introduction to Electrodynamics' by David J. Griffiths. The example explores the electrostatics problem of two infinitely long grounded metal plates, separated by a distance 'a' and connected by metal strips at a distance 'b' from the corners, maintained at a constant potential Vo. Through a step-by-step explanation, we demonstrate how to calculate the potential inside the resulting rectangular pipe formed by the plates. Join us as we unravel the intricacies of this fascinating problem in electromagnetism.
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