# Laplace equation and electrostatic potential

• Jan 20th 2008, 01:04 AM
Pinsky
Laplace equation and electrostatic potential
Can someone please explain the connection between electrostatic potential and Laplace's equation? And what does that have to do with harmonic functions?
• Jan 20th 2008, 02:07 AM
mr fantastic
Quote:

Originally Posted by Pinsky
Can someone please explain the connection between electrostatic potential and Laplace's equation? And what does that have to do with harmonic functions?

This is a real broad brush question. A decent answer would take up several chapters of a book. I can say this with confidence because I have studied the chapters of such books.

In my opinion you're best advised to consult any decent undergraduate textbook on electro-magnetism and study the salient chapters at your leisure.

eg. Lorraine and Corson - Electromagnetic Fields and Waves, Jackson -Classical Electrodynamics.

In a nutshell:

1. In regions of space where there is no charge density the electrostatic potential satisfies Laplaces equation: \$\displaystyle \nabla^2 V = 0\$ where V is the electrostatic potential.

2. Any real function with continuous second partial derivatives which satisfies Laplace's equation is called a harmonic function.

Background knowledge, derivation, details and examples (and answers to the million and one questions the nutshell has prompted) are what constitute the above-mentioned chapters.

Just curious by the way ...... how do you happen to be in the unfortunate circumstance of this question being urgent homework?
• Jan 20th 2008, 02:21 AM
Pinsky
O well, i'll try that then.

It's one of my questions for math theory.

"Appliance of harmonic functions on electrostatic potential"
• Jan 20th 2008, 02:27 AM
mr fantastic
Quote:

Originally Posted by Pinsky
O well, i'll try that then.

It's one of my questions for math theory.

"Appliance of harmonic functions on electrostatic potential"

So you have to write a paper or something?

The nutshell says it all. What you should do:
1. Summarise definitions of terms and concepts in the nutshell.
2. Go from the nutshell to worked examples.
Any decent textbook on electro-magnetism will have all you need.