Boundary conditions for Laplace's equation

Click For Summary
SUMMARY

The discussion focuses on the boundary conditions for Laplace's equation, particularly in the context of the Legendre expansion of the potential due to a unit charge represented by the equation 1/|x-x'|. It is established that the potential satisfies Laplace's equation everywhere except at the point x=x', where the charge is located. The necessity of defining boundary conditions at this singular point is emphasized, highlighting the importance of understanding potential behavior in the vicinity of point charges.

PREREQUISITES
  • Understanding of Laplace's equation and its applications in electrostatics.
  • Familiarity with the concept of potential in physics, particularly in relation to point charges.
  • Knowledge of Legendre polynomials and their role in potential expansions.
  • Basic grasp of boundary value problems in partial differential equations.
NEXT STEPS
  • Study the derivation and applications of Laplace's equation in electrostatics.
  • Explore the properties and applications of Legendre polynomials in potential theory.
  • Learn about boundary value problems and their solutions in the context of partial differential equations.
  • Investigate the implications of singularities in potential fields and how they affect boundary conditions.
USEFUL FOR

Physicists, mathematicians, and engineering students who are studying electrostatics, potential theory, or boundary value problems in partial differential equations.

IFNT
Messages
31
Reaction score
0
I don't seem to grasp the meaning of boundary conditions for Laplace's equation.

Consider the Lagendre expansion of the potential at x due to a unit charge 1/|x-x'|, where x' is the position of the unit point charge.
To do the expansion, we need to consider a volume in space where the potential satisfies the Laplace equation. I can see that the potential satisfies Laplaces equation anywhere in space except at x=x', but what are the boundary conditions? Don't we have to know the potential at x=x', which is a boundary?
 
Physics news on Phys.org
Am I too vague in my question?
 

Similar threads

Graduate The no boundary proposal and quantum gravity
  • · Replies 7 ·
Replies
7
Views
5K
Undergrad Dirichlet problem boundary conditions
  • · Replies 4 ·
Replies
4
Views
2K
Graduate The boundary conditions in reference to Laplace's equation
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad Boundary conditions for a stream function in a hydrodynamics problem
  • · Replies 5 ·
Replies
5
Views
1K
Graduate Why is "method of image current" valid in magnetostatics?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Where is Laplace's Equation Valid in Different Domains?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Solving Laplace's equation in polar coordinates for specific boundary conditions
  • · Replies 3 ·
Replies
3
Views
3K
Python Tackling Boundary Conditions in Python (Griffins Example)
  • · Replies 1 ·
Replies
1
Views
1K
Potential in the three regions of an infinite slab
  • · Replies 1 ·
Replies
1
Views
1K
Potential function of a flow around a stagnation point
  • · Replies 19 ·
Replies
19
Views
3K