Poisson’s and Laplace’s Equation

Pierre-Simon & marquis de Laplace are derived this equation.

marquis de Laplace(1749-1827)

laplaceBorn March 23, 1749, Beaumount-en-Auge, Normandy, France, died March 5, 1827, Paris.  French mathematician, He is an astronomer, and physicist who is best known for his investigations into the stability of the solar system.

Pierre-Simon(1781 – 1840)

PoissonPoisson’s most important works were a series of papers on definite integrals and his advances in Fourier series. This work was the foundation of later work in this area by Dirichlet and Riemann.

Poisson’s  Equation

Electric potentials can be conveniently related to the charge density by using Laplace equation as follows.Poisson’s and Laplace’s Equations are the combination of Gauss’s law with the gradient of the electric field.

Electric field E at any point may be expressed as the -ve of the gradient of potential at that pont.

Ie, E =  – grad.V  = −∇.V

Put this value of E in differential form of gauss’s law

∇⋅E = ρ/ε

∇⋅(-∇V) =   ρ/ε

∇⋅(∇V) =  – ρ/ε

∇² V  =   – ρ/ε …………………………….Poisson’s Equation

This is known as poisson’s equation.∇² is called Laplacean operator.solution of this equation gives the value of the potential V

When ρ is equal to zero,that means all parts of the space contain no electric charge.

∇² V = 0 ………………………………..Laplace’s equation

This is called Laplace’s equation.

Laplace’s equation will be different in different cordinate system depends on ∇²

Example

In cartesian system

∇² = ∂²V/∂X²  +  ∂²V/∂Y²  + ∂²V/∂Z²

∂²V/∂X²  +  ∂²V/∂Y²  + ∂²V/∂Z²   = 0

References

  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press

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