Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Pie Academy

Physics : Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. The electric field is related to the charge density by the divergence relationship

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

and the electric field is related to the electric potential by a gradient relationship

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Therefore the potential is related to the charge density by Poisson's equation

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

In a charge-free region of space, this becomes LaPlace's equation

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

This mathematical operation, the divergence of the gradient of a function, is called the LaPlacian. Expressing the LaPlacian in different coordinate systems to take advantage of the symmetry of a charge distribution helps in the solution for the electric potential V. For example, if the charge distribution has spherical symmetry, you use the LaPlacian in spherical polar coordinates.

Since the potential is a scalar function, this approach has advantages over trying to calculate the electric field directly. Once the potential has been calculated, the electric field can be computed by taking the gradient of the potential.

Potential of a Uniform Sphere of Charge

The use of Poisson's and Laplace's equations will be explored for a uniform sphere of charge. In spherical polar coordinates, Poisson's equation takes the form:

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

but since there is full spherical symmetry here, the derivatives with respect to θ and φ must be zero, leaving the form

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Examining first the region outside the sphere, Laplace's law applies.

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Since the zero of potential is arbitrary, it is reasonable to choose the zero of potential at infinity, the standard practice with localized charges. This gives the value b=0. Since the sphere of charge will look like a point charge at large distances, we may conclude that

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

so the solution to LaPlace's law outside the sphere is

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Now examining the potential inside the sphere, the potential must have a term of order r2 to give a constant on the left side of the equation, so the solution is of the form

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Substituting into Poisson's equation gives

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Now to meet the boundary conditions at the surface of the sphere, r = R

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The full solution for the potential inside the sphere from Poisson's equation is

Poisson and Laplace Equations - Electrostatics, Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Dynamic Test

Content Category

Related Searches

practice quizzes

,

Previous Year Questions with Solutions

,

Electromagnetic Theory

,

Electromagnetic Theory

,

Sample Paper

,

past year papers

,

Summary

,

CSIR-NET Physical Sciences Physics Notes | EduRev

,

mock tests for examination

,

Free

,

video lectures

,

ppt

,

CSIR-NET Physical Sciences Physics Notes | EduRev

,

pdf

,

Important questions

,

study material

,

Poisson and Laplace Equations - Electrostatics

,

Objective type Questions

,

Extra Questions

,

Poisson and Laplace Equations - Electrostatics

,

Electromagnetic Theory

,

Viva Questions

,

MCQs

,

shortcuts and tricks

,

Poisson and Laplace Equations - Electrostatics

,

Exam

,

CSIR-NET Physical Sciences Physics Notes | EduRev

,

Semester Notes

;