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# MODERN PHYSICS

Fine Structure To use laplaces and poissons equation in electricity and magnetism, this equation used in determine formula in solve problem in potential. So, first of all we must understand about potential. a. Introduction to Potential The electric field E is not just any old vector function, it is a very special kind of vector function, one whose curl is always zero. , for example, could not possibly be an electrostatic field; no set of charges, regardless of their sizes and position, could ever produce such a field. In this section were going to exploit this special property of electric field to reduce a vector problem (finding E) down to a much simpler scalar problem. What Im going to do now amounts to a proof of that claim, in the context of electrostatics.

(figure 2.30) Because , the line integral of E around any closed loop is zero (thats follow from stokes theorem). Because , the line integral of E from point a to point b is the same for all paths (otherwise you could go out along path (i) and retrun along path (ii) figure 2.30and obtain ). because the line integral is dependent of path. We can define a function. ( )

Here o is some standard reference point on which we have agreed beforehand; V then depends only on the point r. it is called the electric potential. Evidently, the potential difference between two points a and b is ( ) ( )

## Now, the fundamental theorem for gradient states that ( ) So, ( ) ( ) ( )

Since, finaly, this is true for any points a and b, the integral must be equal;

b. Poissons and Laplaces Equation From the equation above explained that the electric field can be written as the gradient of a scalar potential.

## The question arises: what do the fundamental equation for E, and

Note: E= electric field = charge density = permittivity

( ) Look like in terms of V? well, , so, apart from that persisting minus sign, the divergence of E the laplacian of V. Gauss law then says that

This is known as Poissons equation. In region where there is no charge, so that = 0, poissons equation reduces to laplaces equation

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. c. 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:

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

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

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

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

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

## Substituting into Poisson's equation gives

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

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

Reference: -

## Griffith.1999. Introduction to electrodynamics.United States of America:Prentice Hall

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/laplace.html http://mathworld.wolfram.com/LaplacesEquation.html http://farside.ph.utexas.edu/teaching/em/lectures/node31.html
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