Do we use Laplace and Poisson's equation only when electrostatic condition at some boundary is known?
These equations help us to find out V=> E => D=> J=> I=>C => R.........
due to Surfaces kept at different potential, for eg capacitor.
If charge density is zero, then Lalace eqn is used otherwise Poissons eqn.
V in poissons eqn is Electric potential in case of Electrostatics, Magnetic pot. in Magnetoststics, Temperature in thermaldynamics, Pressure in seepage problems etc.
These equations help us to find out V=> E => D=> J=> I=>C => R.........
due to Surfaces kept at different potential, for eg capacitor.
If charge density is zero, then Lalace eqn is used otherwise Poissons eqn.
V in poissons eqn is Electric potential in case of Electrostatics, Magnetic pot. in Magnetoststics, Temperature in thermaldynamics, Pressure in seepage problems etc.
You should use Poisson's equation when your solution region contains space charges and if you do not have space charges(practically it is impossible) you can use Laplace equation.
Poisson's equation is taking care of volume charge density while Laplace equation does not.
The Laplace/Poisson Equasions are the Helmholts equations when the time derivative is zero (f=0) .
For this case there is no dependance between the magnetic and electrical fields so the Laplace/Poisson equations solutions represent the electrical fields alone.
If there are no electical static charges, and the source exciting the electrical filed comes only from the boundary conditions, the laplace equation is used. Laplace Equation is a private case of Poisson's equation in which the electrical charges are zero.
In both cases (Laplace/Poisson) the sollution is appropriate only for static, electrical fields, and boundary conditions should be applied in both cases of course.
Dear Sandeep,
The Poisson equation div D= roh is one of the basic equations in electrical engineering relating the electric displacement D to the volume charge density.
The electric displacement is the electric flux density. It is proportional to the electric field field E and it is equal to dielectric constant epsilon E, that is D= epsilon E. While the electric field independent of the material popularization, the electric flux density depends on the electric polarization of the material.
So, an other form to the Poisson equation is div E= roh/epsilon.
One of the very important interpretation to last form of the Poisson equation is the the electric field changes only by the presence of electric charges. It is so that electric field lines begin on positive electric charges and terminate on the negative charges. Since the electric field E is negative of the gradient of the electric potential V, then E= -grad V. So, an other form of the Poisson equation is
div grad V=- roh/epsilon.
This is differential equation relating the potential V to density in incremental volume in space. It is used to solve for V as a function of position which is the potential distribution. So, given roh in a space one can get V.
The La Place equation is a special case of Poison equation where roh is equal to zero, meaning space charge free charges.
Electromagnetic equations are the Ampere law relating the magnetic field to the electric current and the Induction law relating the induced voltage to the time rate of change of the magnetic flux.
Best wishes
See to understand the significance you have to understand what actually Laplace's Equation and It's Solution Signify. I don't think there is any other method to understand it more than a visualization. So I have linked a video here..watch the Series..
https://youtu.be/Vj14UWjKkVg
Solving the Poisson equation amounts to finding the electric potential φ for a given charge distribution . The mathematical details behind Poisson's equation in electrostatics are as follows (SI units are used rather than Gaussian units, which are also frequently used in electromagnetism).
Laplace's equation, second-order partial differential equation widely useful in physics because its solutions R (known as harmonic functions) occur in problems of electrical, magnetic, and gravitational potentials, of steady-state temperatures, and of hydrodynamics.
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