Laplace and Poissons equation

1. I &II Semester -2020-21
Unit-6. Introduction to
Electromagnetism
By
Hasan Ziauddin
Assistant Professor
Department of Physics
RIET , Jaipur
Rajasthan

2. Electric Field

3. Electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a
type of physical interaction that occurs between electrically charged particles. The
electromagnetic force is carried by electromagnetic fields composed of electric fields and
magnetic fields, and it is responsible for electromagnetic radiation such as light. It is one of
the four fundamental interactions (commonly called forces) in nature

4. Charge at rest

5. There are many cases where we have uniformly moving particles—for instance, cosmic rays
going through a cloud chamber, or even slow-moving electrons in a wire
Charge with uniform velocity

6. Charge moving with an acceleration
Note: This is what is meant by saying that charges in uniform linear motion do not radiate
but accelerating charges do.

7. Divergence and Curl

8. The Divergence of E

11. Applications of Gauss’s Law

13. The Curl of E

14. Curl Of the Electric Field
(For Stationary Charge)

15. (For Charge moving with uniform speed)
Curl of the Electric Field (Digression)

18. Electric Potential

19. Electric Potential due to a point charge at origin:

20. Electric Potential due to localized charge distribution:

21. Calculating the Electric Field

22. Laplace Equation (Divergence of Gradient)
Gradient Field

23. Laplace Equation in Three Coordinate System
In Cartesian coordinates:
and,
Knowing
Hence, Laplace’s equation is,

24. In cylindrical coordinates:
and,
Knowing
Hence, Laplace’s equation is,

25. In spherical coordinates:
and,
Knowing
Hence, Laplace’s equation is,

26. Laplace’s and Poisson’s Equation
We have determined the electric field 𝐸 in a region using Coulomb’s law or Gauss law
when the charge distribution is specified in the region or using the relation 𝐸 = −𝛻𝑉
when the potential V is specified throughout the region.
However, in practical cases, neither the charge distribution nor the potential
distribution is specified only at some boundaries. These type of problems are known
as electrostatic boundary value problems.
For these type of problems, the field and the potential V are determined by using
Poisson’s equation or Laplace’s equation.
Laplace’s equation is the special case of Poisson’s equation

27. 𝐸 = −𝛻𝑉
𝛻 ∗ (𝐸 ) = 𝜌𝑉/ ∈0
Electric field based on potential
As per Gauss Law of Electric Field
𝛻 ∙ (−𝛻𝑉) = 𝜌𝑉 /∈0
𝛻 ∙ 𝛻𝑉 = −𝜌𝑉/∈0
The Poisson’s and Laplace’s equation can be easily derived from Gauss’s equation
Putting the value of E in Gauss Law,
 This equation is known as Poisson’s equation which state that the potential
distribution in a region depend on the local charge distribution.

28.  In many boundary value problems, the charge distribution is involved on the
surface of the conductor for which the free volume charge density is zero, i.e., ƍ=0. In
that case, Poisson’s equation reduces to,
 This equation is known as Laplace’s equation.
Laplace’s equation

29. Summary

30. Application of Laplace’s and Poisson’s Equation
Using Laplace or Poisson’s equation we can obtain:
1. Potential at any point in between two surface when potential at two surface are
given.
2. We can also obtain capacitance between these two surface.

32. Parallel Plates

34. Q. Consider two concentric spheres of radii a and b, a<b. The outer sphere is kept at
a potential 𝑉0 and the inner sphere at zero potential. Solve Laplace equation in
spherical coordinates to find
1. The potential and electric field in the region between two spheres.
2. Find the capacitance between them.
SOLUTION:
From the given data it is clear that V is a function of r only.
e.i.
߲𝑉
߲ߠ
and
߲𝑉
߲߶
are zero. We know the Laplace's equation in spherical coordinate
system as,
This equation reduces to,
By Integrating we get,

35. Integrating again we get,
The given boundary condition are,
Applying these boundary con to equ (i)
Subtracting equ (i) from equ (ii)
Putting value of A in equ (ii) we get,

36. Putting these values in equ (i), we get potential V as,
We have,