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Laplace Equation

Laplace Equation in Brief

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Laplace Equation

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  1. Electromagnetic Theory I BS Physics (A & B) 22nd April, 2020 Dr. Shahzada Qamar Hussain

  2. Layout of the presentation • Continuous Charge Distributions • Laplace’s Equation (Introduction) • Derivation of Laplace Equation • Laplace equation in one dimension • Laplace equation in Two Dimensions • Laplace equation in Three Dimensions • Applications of Laplace Equation • References

  3. Continuous Charge Distributions ~ ~ ~

  4. Laplace’s Equation (Introduction) • The primary task of electrostatics is to study the interaction (force) of a given stationary charges. Poisson’s Equation • To solve a differential eq. we need boundary conditions. • In case of ρ = 0, Poisson’s eq. reduces to Laplace Equation as Laplace Equation The solutions of Laplace’s eq are called harmonic function So the average of total electric potential outside will be the same at that in the center.

  5. Derivation of Laplace’s Equation Let  E be the electric field,  be the electric charge density, and εo be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states (1) Now, the electric field can be expressed as the negative gradient of the electric potential V (2) Inserting the value of E from Equation (2) into left side of Equation (1) we get Plugging this relation into Gauss's law Equation (1), we obtain Poisson's equation for electricity Let’s suppose charges are there in free space with volume charge distribution of . If we consider point p out the surface charge distribution where there is no , then the Poisson’s equation becomes… Laplace Equation 0

  6. Laplace’s Equation Cylindrical coordinates Spherical coordinates The solutions of Laplace’s eq are called harmonic function.

  7. Laplace equation in one dimension Boundary conditions: So m (Slope) will be given as No Local Maxima or Minima V(x) is the Average of 5 V(x) = -x + 5 Solution of Laplace Equation in 1 dimension

  8. Laplace equation in Two Dimensions • Partial differential equation. • To determine the solution you must fix V on the boundary – boundary condition. V has no local minima or maxima inside the boundary. Rubber membrane Soap film A ball will roll to the boundary and out.

  9. Solution of Laplace equation in Two Dimensions (1)

  10. Solution of Laplace equation in Two Dimensions (2)

  11. Solution of Laplace equation in Two Dimensions (3) Theorem If u1, u2, …, uk are solution of a homogeneous linear partial differential equation, then the linear combination u = c1u1 + c2u2 + … + ckuk where the ci= 1, 2, …, k are constants, is also a solution. 3D view of 2D Laplace equation

  12. Laplace equation in Three Dimensions Partial differential equation. To determine the solution you must fix V on the boundary, which is a surface, – boundary condition. • V has no local minima or maxima inside the boundary.

  13. Applications of Laplace Equation

  14. Applications of Laplace Equation Vector Calculus • gravitation, magnetism, • heat transportation, • soap bubbles (surface tension) … • fluid dynamics

  15. References Recommended Books 1. David J. Griffiths, Introduction to Electrodynamics, 4th Edition Addison-Wesley (2012) 2. J.R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 4th Edition, Pearson (2009) 3. M.N.O Sadiku, Elements of Electromagnetics, Oxford University Press (2009) Video Lectures https://youtu.be/Z9NjA_f2VZw (Laplace Equation) https://youtu.be/I-lKnLnnbY4(Laplace Equation) https://youtu.be/tEx44PrpQn8 (Laplace Equation in 1 Dimension) https://youtu.be/DNo0FmsGZKg (Laplace Equation in 2 Dimension) https://youtu.be/n1hmYagxdS8 (Applications of Laplace Equation)

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