Poissons Equation: Exploring f(x,y)

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In summary, Poisson's equation (uxx + uyy = f(x,y)) and Laplace's equation (uxx + uyy = 0) are both part of Maxwell's theory of electromagnetism, where u is the electric field and f(x,y) represents the density of charge and the rate of change of the magnetic field at a specific point in space-time. In the context of heat distribution on a metal plate, f(x,y) could represent an external heat source applied to each point on the plate.
  • #1
Nusc
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Homework Statement


So poissons equation takes the for uxx + uyy = f(x,y)
Laplace is where f(x,y). What does the f(x,y) physically represent?


Homework Equations





The Attempt at a Solution

 
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  • #2
Laplace equation is when f(x,y)=0. f(x,y) can represent many things physically. the solution of this problem can represent many things for example u could be a steady state temperature of the cross section of a rod with an electrical current.
 
  • #3
What you wrote does not make sense to me, but the question got throught nonetheless.

In Maxwell's theory of electromagnetism, the electromagnetic field is governed by a set of 4 equations and one of them is Poisson's equation where u is the electric field in space-time (x,y,z,t) and f(x,y,z,t) is an expression taking into account the density of charge and the rate of change of the magnetic field at the point (x,y,z,t) in space-time.
 
  • #4
buzzmath said:
Laplace equation is when f(x,y)=0. f(x,y) can represent many things physically. the solution of this problem can represent many things for example u could be a steady state temperature of the cross section of a rod with an electrical current.

But what is f actually doing to this cross section?
 
  • #5
quasar987 said:
What you wrote does not make sense to me, but the question got throught nonetheless.

In Maxwell's theory of electromagnetism, the electromagnetic field is governed by a set of 4 equations and one of them is Poisson's equation where u is the electric field in space-time (x,y,z,t) and f(x,y,z,t) is an expression taking into account the density of charge and the rate of change of the magnetic field at the point (x,y,z,t) in space-time.

And when f = 0 ? What does it mean in this case?
 
  • #6
Well it means that this particular Maxwell's equation ([tex]\nabla^2\vec{E}=0[/tex]) is describing the evolution of the electric field in a region where there are no electric charges and where the magnetic field is constant.
 
  • #7
So say one is concerned with the heat distribution among a metal plate, what would f mean and what would f = 0 mean?
 
  • #8
Maybe I should have written this in the undergraduate physics forum.
 
  • #9
You can ask a mentor to move it.
 
  • #10
I hate to keep saying this but mathematics is not physics! Quantities in a mathematical equation do NOT have any "physical" meaning and do not "physically represent" anything until you apply them to a specific physics problem.

(I guess I am like an ex-smoker. I started majoring in physics, then switched my major to mathematics, though staying in applied math (My doctoral disertation was on computing Clebsch-Gordon Coefficients in general SU(n)) but have steadily moved to more abstract mathematics since.)

That said, if you have [itex]\nabla \phi= \kappa \partial \phi/\partial t+ f(x,y,t)[/itex] specifically applied to heat distribution on a plate, then f(x,y,z) might represent an external heat source applied to every point of the plate.
 

1. What is the Poisson's equation?

The Poisson's equation is a partial differential equation that describes how a potential field is influenced by a given source distribution.

2. What are the applications of Poisson's equation?

Poisson's equation has a wide range of applications in physics, engineering, and mathematics. It is commonly used in electrostatics, fluid dynamics, heat transfer, and quantum mechanics.

3. How is Poisson's equation solved?

Poisson's equation is typically solved using numerical methods, such as finite difference, finite element, or boundary element methods. Analytical solutions are also possible for simple configurations.

4. What is the role of f(x,y) in Poisson's equation?

f(x,y) is the source term in Poisson's equation, representing the distribution of sources or sinks in the potential field. It can be a function of position, time, or other variables depending on the specific problem.

5. What is the significance of exploring f(x,y) in Poisson's equation?

Exploring f(x,y) can provide insights into the behavior of the potential field and its relationship with the source distribution. It can also help in identifying patterns or trends in the solution and determining the boundary conditions for a given problem.

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