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VortexLattice
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Hey guys! So I'm reading Jackson's Classical Electrodynamics, and I'm trying to get a sense of what Green Functions really are. The book says that "the potential due to a unit source and its image(s), chosen to satisfy homogeneous boundary conditions, is just the Green function appropriate for Dirichlet or Neumann boundary conditions. In G(x,x'), the variable x' refers to the location of the unit source, while the variable x is the point at which the potential is being evaluated." (Here, x and x' are vectors.)
So I went through the example they gave, for a point charge in the presence of a grounded conducting sphere. They find the potential by using the method of images, which basically turns the problem into the potential due to two point charges. Then they do some (unexplained and mysterious) substitutions like "q->4πε" from the potential we found, and that gives us the Green function.
So I guess I kind of understood that. However, in the discussion in the previous chapter, they have the potential in terms of an appropriate Green function G(x,x'), where like before x is the point at which we're evaluating the potential, but this time x' is an integration variable. This kinda confuses me in light of what was said above... The part I quoted seems to say that G(x,x') = [itex]\Phi(x)[/itex] with some substitutions.
But in the previous section, they say [itex] \Phi(x) = \frac{1}{4\pi \epsilon}\int _V \rho(x') G_D (x,x') d^3x' - \frac{1}{4\pi} \oint _S \Phi(x') \frac{\delta G_D}{\delta n'} da'[/itex]...so now the Green function isn't just equal to the potential. So I'm a little confused as to what it actually is.
I tried to do a problem from the book to understand this, problem 2.7. It says "Consider a potential problem in the half-space defined by z≥0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). Write down the appropriate Green function G(x,x')."
I didn't really know what they meant here... I know Dirichlet boundary conditions means that the voltage of the plane, rather than the derivative of the voltage at the plane (the charge density). But they don't actually say what the boundary conditions are, just what type they are. I solved it for the plane being a grounded conductor in the presence of a point charge (like their example above), but I don't think this is what they meant.
Can anyone help me? Thanks!
So I went through the example they gave, for a point charge in the presence of a grounded conducting sphere. They find the potential by using the method of images, which basically turns the problem into the potential due to two point charges. Then they do some (unexplained and mysterious) substitutions like "q->4πε" from the potential we found, and that gives us the Green function.
So I guess I kind of understood that. However, in the discussion in the previous chapter, they have the potential in terms of an appropriate Green function G(x,x'), where like before x is the point at which we're evaluating the potential, but this time x' is an integration variable. This kinda confuses me in light of what was said above... The part I quoted seems to say that G(x,x') = [itex]\Phi(x)[/itex] with some substitutions.
But in the previous section, they say [itex] \Phi(x) = \frac{1}{4\pi \epsilon}\int _V \rho(x') G_D (x,x') d^3x' - \frac{1}{4\pi} \oint _S \Phi(x') \frac{\delta G_D}{\delta n'} da'[/itex]...so now the Green function isn't just equal to the potential. So I'm a little confused as to what it actually is.
I tried to do a problem from the book to understand this, problem 2.7. It says "Consider a potential problem in the half-space defined by z≥0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). Write down the appropriate Green function G(x,x')."
I didn't really know what they meant here... I know Dirichlet boundary conditions means that the voltage of the plane, rather than the derivative of the voltage at the plane (the charge density). But they don't actually say what the boundary conditions are, just what type they are. I solved it for the plane being a grounded conductor in the presence of a point charge (like their example above), but I don't think this is what they meant.
Can anyone help me? Thanks!