- #1
peter46464
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I'm struggling here so please excuse if I'm writing nonsense. I'm trying to understand how, for a gravitational field, Laplace's equation (I think that's the right name) equals zero in empty space.
I understand that the gravitational potential field, a scalar field, is given by \phi=\frac{-Gm}{r} where \phi is the gravitational potential energy of a unit mass in a gravitational field g. The gradient of this is (a vector field) g=-\nabla\phi=-\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right) And the divergence of this vector field is \nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho and is called Poisson's equation. If the point is outside of the mass, then \rho=0 and Poisson's equation becomes\nabla\cdot\nabla\phi=0 (Laplace's equation). My question is, how do I express \phi=\frac{-Gm}{r} as a function of x,y,z so I can then end up with \nabla\cdot\nabla\phi=0 in empty space? I would have thought that I could write \phi=\frac{-Gm}{r}=\frac{-Gm}{\sqrt{x^{2}+y^{2}+z^{2}}} but when I try to calculate \nabla\cdot\nabla\phi from this, I don't get zero. I do this by assuming (in the simplest case) that both y and z are zero and then taking second derivative of \phi=\frac{-Gm}{r}which should be zero (shouldn't it?) but isn't zero. What am I doing wrong? As simple as possible please.
Thank you
I understand that the gravitational potential field, a scalar field, is given by \phi=\frac{-Gm}{r} where \phi is the gravitational potential energy of a unit mass in a gravitational field g. The gradient of this is (a vector field) g=-\nabla\phi=-\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right) And the divergence of this vector field is \nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho and is called Poisson's equation. If the point is outside of the mass, then \rho=0 and Poisson's equation becomes\nabla\cdot\nabla\phi=0 (Laplace's equation). My question is, how do I express \phi=\frac{-Gm}{r} as a function of x,y,z so I can then end up with \nabla\cdot\nabla\phi=0 in empty space? I would have thought that I could write \phi=\frac{-Gm}{r}=\frac{-Gm}{\sqrt{x^{2}+y^{2}+z^{2}}} but when I try to calculate \nabla\cdot\nabla\phi from this, I don't get zero. I do this by assuming (in the simplest case) that both y and z are zero and then taking second derivative of \phi=\frac{-Gm}{r}which should be zero (shouldn't it?) but isn't zero. What am I doing wrong? As simple as possible please.
Thank you