Electrostatics & Quadric Surfaces

[bolbteppa]

Hey guys, I'm in some serious trouble - I don't know how to solve a lot of problems in electrostatics. The main reason is that once I seen that you had to be able to find electric fields & electric potentials due to charge distributions on quadric surfaces I panicked & gave up - I was still having trouble with finding the center of mass & moment of inertia of every quadric surface, every geometry from every location of the surface (just to make it hard on myself!) but I just felt there had to be a better way of dealing with these things than geometric considerations, ever prone to mistakes, but didn't know what to do, so I went off & learned a ton of maths...

I had a similar problem with multiple integrals, once I seen the derivation of the area element in polar coordinates or volume elements etc... I just instinctively rejected the idea of polar integration, & would have seriously suffered had I not come to terms with mindless jacobian transformations & the curvilinear differential as a means to derive those things on the spot.

During the year I finally learned how to use the inertia tensor & found that I could deal with every moment of inertia problem trivially compared to what I thought I'd have to do, exactly analogous to the way you can find area & volume elements for any change of variables with a jacobian & no case-specific geometric contortions, just plugging in formulas & being a bit careful about limits of integration.

I'm wondering whether Laplace & Poisson equations can be used to derive all those formulas in an elementary calculus-based physics book? I'm talking Halliday-Resnick level, I would much prefer to be able to derive every formula myself without worrying about geometry too much (beyond setting up boundary conditions in a pde) & then using all those elementary geometric derivations as little exercises as opposed to my main source!

What am I to do? I have a feeling that the derivations of moments of inertia in elementary physics books are done to apparently make it simpler on students who can't deal with tensors & multiple integrals, but for me it only made things immensely harder. I'm hoping Laplace & Poisson's equations will function as the electrostatics version of the inertia tensor, in other words I'm hoping a bit of advanced mathematics will make my life simpler - am I right, is there a source that would go through the common geometries using Laplace & Poisson equations to derive what's in those elementary physics books, or do I really have to go though (by my count) 37 different cases (including intersections of surfaces etc...) using error-prone geometric arguments just to feel I can deal with the electrostatic charge distributions & the electrostatic potential?

Also, could you do all this with ease using differential forms?

Thanks!

 

[fluidistic]

Hey guys, I'm in some serious trouble - I don't know how to solve a lot of problems in electrostatics. The main reason is that once I seen that you had to be able to find electric fields & electric potentials due to charge distributions on quadric surfaces I panicked & gave up - I was still having trouble with finding the center of mass & moment of inertia of every quadric surface, every geometry from every location of the surface (just to make it hard on myself!) but I just felt there had to be a better way of dealing with these things than geometric considerations, ever prone to mistakes, but didn't know what to do, so I went off & learned a ton of maths...

I had a similar problem with multiple integrals, once I seen the derivation of the area element in polar coordinates or volume elements etc... I just instinctively rejected the idea of polar integration, & would have seriously suffered had I not come to terms with mindless jacobian transformations & the curvilinear differential as a means to derive those things on the spot.

During the year I finally learned how to use the inertia tensor & found that I could deal with every moment of inertia problem trivially compared to what I thought I'd have to do, exactly analogous to the way you can find area & volume elements for any change of variables with a jacobian & no case-specific geometric contortions, just plugging in formulas & being a bit careful about limits of integration.

I'm wondering whether Laplace & Poisson equations can be used to derive all those formulas in an elementary calculus-based physics book? I'm talking Halliday-Resnick level, I would much prefer to be able to derive every formula myself without worrying about geometry too much (beyond setting up boundary conditions in a pde) & then using all those elementary geometric derivations as little exercises as opposed to my main source!

What am I to do? I have a feeling that the derivations of moments of inertia in elementary physics books are done to apparently make it simpler on students who can't deal with tensors & multiple integrals, but for me it only made things immensely harder. I'm hoping Laplace & Poisson's equations will function as the electrostatics version of the inertia tensor, in other words I'm hoping a bit of advanced mathematics will make my life simpler - am I right, is there a source that would go through the common geometries using Laplace & Poisson equations to derive what's in those elementary physics books, or do I really have to go though (by my count) 37 different cases (including intersections of surfaces etc...) using error-prone geometric arguments just to feel I can deal with the electrostatic charge distributions & the electrostatic potential?

Also, could you do all this with ease using differential forms?

Thanks!

As far as I know Laplace and Poisson's equations are the equations that the electric potential satisfy in electrostatics. You solve either Laplace or Poisson equation to get the potential and then you can get the electric field via the relation ##\vec E =- \vec {\nabla } \Phi##. So the answer to your question whether from those 2 equations you can derive all the formula in Resnick and Halliday is no.
So saying that you're stuck at finding the potential and E field basically means you are stuck at solving Laplace and/or Poisson's equations.
On the other hand I've heard that you can derive all formula from Maxwell's equations+ Lorentz force's equation.