Grounded conducting cylinder using Laplace

AI Thread Summary
The discussion focuses on solving Laplace's equation in cylindrical coordinates for a grounded conducting cylinder in a uniform external electric field. The user seeks guidance on finding the potential V(r) and the induced surface charge, while mentioning the need to apply separation of variables. It is noted that the z term can be ignored due to the problem's independence from that variable. Boundary conditions are emphasized, particularly that V(a) = 0 for the grounded cylinder. The conversation concludes with the importance of fixing integration constants to obtain a valid solution.
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Hi, I'm having trouble applying Laplace's equation solution in cylindrical coordinates to the problem of the grounded conducting cylinder of radius a in a uniform external field. The cylinder axis is the z axis, and the external electric field is E0 in the x direction. I need to find the potential V(r) and the induced surface charge. Thanks.
 
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What did u do exactly,did u separate variables and got ODE-s for all three variables...?

Daniel.
 
My professor in class said to use separation of variables.

\Phi = \Phi(\rho, \phi, z)
\nabla^2\Phi = \frac{1}{\rho} \frac{\partial}{\partial\rho} (\rho\frac{d\Phi}{d\rho}) + \frac{1}{\rho^2}\frac{\partial^2\Phi}{\partial\phi^2} + \frac{\partial^2\Phi}{\partial z^2}
which is the Laplace's Equation in cylindrical coordinates. And I think he said that we can ignore the z term because this case is z independent.

Then I'm not sure how to obtain the solutions to this equation. And after I get the solutions, how do I apply it to this problem?
 
Well,u have to come up with so-called limit conditions.The general solution will not be good for anything,if u can't use the limit conditions...

Daniel.
 
I think we call them boundary conditions. Well in this case, since the cylindrical conductor is grounded, the limit condition must be that V(a) = 0.
 
Okay,then,separate varaibles and integrate each equation.Though i think you may need another condition.You must fix 2 integraton constants,after all...

Daniel.
 
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