Grounded conducting cylinder using Laplace

[meteorologist1]

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.

 

[dextercioby]

What did u do exactly,did u separate variables and got ODE-s for all three variables...?

Daniel.

 

[meteorologist1]

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?

 

[dextercioby]

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.

 

[meteorologist1]

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.

 

[dextercioby]

Okay,then,separate varaibles and integrate each equation.Though i think you may need another condition.You must fix 2 integration constants,after all...

Daniel.