Solving Electric Configuration Energy and Capacitance of Infinite Cylinders

[Adamecius]

Homework Statement

Hi I am kinda new here, and my english its not the best so ill try to explain my problem in short words.

There is 2 problems which i already solve, but my question is more terorical that practical, these are the two problems:

a) Calculate the electric configuration energy per length of an infinite cylinder of charge density ρ. It turns out that when i use the equation that gives the configuration energy in terms of the potencial energy the problem have a solution , but when i use the equation that give the configuration energy in terms of the electric field the solution its infinite, and i cannot understand why.

b)The second problem is that, when i calculate the capacitance of 2 infinite cylinders that are very very far from each other, of charge and Radius Q1,R1 and Q2,R2 and both with length "l" using capacitance coefficients it turn out that the problem have a coherent solution, which is a capacitance that only depends of the geometry of the system. Hoever when i calculate the capacitance by definition the problem have a solution that depends of the charge which its imposible.

I will be really thanksfull if yopu could help with this, and i apreciate all the help:)

Homework Equations

a)
Electric configuration energy in terms of electric field and potencial:
Uconf= (εo/2)∫(E^2)*dv Over the whole space

Uconf= ∫ρφ*dv over the volume of the object

b)Capacitance:

C= Qtrans/(φ2-φ1) Capacitance by definition

C= C11*C22-(C12^2)/(C11+C22+2C11C22) Capacitance using capacitance coheficients

The Attempt at a Solution

The solution i give to both probles is that i believe that when we are making the aproximation of saying that the cylinders are infinite, for those equation where I am getting an infinite or not coherent solution that aproximation can't be done, probly becuase there are not such thing as infinite cylinders, and that kind of system its not consistent with the Unicity Theorem, and becuase of that we could have multyples solution to the same problem


Thx again and i apreciate your help see you all and i hope to help someon next time

 

[siddharth]

Electric configuration energy in terms of electric field and potencial:
Uconf= (εo/2)∫(E^2)*dv Over the whole space

This is right. Since you said that you're having problem with the integration, could you post your work? It'll be easier to help that way

C= Qtrans/(φ2-φ1) Capacitance by definition

Can you post your actual attempt at the solution?