Capacitance between two crossed wires

[free_electron]

Homework Statement

I have two wires (length L) which are separated by a distance d. They are also oriented perpendicularly (but I am curious about the orientation impact, part of the question). By simplification, I assume them to be cylinders of radius R (in actuality they are likely strips). I expect R ~ d <<L. I would like to find the capacitance of such a system, where the two wires act as the electrodes.

Homework Equations

Gauss law: E=Q/(e0*2*pi*r*L) (r>R) (Field around a wire, cylindrical geometry)

=> V = integral of E vs distance between wires (R->d): V=Q/(e0*2*pi*L) ln (d/R)

The Attempt at a Solution

C=Q/V = (e0*2*pi*L)/ln(d/R)

The question has several little parts:

a) assuming the cylindrical geometry is the above calculation correct?

b) going to a realistic rectangular strip geometry, is there an analytical solution or can it only be numerically computed on a workstation? Can this solution be found at some public site?

c) How does the orientation matter?

 

[mark726]

Hello,

Thank you for your question. It seems like you have a good understanding of the problem and have approached it correctly using Gauss' law. To answer your questions:

a) Yes, your calculation for the capacitance of the cylindrical wires is correct. However, keep in mind that this is an approximation since the wires are actually strips and not perfect cylinders.

b) For a more realistic rectangular strip geometry, there is not an analytical solution and it would need to be numerically computed. This can be done on a workstation using software such as MATLAB or Mathematica. There may be some public sites or resources that have pre-made programs or simulations for this type of problem, so it would be worth doing some research to see what is available.

c) The orientation of the wires can impact the capacitance of the system. If the wires are parallel to each other, the capacitance will be higher compared to when they are perpendicular. This is because the electric field lines are able to pass through the wires more easily when they are parallel, resulting in a higher capacitance.

I hope this helps answer your questions. Keep up the good work with your calculations and problem-solving skills. Let me know if you have any further questions or need any clarification. Good luck with your studies!