To find charge and capactitance of a finite conductors

[batman2020]

Homework Statement

A single infinitely conductor with a diameter of 10mm and a height of 3m above ground is charged to a voltage of 20kV above earth
a. Find the charge/meter on the conductor.
b. Find the capacitance/meter on the conductor.

Homework Equations

I don't know if these are right but to find the capacitance I used C=4(pi)(epsilon)(r)
To find the charge per meter I used charge/meter=(pi*epsilon)/ln(d/a)
where d=height and a=radius.

The Attempt at a Solution

a. Charge/meter=4.876x10-12
b. C=5.563x10-13

 

[KataKoniK]

Hello, thank you for your post. Your equations and attempts at a solution are on the right track, but there are a few things that need to be clarified. Firstly, your equations for capacitance and charge per meter are not quite correct for this situation. For a single infinitely long conductor, the capacitance per meter is actually given by C=2(pi)(epsilon)/ln(d/a), where d is the distance between the conductor and the ground and a is the radius of the conductor. This is because the electric field lines are distributed evenly around the conductor, so the capacitance is half of what it would be for a cylindrical capacitor with the same dimensions.

Additionally, for the charge per meter, you would need to use the total charge on the conductor (20kV) divided by the total length of the conductor (3m), rather than using the radius and height separately. This would give a charge per meter of 222.22 pC/m.

As for your values for the capacitance and charge per meter, they are both a bit off. The correct capacitance per meter is approximately 1.58x10^-11 F/m and the correct charge per meter is approximately 6.67x10^-9 C/m.

I hope this helps clarify your solution. Let me know if you have any further questions.