Minimum Surface Charge Density for Spark Creation

[m84uily]

Homework Statement

A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0x10^6 N/C.
What is the minimum surface charge density for producing a spark?

Homework Equations

int(EdA) = (Qin) / e0

The Attempt at a Solution

I imagine the surface as a cylinder with the electric field going parallel to it.

int(EA) = backface + side + frontface = EA + EA = 2EA
E = 3.0x10^6 N/C
A = (pi)r^2

(2)(3.0x10^6 N/C)(pi)r^2 = (Qin) / e0

(e0)(2)(3.0x10^6 N/C)(pi)r^2 = Qin

Then, I don't want r is! I've probably gone about solving this all wrong, thanks in advance for your help.

 

[ehild]

You know that free charges can not exist inside a metal so what do you mean on Qin?
The surface charge density is not uniform on the surface of a needle, and the tip of a needle is not flat.
You also might remember that Q/ε₀ electric field lines originate from a unit positive charge, and the electric field strength is the number of field lines traversing a normal unit surface.

So the electric field strength given is related directly to the surface charge density on the metal.

ehild

 

[m84uily]

You definitely cleared up a lot for me! Thanks!

So:

Q/ε0 = E

Q = E(ε0)

 

[ehild]

It is not charge (Q) but surface charge density (σ). But the result is OK.

ehild