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**[Solved] Gauss's Law Question**

Well, I've been reading up on Resnick, and have reached the chapter on Gauss' Law.

He provides an example where he uses Gauss's Law to calculate the E field just outside a positively charged conductor (Arbitrary shape):

His reasoning was as follows:

We draw a Gaussian surface in the shape of a small cylinder whose end faces are parallel to the surface of the conductor. Part of the cylinder is just outside the conductor and part is inside. The field is normal to the conductor's surface from the condition of electrostatic equilibrium. (If E had a component parallel to the conductor's surface, the free charges would move along the surface; in such a case, the conductor would not be in equilibrium.) Thus, there is no flux through the curved part of the surface because E is parallel to the surface.

There is no flux through the flat face of the cylinder inside the conductor because here E=0 (There is no net E field inside a conductor)

Hence, the net flux through the Gaussian surface is that only through the flat face outside the conductor, where the field is perpendicular to the Gaussian surface.

Looking at this face, we see that the flux is EA, where E is the electric field just outside the conductor and A is the area of the cylinder's face. Applying Gauss's Law to this surface we obtain:

[tex]\Phi = EA =\frac{q_{in}}{\epsilon_0}=\frac{\sigma A}{\epsilon_0}[/tex]

Solving for the field, we obtain:

[tex]E=\frac{\sigma}{\epsilon_0}[/tex]

(Excerpt from "Fundamentals of Physics")

Now my question is where exactly we used the fact that we are talking about a surface that's very close to the conductor?

What keeps me from using a very long cylinder and saying the the contribution of the small surface element [tex]dA[/tex] I was looking at is perpendicular to the surface, and independent of the distance from it?

With thanks in advance, Anatoli.

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