[Physics] How to apply Gauss’s law to coaxial conducting cylinders

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$$
\vert \vec E\vert =\frac{\lambda}{2\pi \varepsilon_0 r}
$$
So I know this is the magnitude of the electric field of a line of charge using a cylindrical Gaussian surface. But, now let's say I have two coaxial metal conducting cylinders, one with the inner cylinder radius $a$ that is positively charged, and the outer cylinder with a larger radius $b$ which we can just say is negatively charged. My goal is to find the Electric field at a distance $a > r > b$.

Now I know that charge density $\lambda$ = (Charge)/(Length of wire) in the line of charge formula, which is how the length cancels out from the denominator. But when I introduce the inner cylinder with radius $a$, how do I adjust this formula to account for the fact that I now have an inner conducting cylinder with some radius? Does the length portion of the charge density have to change to something that represents the area of the inner cylinder?

Part of me thinks the formula might be the same because, if the electric field just shoots radially outward from the inner cylinder then the radius doesn't matter. Is that the case? Or do I need to account for the size of this inner cylinder?

Best Answer

I think the figure below shows something like the geometry you have in mind: this is a cross-sectional view of an infinitely long cylinder, with inner solid cylinder of radius $a$ coaxial with a hollow cylinder of inner radius $b$.

enter image description here

The key point to observe is that a Gaussian cylinder of radius $a<r<b$ will only enclose the charge of the inner solid cylinder. Hence, a long as $a<r<b$, the $\vec E$ will be that of the inner cylinder alone. When $r$ goes beyond $b$ and encloses some or all of the charge of the outer hollow cylinder, the geometry will not change but the net charge enclosed will be reduced so the field will be reduced accordingly. If the outer hollow cylinder has the same charge per unit length as the solid inner one, then the net charge enclosed for $r>c$ will be $0$ and the field will thus be $0$ outside the arrangement.

[Figure credit: modified from Young and Freedman's University Physics]

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