What is the energy from a hollow cylinder of surface charge density sigma, radius R and charge q?

Energy? DO you mean E, the electric field intensity?

If you mean Energy, it is an odd question, but the answer is that it is the work it took to put the charge q on the surface. In this case, a complex integration of dq.

The question is that there are two cylinders, the inner one with a volume density charge rho and the outer (hollow) one with some surface density charge.

The net charge is zero, and I need to find out what the surface density charge is equal to and the energy per unit lenght of the system. I already calculated what the field is inside the system, but I don't know how to solve for the contribution by the outer shell. I figured I'd need the energy anyway at the end to find the energy per unit lenght right?

The field from the outer is only on the outer side of the cylinder. If the charge is the same, but opposite, then (Gauss Law) the charge enclosed is zero, so field is zero. THe energy is from the field between the two cylinders...Integrate the E squared (energy density) function over the volume between the cylinders.

And the E value I would use is q/(2pi(E0)R) correct?

No. In the numerator, you should have lambda, charge per unit length. You can get it from the given geometry, R, q, and surface charge density.

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