A long, thin straight wire with linear charge density -λ runs down the center of a thin, hollow metal cylinder of radius R. The cylinder has a net linear charge density 2λ. Assume λ is positive. Find expressions for the electric field at the following locations. Use lambda for λ, pi for π, epsilon_0 for ε0, r for r, and R for R.

(b) outside the cylinder, r > R

To find the expression for the electric field outside the cylinder (r > R), we can use Gauss's Law. Gauss's Law allows us to determine the electric field by calculating the total electric flux through a closed surface.

1. Choose a Gaussian surface: In this case, we'll take a cylindrical surface with radius r and length L outside the cylinder.

2. Determine the total charge enclosed: The total charge enclosed by the Gaussian surface is the charge on the thin wire inside it. Since the linear charge density of the wire is -λ and the length of the wire inside the Gaussian surface is L, the total charge enclosed is Q = -λL.

3. Calculate the surface area of the Gaussian surface: The surface area of the curved part of the Gaussian surface is 2πrL, and the area of the flat circular ends is πr^2 (assuming L is sufficiently large). So, the total surface area of the Gaussian surface is A = 2πrL + πr^2.

4. Apply Gauss's Law: Gauss's Law states that the total electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of free space (ε0). The electric field can be obtained by dividing the flux by the surface area:

Flux = Q / ε0
Electric field (E) = Flux / A

Substituting the values, we get:

Electric field (E) = (Q / ε0) / A
= [-λL / ε0] / [2πrL + πr^2]

Simplifying further, we have:

E = -λ / [2πε0(r + R)]

Therefore, the expression for the electric field outside the cylinder (r > R) is E = -λ / [2πε0(r + R)].