Final answer:
To compute the flux of E outward through the cylinder x² + y² = r², for 0 ≤ z ≤ h, we can use Gauss' law and a surface integral. The flux can be computed by integrating the dot product of the electric field E and the differential area element dA over the surface of the cylinder.
Step-by-step explanation:
To compute the flux of E outward through the cylinder x² + y² = r², for 0 ≤ z ≤ h, we can use Gauss' law. Gauss' law states that the flux of an electric field through a closed surface is proportional to the charge enclosed by that surface.
In this case, we have a cylindrical surface defined by x² + y² = r², and we want to compute the flux of E outward through this surface. We can do this by integrating the dot product of E and dA over the surface of the cylinder.
More specifically, the flux is given by the surface integral of E⋅dA over the curved surface of the cylinder and the top and bottom surfaces:
Flux = ∫∫E⋅dA = ∫∫E⋅n dA
where E is the electric field, dA is a differential area element, and n is the outward normal vector to the surface element dA.
This integral can be simplified using the cylindrical coordinate system, where dA = rdrdθ and E = E(r,θ,z). The flux can then be computed as:
Flux = ∫∫E(r,θ,z)⋅(r,θ,z) rdrdθ dz