Question

Consider a long, cylindrical charge distribution of radius R with a uniform charge density ?. Find the electric field at distance r from the axis, where r < R. (Use any variable or symbol stated above along with the following as necessary: ?₀.)

Answer 1

The charge inside the cylinder is equal to the charge distribution times its volume ?V = q_inside. The Electric Flux through any closed Gaussian surface is equal to the integral of EdA = EA = q_inside / ?0.
Where the area is the area of the surface of the cylinder with radius r, perpendicular to the axis = 2*pi*r*l (l = length of cylinder). and the volume is that of the cylinder of radius r = pi*(r^2)*l. Substituting and doing the algebra to isolate E gives you E = (?*pi*(r^2)*l)/(?0*2*pi*r*l). You can do the simplifications and crossings out :)