Asolid insulating cylinder of radius r has a positive uniform volume charge density rho. the cylinder’s axis is along the z-direction, and it can be considered infinitely long in this direction. a) determine the electric field at a point outside the cylinder r > r, where r is the distance from the axis of the cylinder. give both the magnitude and direction of the electric field.

E = ρ r / 2ε₀   r>R

Explanation:

To find the field we can use Gauss's law that says that the flow is equal to the charge of the surface divided by electric permittivity (eo)

       Φ = ∫ E .dA = / ε₀

To use this law we must define a Gaussian surface that takes advantage of the symmetry of the problem, we use a cylindrical surface parallel to the charge cylinder

The charge density for r> R, the entire charge is internal

      ρ = / V

      V = (π r²) L

      = ρ (π r² L)

Field lines are protruding to the cylinder and therefore the scalar product is reduced to the ordinary product

      ∫ E dA = / ε₀

      E (2π r L) = ρ (π r² L) / ε₀

     E = ρ r / 2ε₀           r>R

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