How Do You Find the Electric Field for an Infinitely Long Charged Cylinder?

[Lancelot59]

I need to find the electric field at all values of radius for an infinitely long cylinder of charge. It's in insulator of radius R, and has a volume charge density
$$\rho = \rho_{0}(\frac{r}{R})$$ while r<R.

I need to find the electric field at all points first off. I'm not entirely sure how to go about doing this. I have gauss's law, but I'm not sure how to go about applying it in this case.

 

[DukeLuke]

I would start with the integral form of Gauss's law.
$$\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}$$

In this case your $Q_{enc}$ will be a function of r, and can be found using the volume integral in cylindrical coordinates

$$Q_{enc}= \int \rho d\tau = \int \rho r dr d\theta dz$$

and integrating the radius from 0 to r (since you want an expression that will work for any r). I would definitely use cylindrical coordinates and just integrate in the z direction from 0 to some length l (the z length will cancel out later).

For the left hand side of Gauss's law you want to use a cylindrical Gaussian surface so that the electric field is always perpendicular to the surface. In this case $$\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = |E|A \hat{r}$$ where A is the area of the of the cylinder (again just use the variable l for the length in this area). After doing the integral to find $Q_{enc}(r)$ you should be able to equate the two sides and solve for E(r) for r<R. When r>R you need to think about what $Q_{enc}$ will be, but I think you have plenty here to start the problem.