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**1. Homework Statement**

The problem was to calculate the electric field inside an infinite length cylinder (with radius R) with a non uniform charge density. The charge density depended on r. Its easy enough to solve using a gaussian cylinder with r less than R. But what if I wanted to complicate things and use a gaussian sphere inside the cylinder with r < R?

**2. Homework Equations**

∫E[itex]\bullet[/itex]dA = q / ε° Gauss' Law

ρ = ρ°(1 - r/R) This is charge density distribution

q = ρ[itex]\bullet[/itex]dV

V= 4/3 π r^3

A= 4 π r^2

**3. The Attempt at a Solution**

Since the electric field is not the same everywhere, it can't be removed from the first integral. It is however constant over dθ when r and d[itex]\Phi[/itex] are held constant.

q =∫ ρ*dV = ∫ ρ°(1- r/R) * A* dr

q = ∫ρ°(1-r/R) * 4 π r^2 * dr

q is easy enough to solve. So the problem lies in ∫E*dA

I had a few ideas about this, First one: convert the problem into spherical coordinates and solve it that way. Second one: The electric field is a vector quantity, so could I say the total electric field at a point r is equal to the sum of the partial derivatives of the electric field at that point? In that case, I would need to find an expression for dE/d[itex]\Phi[/itex]. Any guidance on this would be appreciated. Thanks!

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**