For a very long uniformly charged cylinder with a radius R and a linear charge density λ, the volume charge density, ρ, for the length of the cylinder would be expressed as:

$\overline{){\mathbf{\rho}}{\mathbf{=}}\frac{\mathbf{q}}{\mathbf{\pi}{\mathbf{R}}^{\mathbf{2}}\mathbf{L}}}$, where q is the charge on length, L.

The linear charge density, λ:

$\overline{)\begin{array}{rcl}{\mathbf{\lambda}}& {\mathbf{=}}& \frac{\mathbf{q}}{\mathbf{L}}\\ {\mathbf{\rho}}& {\mathbf{=}}& \frac{\mathbf{\lambda}}{\mathbf{\pi}{\mathbf{R}}^{\mathbf{2}}}\end{array}}$

Electric flux:

$\overline{){{\mathbf{\varphi}}}_{{\mathbf{E}}}{\mathbf{=}}{\mathbf{\int}}{\mathbf{E}}{\mathbf{.}}{\mathbf{d}}{\mathbf{A}}}$

Remember A•B = ABcosθ

**a.**

Electric flux from the Gaussian surface is:

$\begin{array}{rcl}{\mathbf{\varphi}}_{\mathbf{E}}& \mathbf{=}& {\mathbf{\int}}_{\mathbf{c}\mathbf{u}\mathbf{r}\mathbf{v}\mathbf{e}\mathbf{d}}\mathbf{E}\mathbf{d}\mathbf{A}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{0}\mathbf{\xb0}\mathbf{)}\mathbf{+}\mathbf{2}{\mathbf{\int}}_{\mathbf{p}\mathbf{l}\mathbf{a}\mathbf{n}\mathbf{e}}\mathbf{E}\mathbf{d}\mathbf{A}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{90}\mathbf{\xb0}\mathbf{)}\\ & \mathbf{=}& \mathbf{E}\mathbf{\left(}\mathbf{2}\mathbf{\pi}\mathbf{r}\mathbf{l}\mathbf{\right)}\mathbf{+}\mathbf{0}\end{array}$

A very long, uniformly charged cylinder has radius R and linear charge density λ.

a. Find the cylinder's electric field strength outside the cylinder, r ≥ R. Give your answer as a multiple of λ/ε0.

Express your answer in terms of some or all of the variables R, r, and the constant π.

b. Find the cylinder's electric field strength inside the cylinder, r ≤ R. Give your answer as a multiple of λ/ε0.

Express your answer in terms of some or all of the variables R, r, and the constant π.

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