- #1

Fronzbot

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## Homework Statement

A spherical distribution of charge [tex]\rho = \rho_{0}[1-(R^{2}/b^{2}][/tex] exists in the region [tex]0\leqR\leqb[/tex]. This charge distribution is concentrically surrounded by a conducting shell with inner radius [tex]R_{i} (>b)[/tex] and outer radius [tex]R_{o}[/tex]. Determine

**E**everywhere.

## Homework Equations

[tex]

\int E \cdot ds = Q_{enc} / \epsilon_{0}

[/tex]

## The Attempt at a Solution

I have the right answers (from the back of the book) but cannot figure out how to get there. Before I write my solution down, here are the correct answers:

[tex]

For 0 \leq R \leq b:[/tex]

[tex]

E_{R1} = \frac{\rho_{0}R}{\epsilon_{0}}(\frac{1}{3}-\frac{R^{2}}{5b^{2}})

[/tex]

[tex]

For b \leq R < R_{i}:

[/tex]

[tex]

E_{R2} = \frac{2 \rho_{0} b^{3}}{15 \epsilon_{0} R^{2}}

[/tex]

[tex]

For R_{i}<R<R_{o}

[/tex]

[tex]

E_{R3} = 0 [/tex]

[tex]

For R > R_{o}:

[/tex]

[tex]

E_{R4} = \frac{2 \rho_{0} b^{3}}{15 \epsilon_{0}R^{2}}

[/tex]

Now I didn't get very far, but here's what I have:

[tex]

Q_{enc} = \int \rho dV = \rho_{0}[1 - \frac{R^2}{b^2}]\frac{4\pi R^{3}}{3}

[/tex]

[tex]

\int E \cdot ds = E_{R1} 4 \pi R^{2}[/tex]

[tex]

E_{R1} = \frac{\rho [1-\frac{R^2}{b^2}]R}{3\epsilon_{0}}[/tex]

[tex]

E_{R1} = \frac{\rho R}{\epsilon}(\frac{1}{3}-\frac{R^2}{3b^2})

[/tex]

I have some work for the next two, but I'd rather go one step at a time here to make sure I completely what's going on. At least this answer is close, but I'm not sure where the book's 5b^2 came from.