Is this the correct set up for the electric field?

[ghostfolk]

Homework Statement

A spherical conductor of radius $a$ carries a charge q and also there is a jelly of constant charge $rho$ per unit volume extending from radius $a$ out to radius $b$.
I'm looking to see if I got the correct set up for the electric field of this spherical conductor for all space.

Homework Equations

$\oint \vec{E} \cdot d\vec{a}=\frac{Q_{enc}}{\epsilon_0}$

The Attempt at a Solution

$\oint \vec{E} \cdot d\vec{a}=4\pi r^2$
$Q_{enc}=\int_a^r 4\pi r'^2 \rho dr'+q=\frac{4\pi}{3}(r^3-a^3)\rho+q$
So then,
$E=\begin{cases} 0, r<a& \\\ \rho \frac{(r^3-a^3)}{3r^2 \epsilon_0}+ \frac{q}{4\pi r^2\epsilon_0} \hat{r}, a<r<b\\ \rho \frac{(b^3-a^3)}{3r^2 \epsilon_0} +\frac{q}{4\pi r^2\epsilon_0} \hat{r}, b\le r \end{cases}$

 

[vela]

Looks good except for the incorrect notation. You should write
$$\vec{E}=\begin{cases}
0 & r<a \\
\left[\rho \frac{(r^3-a^3)}{3r^2 \epsilon_0}+ \frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & a<r<b \\
\left[\rho \frac{(b^3-a^3)}{3r^2 \epsilon_0} +\frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & b\le r
\end{cases}.$$ The way you wrote it, $\hat{r}$ only multiplies the last term, and you'd be adding a scalar to a vector, which doesn't make sense.

 

[ghostfolk]

Looks good except for the incorrect notation. You should write
$$\vec{E}=\begin{cases}
0 & r<a \\
\left[\rho \frac{(r^3-a^3)}{3r^2 \epsilon_0}+ \frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & a<r<b \\
\left[\rho \frac{(b^3-a^3)}{3r^2 \epsilon_0} +\frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & b\le r
\end{cases}.$$ The way you wrote it, $\hat{r}$ only multiplies the last term, and you'd be adding a scalar to a vector, which doesn't make sense.

Yeah I should've added parentheses.Thanks.