Calculating Electric Field of Non-Conducting Sphere | Step-by-Step Guide"

[Azael]

I apologise for any spelling errors or terms missnamed since I am swedish and this course I am reading is only swedish books and terms. But I think I have gotten the translations right. Also my first try with latex.


I have a non conducting sphere with the radius R and the volume charge density

$$\rho (r) = \rho_o (1- \frac{r}{R}$$ when 0<r<R
and $$\rho(r) = 0$$ when r>R
where $$\rho_0$$ is a positive constant.


I want to calculate the field E(r) for 0<r<R and R<r and I want to use this formula

$$E = \int \frac{dQ \hat{r}}{4 \pi \epsilon r^2}$$

This is how I do it

$$E = \frac{\rho}{4 \pi \epsilon_0 } \int_{0}^{r} (1-\frac{r}{R}) sin\theta d\theta dr d\phi$$

I get that to $$\bar{E}= \frac{\rho_o}{\epsilon_0} (r- \frac{r^2}{2R}) \hat{r}$$

is that a correct answere for 0<r<R??

gonna post this now and se if I got the latex right

 

[Azael]

if I use gauss

$$\oint Eds = \frac{{Q_e_n_c_l}}{{\epsilon_o}}$$

with $$Q_e_n_c_l = \int_{0}^{r} \rho_0(1 - \frac{r}{R}) r^2 \sin \Theta dr d \Theta d \Phi = \rho_0 4\Pi (\frac{r^3}{3} - \frac{r^4}{4R})$$

I get as answere
$$\overline{E} = \frac{\rho_o}{\epsilon_0} ( \frac{r}{3} - \frac{r^2}{4R}) \hat{r}$$

why do I get different answeres. What equation do I implement wrong(or do I do them both wrong)?

seems like the latex won't work

 

[whozum]

might want to fix your tex tags. put the tags on the same line as the code

 

[Azael]

found my error. I used capitals by misstake in the [ /tex ]

 

[Azael]

I solved it today fortunaly :)