Question

Electric potential. I have posted the questions as a picture. I hope somebody can help me.

Answer 1

Could you write down Gauss' Law?

Answer 2

Electric field for r>a \[E\sim\frac{1}{r^2}\]

Answer 3

Electric field shall be calculated at all points in the space
Calculate charge enclosed by sphere of radius r

\[Q=\int\limits_{0}^{r}p(r)4\pi r^2dr\]
we find
\[Q'=\frac{Q}{a}r\]
nor apply guass's law for spherical symmetry
\[E=\frac{Qr}{4\pi r^2a \epsilon_0}\]
now
do this
\[V=-\int\limits_{r}^{\infty}Edr\]

Answer 4

There are a few changes You must see through the now

Answer 5

Feel'in lazy to type

Answer 6

Above relation for E is for 0<r<a
for r>a
\[E=\frac{Q}{4\pi \epsilon_0r^2}\]

Answer 7

Now potential
\[Q=-\frac{Q}{4\pi \epsilon_0r}\] for r>a

0<r<a
\[V(r)=-\frac{Q}{4\pi \epsilon_oa}+\frac{Q}{4\pi a \epsilon_o}\ln(\frac{a}{r})\]