Potential of an undefinided cylinder

[Guillem_dlc]

Homework Statement

Determine the potential that creates an undefined cylinder of radius $R$ and density density $\rho$ that is uniformly charged.

Homework Equations

Gauss's law.

The Attempt at a Solution

I know that for this problem I can use gauss because it is a cylinder, now I do not get anywhere right, it is also a question of calculating the potential. Some help? Besides, the indefinite thing confuses me a little ...

 

[TSny]

I venture to guess that the problem statement is to be interpreted as asking for the potential created by an infinitely long cylinder of radius $R$ and uniform volume charge density $\rho$.

Please show the details of your attempt of using Gauss' law.

 

[Guillem_dlc]

I venture to guess that the problem statement is to be interpreted as asking for the potential created by an infinitely long cylinder of radius $R$ and uniform volume charge density $\rho$.

Please show the details of your attempt of using Gauss' law.

Yes, of course I'll pass it to you. I think that's fine by now.

 

[TSny]

It's preferable to type out your equations rather than post pictures. That way, we can easily quote specific parts of your calculations.

In your final expression for $\vec E$, I take it that the symbol circled in green

should be $r$. If so, then it looks OK. But is this the electric field at a point inside the cylinder or at a point outside the cylinder?

 

[Guillem_dlc]

It's preferable to type out your equations rather than post pictures. That way, we can easily quote specific parts of your calculations.

In your final expression for $\vec E$, I take it that the symbol circled in green View attachment 222331 should be $r$. If so, then it looks OK. But is this the electric field at a point inside the cylinder or at a point outside the cylinder?

Yes it is a [ tex ] r [ / tex ] sorry it is not clear. It's the field for the interior I think.

 

[TSny]

It's the field for the interior I think.

OK, that's correct. Are you trying to get the potential at a point inside the cylinder, outside the cylinder, or both?

What is the connection between electric field and electric potential?

 

[Guillem_dlc]

OK, that's correct. Are you trying to get the potential at a point inside the cylinder, outside the cylinder, or both?

What is the connection between electric field and electric potential?

We have seen that $\vec{E}=\rho \dfrac{r}{2\varepsilon_0}\vec{r}$ is in the inside of the cylinder.
Also, $E2\pi rh=\dfrac{1}{\varepsilon_0}\rho V\Rightarrow E=\dfrac{\rho \pi R^2h}{2\pi rh\varepsilon_0}\Rightarrow \vec{E}=\dfrac{\rho R^2}{2\varepsilon_0r}\vec{r}$ is outside the cylinder.
The connection between electric field and electric potential: one is the derivative of the other, is not it? Would I have to integrate in this case? But I do not know anymore.

 

[TSny]

Yes, integrate. See method 2 here https://webhome.phy.duke.edu/~schol/phy152/howtos/potential_howto.pdf

 

[Guillem_dlc]

Yes, integrate. See method 2 here https://webhome.phy.duke.edu/~schol/phy152/howtos/potential_howto.pdf

Thanks, I'm trying, but I'm still not getting the correct results

 

[TSny]

Please show your work.

 

[Guillem_dlc]

Please show your work.

I have solved the $V (r)$ for $r <R$ (inner points).
$V(r)=\int_r^0 \vec{E}\cdot d\vec{l}=\int_r^0\dfrac{\rho r}{2\varepsilon_0}\vec{a}_r\cdot d\vec{l}=\dfrac{\rho}{2\varepsilon_0}\int_r^0rdr=\dfrac{\rho}{2\varepsilon_0}\left[ \dfrac{r^2}{2}\right] _r^0=-\dfrac{\rho r^2}{4\varepsilon_0}$

 

[TSny]

I have solved the $V (r)$ for $r <R$ (inner points).
$V(r)=\int_r^0 \vec{E}\cdot d\vec{l}=\int_r^0\dfrac{\rho r}{2\varepsilon_0}\vec{a}_r\cdot d\vec{l}=\dfrac{\rho}{2\varepsilon_0}\int_r^0rdr=\dfrac{\rho}{2\varepsilon_0}\left[ \dfrac{r^2}{2}\right] _r^0=-\dfrac{\rho r^2}{4\varepsilon_0}$

This looks right. Note that you are taking the potential to be zero at $r = 0$, which is a good choice for this problem.

 

[TSny]

Have you also been able to get the potential at points outside the cylinder?

 

[Guillem_dlc]

No

 

[TSny]

No

What is it about points outside the cylinder that is making it difficult to find the field and potential?