Electrostatics, infinite charged hollowed cylinder

[Telemachus]

Hi. I want to know if I did this on the right way. The exercise says: An infinite hollowed cylinder, with the cavity being another concentric cylinder has a uniform charge density. Find the electric field and the potential over all space.

And this is how I proceeded. I've called the inner radius for the cylinder a, and the outer b. Then:
$$\rho=cte=\sigma$$
$$Q=\int _V \rho d\tau=\sigma(b^2-a^2)\pi l$$
Being l the length of the cylinder (infinite for the case, this is one of the things that bothers me). Then I've applied Gauss law to get the electric field for the region outside the cylinder, with the radius bigger than b.

$$\oint E da=\frac{Q}{\epsilon_0}$$
$$E2\pi r l=\frac{\sigma(b^2-a^2)\pi l}{\epsilon_0}$$
$$E=\frac{\sigma(b^2-a^2)\hat r}{2\epsilon_0 r}$$

Then the potential for the same region, I choose the origin at the point a. I can't choose infinity, because it isn't a localized charge, and at zero it blowed up too, so I just choose a:
$$V(r)=-\int_o^r Edl=-\int_a^r\frac{\sigma(b^2-a^2)\hat r}{2\epsilon_0 r}=-\frac{\sigma(b^2-a^2)}{2\epsilon_0}\ln \left ( \frac{r}{a} \right )$$

I proceeded similarly for the other regions. The answer I get looks some kind of weird, the natural logarithm doesn't looks good, and it disconcerted me that I couldn't choose the center of the cylinder as the origin, so I think I'm probably making some mistake here. Besides the potential seems to increase when one gets further away from the cylinder, which makes no sense at all.

What you say?

Thanks in advance.

 

[pabloenigma]

The answer I get looks some kind of weird, the natural logarithm doesn't looks good, and it disconcerted me that I couldn't choose the center of the cylinder as the origin, so I think I'm probably making some mistake here. Besides the potential seems to increase when one gets further away from the cylinder, which makes no sense at all.

What you say?

Thanks in advance.

Your calculation is correct.It makes no sense at all,because even infinite cylinders don't make any sense.In words of Griffiths,"Notice that the difficulty occurs only in textbook problems; in "real life" there is no such thing as a charge distribution that goes on forever, and we can always use infinity as our reference point."

You are correct,no need to lose sleep over it.

 

[Telemachus]

Thank you Pablo. I just read that words from griffiths like an hour ago :P curious considence. But anyway, it doesn't make any sense I think the expression I get for the potential, the logarithm increases to infinity as the radius increases. Compared to an infinite plane... oh, wait, I was thinking on the field of the plane, now I think that it makes sense, because the logarithm increases slower than the expression that I could get for an infinite plane.

Thanks.

 

[ehild]

Your derivation is correct. You cannot choose the zero of potential at infinity in case of cylindrical charge distribution. Think: Infinite long cylinder does not exist. You can consider it infinite and use symmetry to calculate the electric field if the distance from the axis of the cylinder is much smaller than the length of the cylinder. If the cylinder is finite and you want to determine the electric field and potential at a point, farther than the length of the cylinder, you can not ignore the ends. And very far away the cylinder will behave more an more like a point charge.

ehild

 

[paranormal]

Hello,

Thank you for sharing your approach to this exercise. Overall, your method and equations seem correct. However, there are a few things to consider:

1. In an infinite cylinder, the length (l) does not matter as it extends infinitely in both directions. So, you can simply use the charge density (rho) instead of calculating it through integration.

2. The electric field equation you used is for a point charge, not for a charged cylinder. For a charged cylinder, the electric field at a point outside the cylinder is given by E = (lambda / 2*pi*epsilon_0*r) where lambda is the linear charge density (rho*l) and r is the distance from the center of the cylinder.

3. Your choice of origin for the potential is correct. Choosing infinity or the center of the cylinder would result in infinite potential.

4. The potential at infinity is usually taken as zero, so you can use this as a reference point instead of the origin.

5. The potential should decrease as you move further away from the cylinder, not increase. This could be due to a sign error in your equation or a mistake in your calculation.

I suggest double-checking your calculations and equations to see if there are any errors. Also, try plotting your potential equation to see if it makes sense. If you are still unsure, I recommend consulting with a colleague or professor for further clarification.

Overall, your approach seems correct, but it is always good to double-check your work to ensure accuracy. Best of luck with your studies!