Problem with electric field and potential

[Frank Einstein]

Homework Statement

Hello, I have been assigned with a electrodynamics problem with which I have some problems:

I have a conductor shaped as an infinitely long cylinder of radius a; it is surrounded by a cylindrical surface which has a uniform density of charge RO and radius b.

I have to find the electric field and the electrostatic potential in all points of space.

Homework Equations

Gauss equation integrate E(electric field)*dS(differential of surface)=Q/epsilon₀

The Attempt at a Solution

The attempted solution is to use the Gauss theorem, picking another cylinder o radius x, height h and calculating the integral to obtain E*2*PI*x*h =Q/epsilon₀ . Then Q=density(RO)*2*PI*h*b.

I arrive to E(vector)=(RO*b)/(epsilon₀ *x) u^_r outside of the shell of radius b and 0 inside of it, for both the space between the conductor and charged surface and the inside of the conductor itself.

To calculate the potential, the only thing I can think is to calculate the potential as the integral of the energy multiplied by minus one; but that is not applicable since this is an infinite system.

 

[Orodruin]

Is anything more said about the conductor? For example its potential?

 

[Frank Einstein]

Yes, sorry, I forgot to say that it has no charge.

 

[Orodruin]

Is this your inference or part of the problem formulation?

 

[Frank Einstein]

No charge is part of the problem's formulation, I haven't written the exact problem definition because it is not in english and I am not a native english speaker. Sorry if I have caused any trouble.

 

[Orodruin]

So does the problem say something about the potential of the conductor? Is it perhaps grounded? It just seems strange to me that they would include it at all if it was charge-less and did not have a defined potential.

 

[Frank Einstein]

having an uncgarged cylinder of radius a and infinite lenght. Said conductor is surrounded by a cylindric shell of radius b and uniform charge density RO.
1) find the electrostatic field in all points of space
2)find the electrostatic potential in all points of space
3) analyze the behaviour of the field and potential at big distances and discuss its meaning
That is the best I can translate the problem. I hope it can help.

 

[Orodruin]

Well, if that is what it says it is what it says. The potential at a point is given by the line integral of the field (not energy, but I guess you were confunded by the field being represented by an E) along a curve from a reference point to that point. Since no potential has been mentioned, you are free to chose a reference. Normally you would chose infinity, but that does not work in this case since the integral will diverge so simply pick a different reference point.

 

[Frank Einstein]

So I can pick a randon point into the conductor and take it as the orygin?. If I do that, to preserve the continuity of potential I should give it the same value as the potential in the surface of the shell of radius b?.

 

[BvU]

1. Yes. picking anything except the axis as r=0 is unwise (you already used that to get E outside the shell). But as a reference point for V, anything with r $\le$ a will do: anywhere within the conductor you have the same potential by definition. You probably meant 'reference point', not 'origin'.

2. continuity is required everywhere, but what you have in mind (I sincerely hope) is continuity at r = a. And that continuity you can enforce, no matter what value for V you choose at the reference position. (Gauss gave you E, which gives V up to an integration constant).

'in the surface of the shell of radius b' is a bit ambiguous (could be r=b, could be r=a). Definitely V(r=0) $\ne$ V(r=b)!

Potential is defined up to a constant. Only differences count. That really means that you can really choose any value you desire. It's just that we are so used to picking V=0 at infinity that we have to re-consider this choice for an academic charge configuration such as in this exercise, where you can't pick 0 at infinity.