Current density as a function of distance from the axis of a cylinder

[Physics lover]

Homework Statement

In a uniform solid cylinder of radius r conductivity increases linearly with the distance as we move from the the axis to the surface of the cylinder from sigma1 to sigma2.Current i enters the cylinder from one end and leaves from the other end .Find the current density as a function of distance from the axis of cylinder at any cross section.

Relevant Equations

J=sigma E
E=V/L
V=IR
Where R is resistance

I first took out the variation of conductivity along the radius of cylinder.Also we know that J=sigmaE.Therefore i have to find variation of E also.But how will i find that as potential is also not given.Help.

 

[haruspex]

I would treat it as concentric cylindrical tubes. Consider end to end potentials.

 

[Physics lover]

I would treat it as concentric cylindrical tubes. Consider end to end potentials.

Can i assumee it to be V.
But what will i do next as V is not in the final answer.

 

[haruspex]

Can i assumee it to be V.

Yes, but what exactly do you mean by "it"? I wrote potentials, plural.

 

[Physics lover]

Yes, but what exactly do you mean by "it"? I wrote potentials, plural.

Ok and how can i write that.

 

[haruspex]

Ok and how can i write that.

If you treat it as separate concentric tubular conductors, what can you say about the end-to-end potential drop along each?

 

[Physics lover]

If you treat it as separate concentric tubular conductors, what can you say about the end-to-end potential drop along each?

Is it IR where R is resistance of conductors.

 

[haruspex]

Is it IR where R is resistance of conductors.

Yes, but how do the potentials compare with each other?

 

[Physics lover]

Yes, but how do the potentials compare with each other?

Sorry sir,but i can't get what are you trying to ask.Please explain it somewhat more.

 

[JD_PM]

Do you know what's $E$ for the cylinder? Hint: Construct a Gaussian cylindrical surface.

 

[haruspex]

Sorry sir,but i can't get what are you trying to ask.Please explain it somewhat more.

Consider two concentric cylindrical tubes at different radii. How does the end to end potential difference along one compare with that along the other?

 

[Physics lover]

Consider two concentric cylindrical tubes at different radii. How does the end to end potential difference along one compare with that along the other?

Ok so you are asking about variation of potential along radius,V=EL but i don't know about E.What should i do.I am totally stuck.

 

[Physics lover]

Do you know what's $E$ for the cylinder? Hint: Construct a Gaussian cylindrical surface.

I think it will not help as charge on cylinder is not known.

 

[JD_PM]

Consider two concentric cylindrical tubes at different radii. How does the end to end potential difference along one compare with that along the other?

As haruspex suggested, you need to find an expression for the potential difference between the cylinders. Let's say the inner cylinder has a radius $a$ and the outer $b$. Then:

$$V = - \int_b^a E \cdot dl$$

Where $V$ is the the work you have to do for bringing a test charge $q$ from infinity to distance $r$ with respect to the reference charge (this is what is called electric potential energy) per unit test charge. (i.e. $V = E_p/q$).

But you need to know $E$ to solve the line integral.

Is it clear till this point?

 

[haruspex]

you are asking about variation of potential along radius

Yes, but think what the ends of the cylinders are connected to.

As haruspex suggested, you need to find an expression for the potential difference between the cylinders.

No, the potential difference along the cylinders, and how that compares between one cylinder and another. Remember, the current flows along the cylinders.

 

[Physics lover]

Yes, but think what the ends of the cylinders are connected to.

No, the potential difference along the cylinders, and how that compares between one cylinder and another. Remember, the current flows along the cylinders.

Current is flowing in cylinder so can we assume that it is connected to a battery.

 

[Physics lover]

I think i will first have to find R by taking an element and then integration,and then V by V=IR,which will give me V,and then E by dV/dL.Am i correct.

 

[haruspex]

I think i will first have to find R by taking an element and then integration,and then V by V=IR,which will give me V,and then E by dV/dL.Am i correct.

No, it is I, as a function of r, that you are trying to find.

Current is flowing in cylinder so can we assume that it is connected to a battery.

Right, and all connected to the same battery... so what does that tell you about how the potential drop along one cylindrical tube compares with that along another?

 

[JD_PM]

No, the potential difference along the cylinders, and how that compares between one cylinder and another. Remember, the current flows along the cylinders.

My idea is that we're dealing with two coaxial cylinders,separated by material of conductivity $\sigma$. The current is flowing along the axis, and there's an electric filed pointing radially outwards.

Isn't this the scenario?

 

[Physics lover]

No, it is I, as a function of r, that you are trying to find.

Right, and all connected to the same battery... so what does that tell you about how the potential drop along one cylindrical tube compares with that along another?

Will it vary like Vr/R

 

[haruspex]

My idea is that we're dealing with two coaxial cylinders,separated by material of conductivity $\sigma$. The current is flowing along the axis, and there's an electric filed pointing radially outwards.

View attachment 245578

Isn't this the scenario?

No, I don't see how you can read it that way. It says the conductivity varies continuously with radius. If there were a radial field then there would be radial current, but it says the current is along the cylinder.

 

[haruspex]

Will it vary like Vr/R

What is V there? Don't just say voltage, voltage where?

Maybe you assume I am asking a difficult question, but it's very easy.
Say the voltage at one end of the axis of the cylinder is V₀. What do you think the voltage is at the periphery of the cylinder at that end?

 

[Physics lover]

What is V there? Don't just say voltage, voltage where?

Maybe you assume I am asking a difficult question, but it's very easy.
Say the voltage at one end of the axis of the cylinder is V₀. What do you think the voltage is at the periphery of the cylinder at that end?

As current is flowing therefore there will be a voltage difference.So is it V₀ -ER where R is Radius.Am i correct?

 

[haruspex]

As current is flowing therefore there will be a voltage difference.So is it V₀ -ER where R is Radius.Am i correct?

You seem to be asuming that the external source of voltage is connected to the cylinder at a single point, at the axis. That is not given, seems unlikely, and would make the question very difficult. Assume the connection spans the entire circle at each end.

 

[Physics lover]

You seem to be asuming that the external source of voltage is connected to the cylinder at a single point, at the axis. That is not given, seems unlikely, and would make the question very difficult. Assume the connection spans the entire circle at each end.

If external source is connected to the entire circle then the potential at the periphery will be V₀

 

[haruspex]

If external source is connected to the entire circle then the potential at the periphery will be V₀

Right. Similarly at the other end.
So you can answer post #11 now, yes?
Next, what is the resistance of the cylindrical tube radius r, thickness dr?

 

[Physics lover]

Right. Similarly at the other end.
So you can answer post #11 now, yes?
Next, what is the resistance of the cylindrical tube radius r, thickness dr?

Are you asking in terms of V₀ or i have to calculate it using R=pL/A.

 

[haruspex]

using R=pL/A.

Yes.

 

[Physics lover]

Yes.

Thanks for the help sir the problem is solved now.