Two concentric conducting spherical shells and resistor in between

[zenterix]

The only thing missing from the statement is "Assume that when we apply voltage in the inner and outer surface, the resistor reaches the steady state where the current is the same through every concentric surface of the configuration"

There would also need to be information about why we can disregard the assumptions violated by that setup.

The assumptions about $\vec{J}$ seem to be violated without explaining how charge gets into the spherical setup in the first place.

 

[Delta2]

There would also need to be information about why we can disregard the assumptions violated by that setup.

The assumptions about $\vec{J}$ seem to be violated without explaining how charge gets into the spherical setup in the first place.

Yes it seems like there should be charge accumulation in the surface of the inner shell if that's what you mean, unless we get an explanation of how the current circulates from the inner surface back to the outer surface.

But if you think that you think more like an engineer than a scientist, I classify this as an engineering detail.

 

[zenterix]

Yes it seems like there should be charge accumulation in the surface of the inner shell if that's what you mean, unless we get an explanation of how the current circulates from the inner surface back to the outer surface.

But if you think that you think more like an engineer than a scientist, I classify this as an engineering detail.

I think I have the capability to think like both.

I think you could also argue that disregarding such nuance would be the approach of some engineers, and that such details would be called academic by them.

In fact, I think that unless we go into the physics nuance this whole problem is useless unless you are practicing integration.

I still don't see the point of this problem really.

 

[haruspex]

It doesn't seem to be backwards.

It is not backwards in the context of defining resistance, but it is backwards in the context of the problem in this thread (where the resistivity is a given).

 

[haruspex]

@haruspex do you know how to make sense of the formula $R=\rho L/A$ in the case of the spherical shell resistors that make up the integral that solves this problem? Or at least derive it for this case?

I don't understand your difficulty. A shell radius r, thickness dr, with radial current, is a resistor length dr and area $4\pi r^2$. Plug that into the formula.

 

[Delta2]

I think I have the capability to think like both.

I think you could also argue that disregarding such nuance would be the approach of some engineers, and that such details would be called academic by them.

In fact, I think that unless we go into the physics nuance this whole problem is useless unless you are practicing integration.

I still don't see the point of this problem really.

Yes ok you might have both an engineering and scientific mind. In my opinion it is more engineering than scientific on how the current circulates but ok i might be wrong here.

Its a good problem if you ask me, unusual resistor configuration and current density distribution. You should be intrigued by unusual problems.

 

[PhDeezNutz]

I just want to make sure…..this thread is not so much about how to solve the problem but rather about the premise of the problem. Correct?

Edit: I did a google search for “spherical resistors” to see if they were commercially available and apparently they are not. The only thing that showed up was problems like the one in the OP. So maybe it is purely theoretical.

 

[Delta2]

Slight problem not exactly with the premise but the statement of the problem. For me at least it wasn't so obvious that the current inside this resistor will reach a steady state but if one has the necessary insight can see it and this steady state doesn't need to be explicitly stated in the problem like an assumption.

So maybe it is purely theoretical.

Yes maybe that engineering detail on how to complete the circulation of current is a big problem. The theoretical interest I find in this is that it is a resistor structure such that the current is spatially constant within the resistor but the current density again inside this resistor isn't spatially constant.

 

[haruspex]

Is this resistor purely theoretical?

Not necessarily. I can imagine how it could be a reasonable approximation to something real, perhaps in the context of a living cell.
You would have been happier, presumably, with concentric cylinders. The same concentric shell analysis applies.

 

[zenterix]

I just want to make sure…..this thread is not so much about how to solve the problem but rather about the premise of the problem. Correct?

Edit: I did a google search for “spherical resistors” to see if they were commercially available and apparently they are not. The only thing that showed up was problems like the one in the OP. So maybe it is purely theoretical.

The OP was about solving the problem. I did not foresee any of the ensuing doubts about the solution you provided because I did not think that we could have current flowing radially through the spherical setup. I then asked if that was realistic and then we got to this topic of if this type of scenario is possible in real life.

 

[zenterix]

I don't understand your difficulty. A shell radius r, thickness dr, with radial current, is a resistor length dr and area $4\pi r^2$. Plug that into the formula.

I guess you're right it is that simple.

 

[PhDeezNutz]

I guess you're right it is that simple.

I don’t think you’re wrong for asking these questions. You’re right I was a bit on “auto-pilot” when answering the question. Maybe that’s a bad habit I developed to merely get through school.

 

[SammyS]

I insist you must edit the first sentence of your post to "Just because the current density is steady, does not mean is the same everywhere"

Insist ?

Really, did you say Insist ? !!!!

 

[haruspex]

I would define steady state here as the state that in which the total current through a conceptual spherical shell of any radius r, is independent of r.

No, the latter can be true purely by symmetry even before steady state is reached.

For me at least it wasn't so obvious that the current inside this resistor will reach a steady state

We are asked for the resistance. How are you going to define that while not in steady state?
Charge flow follows the diffusion equation, and it is in the nature of that equation to tend to a steady state, though in theory it might never get there.

 

[Delta2]

No, the latter can be true purely by symmetry even before steady state is reached.

Hmm you are mostly right here, but I think there is not complete symmetry , it depends on the way we apply the source voltage to the inner and outer surfaces, if it is applied by thin wires that touch the surfaces (inner and outer) at specific points, there is not complete symmetry IMO.

Anyway you know how much I hate symmetry arguments though I have to admit your are mostly right here.

We are asked for the resistance. How are you going to define that while not in steady state?
Charge flow follows the diffusion equation, and it is in the nature of that equation to tend to a steady state, though in theory it might never get there.

Yes again you are mostly right here maybe something 95% right. I don't know I just feel the problem statement is kind of incomplete without stating the steady state assumption.