Difficult capacitance problem -- 3 long concentric metal cylinders

[phantomvommand]

Homework Statement

Three long concentric conducting cylindrical shells have radii R,2R and (root2)*R. Inner and outer shells are connected to each other. The capacitance across middle and inner shells per unit length is:

Relevant Equations

Capacitance per unit length for cylindrical shells:
C = 2pi e0/ ln(r2/r1)

A solution I found online claims that the effective capacitance between the middle and inner shell can be seen as:

C (effective) = C1 + C2,
where C1 is the capacitance between the inner and outermost shell, and C2 is the capacitance between the middle and outermost shell. Apparently C1 and C2 are arranged in parallel, and the capacitance between the middle and inner shell is the effective capacitance across this parallel arrangement.

Does anyone know why this is so?

Thank you!

 

[Delta2]

Something confuses me here. Is there supposed to be a straight linear conductor located at the center of all shells? Otherwise how can we define the capacitance C1 between the inner and outer shell since they are connected , hence they are at same potential , hence the potential difference $V$ is zero so $C_1=\frac{Q_1}{V}=\frac{Q_1}{0}$??

 

[phantomvommand]

Something confuses me here. Is there supposed to be a straight linear conductor located at the center of all shells? Otherwise how can we define the capacitance C1 between the inner and outer shell since they are connected , hence they are at same potential , hence the potential difference $V$ is zero so $C_1=\frac{Q_1}{V}=\frac{Q_1}{0}$??

Exactly. I am not sure either. How would you solve this problem?
The link below is where I found the problem. Unfortunately, the teacher is speaking in hindi (i think), which I do not understand. Nonetheless, his workings can be roughly understood.
https://doubtnut.com/question-answe...2r-and-2sqrtr-inner-and-outer-shells-12228579

The following links explain the same problem too:
https://www.getpractice.com/questions/126640
https://www.toppr.com/ask/question/three-long-concentric-conducting-cylindrical-shells-have-radii-r2-r-and-2sqrt2-r-inner-and/
https://brainly.in/question/11974351

 

[Steve4Physics]

If you think about it like this, it might help…

Call the inner cylinder A, the middle cylinder B, and the outer cylinder C.

$C_1$’s ‘plates’ are the outer surface of A and inner surface of B.
$C_2$’s ‘plates’ are the outer surface of B and inner surface of C.

The connection between A and C form a junction between $C_1$ and $C_2$.
The body of B forms another junction between $C_1$ and $C_2$.
So $C1$ and $C2$ are in parallel.

 

[phantomvommand]

If you think about it like this, it might help…

Call the inner cylinder A, the middle cylinder B, and the outer cylinder C.

$C_1$’s ‘plates’ are the outer surface of A and inner surface of B.
$C_2$’s ‘plates’ are the outer surface of B and inner surface of C.

The connection between A and C form a junction between $C_1$ and $C_2$.
The body of B forms another junction between $C_1$ and $C_2$.
So $C1$ and $C2$ are in parallel.

This has been very helpful. However, it the solution online claims that the capacitance between AB = Capacitance of AC + BC. Why is this so? I can't see how a similar argument is applied.

 

[phantomvommand]

This has been very helpful. However, it the solution online claims that the capacitance between AB = Capacitance of AC + BC. Why is this so? I can't see how a similar argument is applied.

I have since looked at each of the links in detail. Apart from the original link, the remaining links describe a problem where the radii are R, 2R and sqrt8 R, while the original problem describes a problem where the radii are R, sqrt2 R and 2R.

However, they all get the same answer of 6e0/ln2. I suspect the original link solved the problem wrongly. Following Steve4Physics's logic, the 6e0/ln2 can indeed be obtained for the problem where the radii are R, 2R and sqrt8 R.

 

[Steve4Physics]

However, it the solution online claims that the capacitance between AB = Capacitance of AC + BC. Why is this so? I can't see how a similar argument is applied.

I can’t follow the online solution. It basically says:
C1 is formed by the outer surface of A and the inner surface of C.
C2 is formed by the outer surface of B and the inner surface of C.

But this means the inner surface of C has two (conflicting) functions – simultaneously acting as a plate belonging to 2 different capacitors. This doesn’t make sense.

Also, C1 isn’t a simple capacitor because it has cylinder B sitting inside.

So the online solution has some problems.

 

[hutchphd]

If you think about it like this, it might help…

Yes. Perhaps it is easier to think of the corresponding parallel plate capacitor having layers A,B,C.
If we attach a wire to A,C as one port and to B as the other this will also give $C_{AB}\parallel C_{BC}$.