Resistance calculation of odd-shaped plate

[Roger44]

Hello
If the resistivity rho and the dimensions are known, any schoolboy can calculate the end-to-end resistance of a strip of metal. But how do you do it if the strip is not regular as in the image?

 

[shawl]

If the problem is in the mesoscopic region, the additional piece can be treated as a Buttiker scattering.

If in the macroscopic, one can ignore the additional piece, because it will not affect the current significantly.

The above reply is limited to your model

 

[Roger44]

Shawl, it's macroscopic bur forget this example.

Could you consider a diamond shaped object connected to a + and a - DC at each end and just confirm that there is no easy "1960's university physics level" approach to calculating exact end-to-end resistance.

 

[shawl]

Considering that an irregular object is consists of many regular sub-objects which are paralleled connected.

The length of sub-object varies continuously from the shortest length (SL) to a longest length (LL).

The SL is a distance between a+ and a- DC contacts.

The LL in a diamond example is 2*(side length).

the resistivity/unit area for the each sub-object is rho/A. (A is the unit area)

The total resistance is calculated by an integration of a function (rho*L/A) from SL to LL.

This method is also suitable for other irregular shaped objects.

And This will be mathematic problem, but not the physical image.

 

[shawl]

If you want to measure a irregular object.

Generally, we choose the Van der pro method, which is 4 or 5 -terminents measurement.

 

[Dadface]

Just looking at the diagram shows that it can be considered as three resistors in series.

 

[marcusl]

Could you consider a diamond shaped object connected to a + and a - DC at each end and just confirm that there is no easy "1960's university physics level" approach to calculating exact end-to-end resistance.

Not sure where 1960 came from, people were plenty bright in that year! If your question is whether the calculation is non-trivial, then the answer is yes, non-trivial. You can analyze it exactly using conformal mapping, finding the potential and current stream lines in terms of Jacobi elliptic functions. The expression for resistance is then fairly simple, but it does depend on the form of the contact at each end (injecting current at an idealized point located at the diamond's corner creates mathematical difficulty with infinite current density, etc.) Find this in Lawden, Elliptic Functions and Applications. The resistance of flat plates of various irregular shapes can also be calculated approximately using an energy variational approach, as shown by Hammond in Energy Methods in Electromagnetism. I'm sure there are many other references (and approaches). I think none will be trivial.

 

[Roger44]

Thanks for your suggestions and I'll read up tonight on you references.

Could you have a look at this figure where the heat (or current) enters along a short length on one side and flows to an equal temp (or potential) on the opposite side.

What approach should be used to resolving this case?

Thanks again

 

[marcusl]

The fastest and most practical approach is not analytic at all. Use a finite-element code to solve it. There are many for E&M and for thermal simulations, and at least one (EMAS) that does both in the same package at the same time.

 

[Roger44]

Good evening

Yes, a non-trivial finite element approach was not on the cursus of Bristol university back in the sixties to my knowlege, whereas slicing up a diamond shape and integrating was six form level. But is it valid? I ask this question for the following reason:

An electric field will exist between any two spaced electrodes. A long metal band placed equally between these two electrodes without contact at either end will alter the electric field because its dialectric properties are different from that of air. If we now make contact, charged carriers will "flow", of value depending on other physical properties of the medium. Is there any reason why the potential (what we measure with a perfect volmeter) should decrease linearly from end to centre point? Other than blind obediance of Ohms law.

Please excuse me if your above references that I haven't had time to study already answer this question.