Current through Flat Sheet or plate

[KuriousKid]

TL;DR Summary

How current flows in Wide sheet or plate. Is it uniform pattern or something else?

Current flow through a wire is considered (or assumed) to be flown uniform across the cross section of the wire (or is it more near skin? or at core?). However I wonder if that's the case with a sheet or plate or circular disc. e.g. If I apply voltage at two opposite corners of the square or exactly in the middle of the plate. How the current will flow?

I know in transformer the thin plates are used to reduce the eddy current effect, but this is not what I'm tying to understand. I want to know how current will flow through the thick and wide object.

I've attached few options and I was wondering if someone have experimented this and knows the real answer? Assume that Rectangle plate of 3 mm thickness and 18 x 12 inches long/wide is applied AC voltage in the middle as shown below. In first case I showed dotted lines as current flow path... which is parallel to each other across the length of the sheet. In second case, it is somewhat elliptical or something like that. Just try to imagine it as it doesn't reach the corners uniformly but flows as it finds its way in sheet.

Or is there any other pattern, the current will flow? Anyone have idea or have experimented this?

 

[DaveE]

TL;DR Summary: How current flows in Wide sheet or plate. Is it uniform pattern or something else?

Current flow through a wire is considered (or assumed) to be flown uniform across the cross section of the wire (or is it more near skin? or at core?).

Assumed is the key word here. It makes homework problems easier and is essentially true at low frequencies. It is absolutely not true at high frequencies. Look up "skin effect" for more info about this.

TL;DR Summary: How current flows in Wide sheet or plate. Is it uniform pattern or something else?

Or is there any other pattern, the current will flow? Anyone have idea or have experimented this?

More like your second picture. The current will flow everywhere across the sheet. But the paths that deviate from the center will have less current because their longer path length creates more resistance. You don't need an experiment, it can be calculated (but not easily). IRL it would be a computer FEA program that would solve it.

Maybe a (not so great) analogy is a crowd of people trying to walk across the sheet. When the direct path gets crowded, they move to the outside.

 

[KuriousKid]

So to make it flow uniformly, in parallel, can we split current using connections as shown in below diagram?

 

[Baluncore]

However I wonder if that's the case with a sheet or plate or circular disc. e.g. If I apply voltage at two opposite corners of the square or exactly in the middle of the plate. How the current will flow?

It is best to make contact with the conductor over a defined equipotential "surface" contact, mainly because a "point" contact will have an infinite current density, infinite resistance, and an infinite voltage drop.

The equipotentials used for the contacts should fit the same geometry as the plate, since the field solution will then have compatible boundary conditions. As an example, a circle will have a central equipotential terminal that is also a finite circle. The other terminal will be an equipotential annulus that covers the circumference. All currents are then radial.

The challenge, with the diagonal corners of a square, is to find some way of transforming the square field and the terminals. Obviously, a square equipotential at the corner terminals will be messy, but a quadrant of a circle in the corner will act as a radial distributor. There is an axis of symmetry between the two adjacent corners, where the current flow will be perpendicular to the axis. That line is also an equipotential.

 

[Borek]

Draw a square net on the plate. Connect each vertex with the neighboring ones with a resistor - that's a simplified model of the plate. Some paths are longer and have higher resistance, so the current will be smaller (this can be proven used Kirchoff's laws).

Does this produce a uniform current in every place of the plate?

 

[DrClaude]

I've simulated it and here is the result for the current density norm

and its logarithm

(A potential difference was applied to the outside edges of the "wires.")

 

[KuriousKid]

This is Awesome @DrClaude. You nailed it. Is there a way we can see the % of current value in this simulation?

 

[DaveE]

So to make it flow uniformly, in parallel, can we split current using connections as shown in below diagram?

View attachment 318714

yes. but close to each contact point you still get the spreading out effect as before.

 

[DrClaude]

This is Awesome @DrClaude. You nailed it. Is there a way we can see the % of current value in this simulation?

You mean relative to peak current (i.e., current in the wires)?

 

[KuriousKid]

Yes. Sort of 10 Amp is input current and then how it is spread in sheet and then converges back to 10 amp.

 

[DaveE]

Yes. Sort of 10 Amp is input current and then how it is spread in sheet and then converges back to 10 amp.

OK, but in the sheet the current is 10A. You need to think in terms of current density in a local region, or electron flux through an area. Notice in @DrClaude's plot the scale is A/m^2. The problem is linear, so the current density at location for a 10A source would just be 10x the answer for a 1A source. So, percentage of the source current is a good way to describe it. That's sort of what he posted already. The pictures won't change, just the scale on the side. The log scale is needed for us to see anything interesting since, as others said, the current density is really big near the contact point.

 

[KuriousKid]

From the First image by @DrClaude, based on Color, I feel that the 90% of sheet have almost same current distributed equally (which could be somewhere around 1 amp or less than that). So my point was, if the simulation software can show numbers on the sheet (may be on current paths/bands), then it would be more informative. Or may be the paths are so thin that it may show very small current running parallelly and then it converges to total input value.

 

[Baluncore]

So my point was, if the simulation software can show numbers on the sheet (may be on current paths/bands), then it would be more informative.

There are a huge number of current paths on the plate, all changing width. If instead of asking for current paths, you ask for equipotentials, then those will be orthogonal to the currents. The currents were probably computed from the equipotentials in the first place.

 

[DrClaude]

Here is a contour plot of the current density

 

[alan123hk]

It is also worth noting that in the actual environment, the part with large current will heat up, so the resistance will increase, and the current distribution will also change accordingly, so the final current distribution may be related to power, temperature, heat dissipation and time.