Eddy current calculation

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Garden shed/back garden experiments have already been carried out with modest currents regarding this issue. I expect the plate to get hotter than the ladder in your case.

https://www.youtube.com/watch?v=hg5eZkq2KgE

 

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Section 3.1 discusses power losses due to Eddie currents.

https://www.princeton.edu/ssp/joseph-henry-project/eddy-currents/eddy_wiki.pdf  It looks a bit involved though.


Z.

mapj1:

The problem is that it is critically dependant on the magnetic losses in the steel when taking it from magnetised one way, through a cycle of being magnitised the other, and back again, in time with the mains cycle, this in turn is a function of the impurities present in the steel, and the grain size, neither or which are well controlled quantities in general purpose construction steel, which is normally designed to meet a mechanical spec, not a magnetic one. I suspect that neither you nor the makers of the box actually have the magnetic loss tangent or even the permiability information

If you need to know it, it is usual to actually measure a sample for heating in a known magnetic field and then calculate back to the material properties.

mapj1:


As a point of note it will not be eddy currents as such that are your main worry, (the induced current is in the direction of the main current, and that is the thin way in the metal plate) rather it is magnetisation losses - think of atomic scale compasses swinging round to form circles around the wire, first one way and then the other, but the action of rotating those compasses causes frictional losses, and heating. 

Because the steel has thousands of times more magnetic response than the air (or epoxy or a fillet of braze metal, if you need to keep the containment blast proof), a very small air gap  that cuts all these circles acts rather like a much longer steel path ,  say 1mm of air , is worth a 1000mm of steel - to all intents and purposes, the magnetisation stops dead at all but the smallest gap , and the bulk of the field lines try to  go another way round that is magnetically shorter.

The further out circles  experience a weaker effect, as the same magnetisation is spread over the circumference of a larger circle .

Quite a lot of text books that should know better get this wrong too, confusing lines of current with lines of magnetic flux, and it makes visualising the correct treatment  and which way to slice things difficult. Eddy currents would be interrupted by by making the sidewall of the box from many thin layers, but this is not that, and is a magnetics problem.

If the wire is a rattling fit in the hole, this reduces the losses,  as then as the magnetic flux is becoming more spread out at greater radii from the wire, (falling more or less as one over radius for an idealised long straight wire at right angles to the plate) and there is no material close to the wire to experience the greatest field.



Given it is up and running, while it may not be great, unless it is actually running hotter at the steel than in free air, it is OK -  and that will be luck with that steel alloy, rather than any magic - if it already works, it is not going to suddenly change chemistry.

In a bad case the wire insualation will melt and the paint blister on the steel, and it will be obvious what needs doing, usually to change the plate for another material, or join the dots of the holes.

 

It might actually be okay in some circumstances in practice but it's too complicated to be proven so.

Is a pretty good summary. You can get so far by estimating the B field at some distance from the conductor and then there are rules of thumb in watts per kilogram of material per tesla of field, but this ranges from about 2 watts of heat per kilo of material in a field of 1 tesla RMS at 50Hz for state of the art transformer cores with 3% silicon steel and optimum grain oriented , to about 100 times worse for random bits of old scaffold pole and snapped off drill bits. (These make really bad DIY transformer cores)

so, taking that awful case,  < 100A RMS at 50Hz  un-cancelled through a 1 inch hole or larger in a typical  thin walled box is not likely to cause dangerous heating.

(or 1/8 of the current at 400Hz etc.)


Either fit multi core cables so the net current per hole is near zero, or replace the plate with a non-magnetic material, or 'dog bone' the holes with cuts of a mm or two, to join holes to include all cancelling currents, and if need be use braze metal or epoxy to back fill.