This is a cautionary tale about how our understanding of electromagnetic radiation can go wrong, despite everyone's best intentions.

My task was to prove that a box full of electronics was safe when it came to electromagnetic radiation hazards. The box obviously *was* safe, on the sensible gronds that what you are checking for is radio transmitters, and the box didn't contain any.

But what it did have a tiny test signal (as weak as the remote door lock button on a car key) which it used occasionally to confirm that its insides were still working correctly, and the existence of that tiny test signal caused us immense grief when our customer had nobody with enough technical understanding to agree that it really didn’t matter.

Now, I don’t have a problem with customers not knowing as much as we do – for the obvious reason that if they did, then they wouldn’t have needed to give us a job in the first place, they could have done it all themselves. But the problem is what you do about it.

When I first met the problem, our people were about to spend an immense amount of money getting measurements made to prove that the box was safe to not fry people. So our peole didn't have enough understanding either. I knew that doing the tests was stupid, and I realised that there was a way to prove that it was – the box had already been measured to make sure that it didn’t generate too much radio interference, and that’s the same thing as radiation hazards – both mean radio transmissions. It was safe because the standard for radio interference is maybe tens of thousands of times lower than the radiation hazard standard.

So I tried explaining this to the customer’s experts – and lost out again, because they still couldn’t understand it well enough to be confident about accepting my explanation.

At this point one can either start arguing with the customer (which may feel good but seldom achieves much), or just accept their limitations and get the job done. There is really only one way to get past this sort of problem, which is to give them the maths or whatever else is needed to prove it. So I set out to calculate the level of the radiation hazard, and ended up solving the similar problem of how to prove that you wouldn’t get killed by exploding fuel either.

The radiation hazard wasn’t too hard to do because the interference measurements that were already made are in almost the same units – there are a few differences such as the range of frequencies and how far away you are when you do the measurements, but when you’ve got a safety factor of tens of thousands of times, all of these things can be allowed for without too much trouble. But fuel is more difficult, especially when my own safety expert (with no understanding of radio) threatened to remove the box because it hadn’t been through an explosion test.

There are two ways that fuel can catch fire - heat or a spark. Obviously nothing should get hot enough to cause any fuel vapours to go bang, so all that you need to do is to calculate the temperature rise. Here, some physics comes to the rescue, in the form of the Stefan–Boltzmann Law which gets its name from having been worked out from measurements by Josef Stefan in 1879 and then derived by theory by Ludwig Boltzmann in 1884. The law says that the heat radiation from any body is given by:

** j*** = ε σ T

A ** j*** = A ε σ T

Now we need some algebra - so skip past this if algebra isn't your thing - but if you can cope with it, here is how it works. In the real world, we don't feel the heat radiation from any object that's the same temperature as us because, just like us, the object radiates as much heats as it absorbs, so everything cancels out. If the object is hotter than us then we can feel it radiating. The Stefan Boltzmann Law tells us the relationship between the heat being radiated and the temperature rise that occurs:

A ** j*** + heat input = A ε σ (T + temperature rise)

Subtracting the first equation from the second equation and using a first order approximation gives us:

heat input = A ε σ (T + temperature rise)^{4} - A ε σ T^{4} = 4 A ε σ (temperature rise) T^{3}

so we can turn that equation the other way around and calculate the temperature rise as:

temperature rise = (heat input) / 4 A ε σ T^{3}

The result for my box turned out to be a few thousandths of a degree temperature rise, which even the most doubtful expert could agree is no problem at all.

Sparks are a bit more complicated, but not much. A spark occurs if a contact is broken, or if two objects get close enough for a spark to jump across. For this box, like most electronics, there was nothing in it that could make an electrical contact come apart to cause a spark. Solving the problem of a spark jumping across a gap seems intuitively to be more difficult, because you would think that as a gap gets smaller and smaller, so the spark needs less and less volts to get started - but strangely, this is not how it works.

In 1889, Friedrich Paschen found that the minimum voltage for a spark never goes below a bit more than 200 volts, regardless of how small the gap is, and regardless of air pressure (which effectively means altitude). He found this by measurement, and the theory was soon developed to show why it happens. Of course, 200 volts is more than you'll find inside any normal box of electronics. So, science to the rescue again, and 19th century science at that (again).

Mission accomplished – pointless measurements called off, and basic algebra to the rescue. A win for sanity. Job done.