## Einstein and your Gas Bill

Taking refuge in my office this lunchtime for a sandwich and a cup of coffee I turned to the latest edition of Physics World and came across an funny little story about a physicist (who is completely new to me) with the splendid name of Fritz Hasenöhrl.

The news story relates to a paper on the arXiv, part of the abstract of which I’ve copied below:

In 1904 Austrian physicist Fritz Hasenohrl (1874-1915) examined blackbody radiation in a reflecting cavity. By calculating the work necessary to keep the cavity moving at a constant velocity against the radiation pressure he concluded that to a moving observer the energy of the radiation would appear to increase by an amount $E=(3/8)mc^2$, which in early 1905 he corrected to $E=(3/4)mc^2$

Since I’ve been doing a bit of dimensional analysis with first-year students, I’m a bit surprised that the authors of this paper read so much into the fact that Hasenöhrl’s formula bears a superficial resemblance to Einstein’s most famous formula $E=mc^2$, probably the best known and at the same time worst understood equation in physics. In fact any physicist worth his or her salt no matter how incorrect their reasoning would have to get something like $E =\alpha mc^2$, with $\alpha$ some dimensionless number, simply because the answer has to have the correct dimensions to be an energy.

Expressing energy in terms of the basic dimensions mass $M$, length $L$ and time $T$ is probability easiest to do when you think of mechanical work (force×distance). Since Newton’s laws give a force equal to mass×acceleration, a force has dimensions $MLT^{-2}$, so work (a form of energy) has dimensions $ML^{2}T^{-2}$. Now try to make this out of a combination of a mass ($M$) and a velocity ($LT^{-1}$) and you’ll find that it has to be mass×velocity2. You can’t get the dimensionless constant this way, but the combination of $m$ and $c$ must be the way it is in Einstein’s formula.

Anyway, all this suddenly reminded me of a day long ago when I appeared on peak-time television in the consumer affairs programme Watchdog, explaining – or, rather, attempting to explain – the physics behind the way gas bills are calculated. Apparently someone had written in to the programme asking why it was that they weren’t just being charged for the volume of gas that had flowed through their meter, but that the cost involved a complicated calculation involving something called the calorific value of the gas.

The answer is fairly obvious, actually. The idea is that to make competition fairer between different forms of energy (particularly gas and electricity) the bills should be for the amount of energy you have used rather than the amount of gas. Since the source of fuel varies from day to day so does its chemical composition and hence the amount of energy that can be extracted from it when it is burned. Gas companies therefore monitor the calorific value, using it to convert the amount of gas you have used into an amount of energy.

On the programme I was confronted by the curmudgeonly Edward Enfield (father of comedian Harry Enfield) who took the line that it was all unnecessarily complicated and that the bill should just be for the amount of gas used, rather in the same way that petrol is sold. When I tried to explain that the way it was done was really fairer, because  it was really the energy that mattered, it quickly became obvious that he didn’t really understand what energy was or how it was defined.  He didn’t even get the difference between energy and power. I suspect that goes for many members of the general public.

It was all a bit tongue-in-cheek, but I enjoyed the sparring. Eventually he came out with a question about why energy was given by $E=mc^2$ rather than $mc^3$ or something else. So I launched into an explanation of dimensional analysis and why $mc^3$ couldn’t be an energy because it has the wrong dimensions. His eyes glazed over. The shoot ended. My splendidly erudite and logically rigorous exposition of dimensional analysis never made it into the broadcast programme.

My brief career on BBC1 was over.

### 6 Responses to “Einstein and your Gas Bill”

1. Anton Garrett Says:

Maybe if you’d said “the amount of heat” rather than “the amount of energy” then he’d have got it.

The density of gas also varies with temperature – do they take that into account?

I came across Hasenöhrl when I was looking at classical theories of the electron, a long time ago.

2. I’m surprised they go to that much trouble with gas; when I was living in England, I was surprised that the water bill was calculated based on the number of people in the house, or the value of the property, or some such quantity which is definitely correlated with the amount of water used but with a large scatter—especially since it is easy to meter water.

• telescoper Says:

Quite a few people have water meters nowadays. The reason for the palaver with gas was when energy utilities were privatized in the 80s there was a fantasy that it would lead to a “market” which would drive down energy prices. This obviously suggested to someone that customers should be charged for energy rather than raw materials, hence the complicated way of calculating gas bills.

One of the interesting things about this is that gas companies nowadays hardly ever bother to read your meter anyway. Instead they send you an “estimated” bill which is usually for about four times the energy you have ever used over any corresponding period in the past, a blatant attempt to induce you to give interest-free loans to the company concerned until the time they finally get around to reading the meter and let on that they’ve been drastically overcharging you, possibly for years.

• Anton Garrett Says:

Estimated bills are OK – you can always demand that they come to read it, or phone across your actual reading. The danger comes when estimated bills are combined with payment by direct debit (for which they offer small discounts). Then they can hoover as much out of your account as they want. If ever you leave a house empty, cancel the direct debit first because the bills will be based on your previous occupied usage, your imformation that it is empty will be ignored, and you will have to go to a lot of trouble to get your money back.

3. Maybe you’d have done better if you’d just pointed out that a litre of some gas can heat up 5 cups of tea, and a litre of some better quality gas can heat up 10. And would you realy want to pay the same amount for each sort of gas?

I think the tangent onto dimensional analysis was always beyond hope.

• telescoper Says:

Actually, to be honest, what they really wanted was for me to make it sound so technical that it made their point for them that it was too complicated for ordinary folks to understand. They didn’t like it when I actually tried to make sense of it in layman’s terms – they really wanted me to talk about Einstein.

What I should have done was explain that burning gas to produce energy really is rocket science….