I remember as a student being presented with the proof that the shortest distance between two points is a straight line (at least, on a 2 dimensional flat surface). Although it’s almost blatantly obvious, it can be formally proved through Calculus of Variations.
However, the quickest route between two points is not necessarily a straight line. Take the example of my walk from my office, on the 3rd floor of EF block at Waikato, to the lecture room G.3.33 on the third floor of G block. Normally it is a short (1 minute) stroll down the third floor corridor in a straight line – the walkway goes from EF building into F building into G building and G.3.33 is straight in front of me. (Those ‘in-the-know’ will realise I’ve over-simplified the situation – there are a couple of 22.5 degree bends in the corridor and a few steps to negotiate, but basically it is a straight line between the two.) But not now. There is building work going on in FG link block. This means that the straight-line route is unavailable. Instead, I have to go down three floors, out the door, walk about 100 metres through the rain around the back of F and G buildings to the G front door, then up three flights of stairs, and walk back through the corridor to G.3.33. Total time more like 6 minutes for a distance of about 50 metres.
Here’s another example: You are a lifeguard on a beach, say 50 metres from the water, and you see someone in trouble in the sea, say 25 metres off shore and 50 metres along the shore from where you are. Which direction do you run in? You don’t run straight towards them, because you know you can run faster than you can swim. Covering the shortest distance in total would mean that you are swimming a longer distance than necessary. It is better to run more along the shore, and enter the water at a point closer to the man in trouble. You cover a longer distance in total, but less of that distance is spent in the water where you are going slowly.
The problem of refraction of light is analogous to the lifesaver description above. The light travels the path that gives the shortest time between two points. Suppose a light ray starts at point A, in air, and ends at point B, in a block of glass. Now, we know that glass has a refractive index. This means that the speed of light in the glass is lower than that in air. The light doesn’t travel in a straight line between A and B, but rather in two straight line segments – in a straight line from point A to point C on the surface of the glass, then in a straight line from point C to point B. Where is point C? It turns out that it’s such that the total time the ray takes from A to B is minimized. Clever or what? How does it ‘know’ that this is the shortest path?
Many problems in physics are similar minimization problems.
It’s been great having time off work with Karen and Benjamin, but, as they say, all good things come to an end. So today it’s back to work, with my first class in about an hour and a half. I spent my final afternoon of parental leave probing the underlying geology of our driveway, trying to work out why it was flooding in patches. (The fact that it had deluged for two days pretty-well non-stop was clearly half the reason, but why wasn’t the water clearing?) As it turned out, the gravel top surface was hiding sizeable potholes underneath, which were just filling with water, which wasn’t draining. When you’ve got time to do that, you know that you don’t need to be on parental leave anymore!
But that’s not the subject of today’s blog. It’s Microwave sterilizers. They are a handy device for rapid sterilization of bottles and other baby parephanalia. Fill with 200 ml of water, zap on full power (in our case 900 W) for 5 minutes, and it’s done.
A quick physics calculation will show what’s happening. A 900 watt microwave will put out a total of 270 thousand joules of energy in five minutes (that’s 900 joules per second times 300 seconds). A paltry 4.2 joules will take 1 ml of water up in temperature by 1 degree C (that’s the specific heat capacity). So to take 200 ml of water from the tap (at say 15 C) and take it to 100 C (boiling) requires 85 times 200 times 4.2 equals 71 400 J of energy. That’s about a quarter of what is being supplied.
So the water will reach boiling point. But where does the extra energy go (there’s about 199 thousand joules of it)? This is put into turning the water at 100 C into steam at 100 C. It requires a considerable energy input to make this step. The latent heat of vaporization of water is about 2300 joules per gram (or ml of water). This means that the extra 199 thousand joules will turn about 80 or so ml into steam. That means there should be plenty of water left at the end of the proceedings.
Of course, the steam will start condensing on the bottles and internal surface of the sterilizer – these have a specific heat capacity too and will take some energy to come up to a similar temperature. But the rough calculation above I hope shows that the 5 minutes for 900 W statement by the manufacturer is pretty sensible.
Oh, and it has now stopped raining.
I shan’t apologise for the lack of entries – other things have been on our mind in recent days. But Benjamin is sleeping just now, which gives a few minutes for some blogging. He’s already been doing a bit of physics – studying the continuity equation. That is, the change in mass equals the mass flow in minus the mass flow out. Our midwife measured how much he’d taken from the breast at a feed simply by weighing him beforehand, and weighing him afterwards. Simple as that. Straightforward continuity equation stuff. The answer, unfortunately, is not enough, which means he’s going to be harder work than many babies.
So 3am? In the last few days, it’s cold, dark, quiet (at least outside) and foggy. A winter anticyclone can really send the temperatures plunging. Having to go out in the morning to make sure the chickens have some water that isn’t ice is a bit of a chore, but it’s one that needs doing as well.
A beautiful son, Benjamin John Wilson, arrived on Thursday 28 June, early in the morning. He was a hefty 4.3 kg. He’s still struggling with getting his head around relativistic electrodynamics and quantum many body theory, but, give it a few years, and he’ll get there. Maybe he’ll even sleep for a bit before then!