Senior Physics Challenge Day 4

Yesterday I didn't write up a blog post for day 4, because I had to pack up my things for today and that cut down the time I had to write a post. Nonetheless, it was a very exciting day and we finished off dealing with infinite square wells, classical potentials and investigated Stefan's law. We also had a lecture on chains from Prof. Mark Warner and another on the physics of juggling by a current PhD student at Cambridge.

The mechanics (both classical and quantum) was just a continuation of what we've been doing all week - while it is very interesting and stretching, you probably don't want a long winded explanation. The experiment from Wednesday, however, was very cool. And by cool, I mean hot. Take your typical 12V lamp and connect in the typical way to a variable power supple, ammeter and voltmeter. Plot the I-V characteristics by varying the potential difference in a way called a binary chop - this method involves graphing as you go and only taking data in areas of the graph that you are unsure about. Once you have enough data points in one region of the graph, you stop taking data points there (it's called a binary chop because the data you collect is between the two values you've already taken).

You have just performed a GCSE experiment and found the I-V characteristics of a lamp. At GCSE you would conclude that the lamp is non-Ohmic (a term I disagree with since Ohm's law specifally states that constant temperature is required for the propoertionality to be true). At A level, you might calculuate the resistance at various points on your graph by dividing the voltage by the current that flowed. This is one step more impressive, but isn't especially interesting in and of itself - if you do it for all data points you collected you get a 'teaspoon curve':

(It's a teaspoon because of the long handle and flat 'spoon head' - the line of best fit shouldn't actaully curve here, it's because the line of best fit isn't particularly correct for this or the previous graph.)

The enables you to do something interesting - say you know that the lab temperature was 28.5C at the time, or 301.7K. The at room temperature the resistance was 20.7 Ohm, and you can use the following relationship:

R=p (L/A)

along with a value of p for Tungsten at 301.7K.

Use can use this to solve for (L/A). This is a much better way of obtaining this value than actuall measurement of the filament, because the filament is coiled and a confusing shape. Also, the funny shape might (I'm not sure) affect the simple resistance formula somewhat, but by experimentally finding this value, it shouldn't matter so long as R=kp.

The next job is to find the resistivity of all of your points by dividing the resistance (on your R-I curve) by the experimental value of (L/A). Convert the resistivities to temperatures using the above curve, and then finally calculate the power output of the lamp as a function of temperature - power simply being given as P=IV.

Now, reading data off of that curve is somewhat annoying, so taking about half a doxen readings reasonably spaced out gives you this curve:

That doesn't tell you much unfortunately, because we can't anaylse graphs easily if they're not straight. However, what if we make a guess based on this graph that P=k (T^n) ? Well then, there's a neat trick we can use: take logs. we get log(P) = log(k)+nlog(T). Plotting log(P) vs log(T) gives us n as the gradient:

A gradient of 4.491 was obtained experimentally. This is slightly different from the theoretical value of n=4, but it's correct to the nearest integer which we should perhaps feel alright about. The fourth power dependance on temperature is called Stefan's law, and the inaccuracies come from a number of sources, mainly that not all of the power is radiated away but some is conducted.