Units – they just don’t go away

One thing that’s become really clear to me in teaching physics is that dimensions and units are not straightforward concepts for students. I might hazard the assertion that they are ‘threshold concepts‘ – ones where grasping what they are about transforms you way of thinking. Most people at least half-understand the idea of units – if we measure a length, we can’t say it’s 26.5.  It’s 26.5 what? nanometres? kilometres? light-years? There’s big differences between them.

There’s also this mathematical conundrum to through out which illustrates the idea of dimensions: "I have a cube. What length does the side of the cube need to be in order for its surface area to equal its volume? "

The algebra-happy student will have no problem with this question. If the side of the cube is x, then the area of one face is x^2 (that is, x squared), so the total surface area is 6 x^2 since there are six faces. The volume is obviously x^3 (x cubed).  So we have 6 x^2 = x^3 and we can cancel a factor of x^2 from left and right to give 6 = x, and so there’s our answer. Six.

Six what?  Nanometres? kilometres? light-years? There’s a big difference. The solution of the conundrum is straightforward. You can never have the surface area equal to the volume (unless you have no cube at all – i.e. x is zero). Surface area and volume are fundamentally different things (dimensions).  In S.I. units, the former would have units metres squared, the latter metres cubed. So the question is ‘wrong’.

This afternoon I was talking with my students about the ‘electron-volt’ unit for energy. There’s clearly some difficulty in grasping this. Literally, it is a volt (which is a measure of energy per unit charge) times the charge on the electron.  The context in which it came up is with contact potentials in solid state physics. If I put a material with work function A, in contact with one of work function  B, then the contact potential is just (A-B)/e, where e is the charge on the electron. Work in electron volt units, eV, and it’s dead straightforward.  For example, if A is 7.0 eV, and B = 5.5 eV, then the contact potential is (7.0 eV – 5.5 eV) / e = 1.5 V.    On the numerator there’s the charge on an electron times a voltage, on the denominator there’s the charge on the electron, and so the charge on an electron cancels and we get a voltage. But many students felt that it couldn’t be that simple – they felt they needed to put in a numerical value for e rather than just cancelling it.

Units and dimensions are tricky things.