# Units – they just don’t go away

By Marcus Wilson 24/05/2013

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.

### 0 Responses to “Units – they just don’t go away”

• Michael Edmonds says:

I’ve always found units to be a very straightforward way of checking if I’ve done the calculation correctly, something I try and teach to my students.
{"ft_lang_label":"__label__en","ft_lang_prob":0.9419593,"math_prob":0.9838604,"size":2745,"snap":"2021-43-2021-49","text_gpt3_token_len":660,"char_repetition_ratio":0.10616563,"word_repetition_ratio":0.02020202,"special_character_ratio":0.23934427,"punctuation_ratio":0.113175675,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9849743,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-24T06:54:43Z\",\"WARC-Record-ID\":\"<urn:uuid:2447c6bb-e5d7-49d7-953a-cbf3c529d53f>\",\"Content-Length\":\"76406\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:be6f6bf9-0349-4f8a-a587-194b6d3092e0>\",\"WARC-Concurrent-To\":\"<urn:uuid:f43d7e78-0c78-42e2-9f83-bdd55e86ad30>\",\"WARC-IP-Address\":\"202.50.95.104\",\"WARC-Target-URI\":\"https://sciblogs.co.nz/physics-stop/2013/05/24/units-they-just-dont-go-away/\",\"WARC-Payload-Digest\":\"sha1:3HH3GDAYEGLNA3362VNJLHX36GD32XLA\",\"WARC-Block-Digest\":\"sha1:H66NRLL2NBJUTA6CZV26ME6ONC4B7VBN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585911.17_warc_CC-MAIN-20211024050128-20211024080128-00028.warc.gz\"}"}