Using Units To Help With Equations In Science

In science subjects such as physics and chemistry, units often give students a lot of trouble. In this article is a demonstration of how a knowledge of how they work in formulae can help you build an equation if you forget whether a quantity goes on the top or bottom of an expression.

The resistance of a wire is a question that frequently crops up in A level physics exams. The question will be something like:

A copper wire of length 2.20m, resistivity 1.68*10-8 ohm m has its resistance measured as 0.0516ohms. What is the cross sectional area of the wire?

Typically one would either know the equation relating these quantities off by heart, or use the formula book. However through common sense and ensuring the units are correct it is fairly simple to come up with the correct equation.

First it is easy to understand that the longer the wire, the greater it`s resistance will be. So our equation starts with

Resistance = length * [other quantities] Next let`s take resistivity, which is a quantity that shows how resistant to charge flow a particular material is. This has its units given to us in the exam paper, and it is in ohm m (ohm meters).

Here is the simple trick. Both sides of the equation must have the same units. As we are working out a resistance, both sides must only have the units, ohms.

The left side clearly does (it is just the quantity resistance, which is in ohms by definition). The right side contains a quantity called resistivity in ohm meters, a length, and an area. These must be combined in a way to leave just ohms, to match the left hand side. As resistivity is in ohm m, not ohm-1 m, it must also be, like length, proportional to resistance.

So now we have Resistance (ohms) = length (m) * resistivity (ohm m) * [other quantities]

The right hand side now has units of ohm m2, but we want only ohms. We still have to fit area into this equation, which has units of m2. We can now see this must be placed on the bottom of the right hand side, to cancel out the m2 in the ohm m2 and leave us with just ohms. (The answer is 7.16*10-7m by the way)

Resistance (ohms) = length (m) * resistivity (ohm m) / area (m2) Units = m * ohm m / m2 = ohms

This works for all equation with physical quantites. It helps because now you know:

1) Whether you should use the length in m, cm, mm the volume of water in litres or cm3 and so on. If x in the equation is in g cm-1, then to obtain a result in g you must multiply x by a quantity in cm. If a concentration in chemistry is given in moles per litre, to obtain the number of moles present, you must multiply by the volume in litres (a litre is 1000cm3, so if the question says 25cm3 you have 25/1000 litres). 2) To know instantly that a quantity must go on the top or the bottom of an equation. In this case we knew that area must go on the bottom because of the units of resistivity (which remember is given in an exam!)

Practise this by looking at simple equations such as

speed (m/s) = distance (m) / time (s)

Hope this helps!

Andrew