overgeneralisation – Science-Education-Research

When is V=IR the formula for Ohm's law?

"Resistance is current over voltage, I think"

Adrian was a participant in the Understanding Science Project. When I interviewed him in Y12 when he was studying Advanced level physics he told me that "We have looked at resistance and conductance and the formulas that go with them". So I asked him was resistance was:

So what exactly is resistance?
Resistance is, erm (pause, c.3s) Resistance is current over, voltage, I think. (Pause, c.3s) Yeah. No.
Not sure?
I can’t remember formulas.

So Adrian's first impulse was to define resistance using a formula, although he did not feel he was able to remember formulae. He correctly knew that the formula involved resistance, current and voltage, but could not recall the relationship. Of course if he understood qualitatively how these influenced each other, then he should have been able to work out which way the formula had to go, as the formula represents the relationship between resistance, voltage and current.

So, I then proceeded to ask Adrian how he would explain resistance to a younger person, and he suggested that resistance is how much something is being slowed down or is stopped going round. After we had talked about that for a while, I brought the discussion back to the formula and the relationship between R, V and I:

And what about this resistance in electricity then, do you measure that in some kind of unit?
Yes, in, erm, (pause, c.2s) In ohms.
So what is an ohm?
Erm, an ohm is, the unit that resistance is measured in.
Fair enough.
It comes from ohm's law which is the, erm, formula that gives you resistance.

V=IR is the formula that gives you resistance, but it is a common misconception, that Ohm's law is V=IR.

Actually, Ohm's law suggests that the current through a metallic conductor (kept at constant conditions, e.g., temperature) is directly proportional to the potential difference across its ends.

So, in such a case (a metal that is not changing temperature, etc.)

I ∝ V

which is equivalent to

V ∝ I

which is equivalent to

V = kI

where k is a constant of proportionality. If we use the symbol R for the constant in this case then

V= RI

which is equivalent to

V = IR

So, it may seem I have just contradicted myself, as I denied that V=IR was a representation of Ohm's law, yet seem to have derived V=IR from the law.

There is no contradiction as long as we keep in mind what the symbols are representing in the equation. I represented the current flowing through a metallic conductor being held at constant conditions (temperature, tension etc.), and V represented the potential difference across the ends of that metallic conductor. If we restrict V and I to this meaning then the formula could be seen as a way of representing Ohm's law.

Over-generalising

However, that is not how we usually understand these symbols in electrical work: V generally represents the potential difference across some resistive component or other, and I represents the current flowing through that component: a resistor, a graphite rod, a salt bridge, a diode, a tungsten filament in a lamp…

In this general case

V = IR

R = V/I

is the defining equation for resistance. If R is defined as V/I then it will always be the case, not because there is a physical law that suggests this, but simply because that is the meaning we have given to R.

This is a bit like bachelors being unmarried men (an example that seems to be a favourite of some philosophers): bachelors are not unmarried men because there is some rule or law decreeing that bachelors are not able to get married, but simply because of our definition. A bachelor who gets married and so becomes a married man ceases to be a bachelor at the moment they become a married man – in a similar way to how a butterfly is no longer a caterpillar. Not because of some law of nature, but by our conventions regarding how words are used. If V and I are going to be used as general symbols (and not restricted to our carefully controlled metallic conductor) then V = IR simply because R is defined as V/I and the formula, used in the general case, should not be confused with Ohm's law.

In Ohm's law, V=IR where R will be constant.

In general, V=IR and R will vary, as Ohm's law does not generally apply.

It would perhaps be better for helping students see this had there been a convention that the p.d. across, and the current through, a piece of metal being kept in constant conditions were represented by, say V_ⓜ and I_ⓜ, so Ohm's law could be represented as, say

V_ⓜ = k I_ⓜ

but, as this is not the usual convention, students need to keep in mind when they are dealing with the special case to which Ohm's law refers.

A flawed teaching model?

The interesting question is whether:

teachers are being very careful to make this distinction, but students still get confused;
teachers are using language carefully, but not making the discrimination explicit for students, so they miss the distinction;
some teachers are actually teaching that V=IR is Ohm's law.

If the latter option is the case , then it would be good to know if the teachers teaching this:

have the alternative conception themselves;
appreciate the distinction, but think it does not matter;
consider identifying the general formula V=IR with Ohm's law is a suitable simplification, a kind of teaching model, suitable for students who are not ready to have the distinction explained.

It would be useful to know the answers to these questions, not to blame teachers, but because we need to diagnose a problem to suggest the best cure.

A wooden table is solid…or is it?

Keith S. Taber

Wood (cork coaster captured with Veho Discovery USB microscope)

Bill was a participant in the Understanding Science Project. Bill (Y7) was explaining that he had been learning about the states of matter, and introduced the notion of there being particles:

So how do you know if something is a solid, a liquid or a gas?
Well, solids they stay same shape and their particles only move a tiny bit
So what are these particles then?
Erm, they're the bits that make it what it is, I think.
Ah. So are there any solids round here?:
Yeah, this table. [The wooden table Bill was sitting at.]
That's a solid, is it?:
Yeah

Technically the terms solid, liquid and gas refer to samples of substances and not objects. From a chemical perspective a table is not solid. A wooden table (such as those in the school laboratory where I talked to Bill) is made of a complex composite material that includes various different substances such as a lignin and cellulose in its structure.

Wood contains some water, and has air pockets, so technically wood is not a solid to a chemist. However, in everyday life we do thing of objects such as tables as being solid.

Yet if wood is heated, water can be driven off. Timber can be mostly water by weight, and is 'seasoned' to remove much of the water content before being used as a construction material. Under the microscope the complex structure of woods can be seen, including spaces containing air.

Bill also suggested that a living plant should be considered a solid.

I think teaching may be a problem here, as when the states of matter are taught it is often not made clear these distinctions only apply clearly to fairly pure samples of substances. In effect the teaching model is that materials occur as solids, liquids and gases – when a good many materials (emulsions, gels, aerosols, etc.) do not fit this model at all well.

Particles are further apart in water than ice

Keith S. Taber

Bill was a participant in the Understanding Science Project. Bill, a Y7 student, explained what he had learnt about particles in solids, liquids and gases. Bill introduced the idea of particles when talking about what he had learn about the states of matter.

Well there's three groups, solids, liquids and gases.

So how do you know if something is a solid, a liquid or a gas?

Well, solids they stay same shape and their particles only move a tiny bit.

This point was followed up later in the interview.

So, you said that solids contain particles,

Yeah.

They don't move very much?

No.

And you've told me that ice is a solid?

Yeah.

So if I put those two things together, that tells me that ice should contain particles?

Yeah.

Yeah, and you said that liquids contain particles? Did you say they move, what did you say about the particles in liquids?

Er, they're quite, they're further apart, than the ones in erm solids, so they erm, they try and take the shape, they move away, but the volume of the water doesn't change. It just moves.

Bill reports that the particles in liquids are "further apart, than the ones in … solids". This is generally true* when comparing the same substance, but this is something that tends to be exaggerated in the basic teaching model often used in school science. Often figures in basic school texts show much greater spacing between the particles in a liquid than in the solid phase of the same material. This misrepresents the modest difference generally found in practice – as can be appreciated from the observations that volume increases on melting are usually modest.

Although generally the particles in a liquid are considered further apart than in the corresponding solid*, this need not always be so.

Indeed it is not so for water – so ice floats in cold water. The partial disruption of the hydrogen bonds in the solid that occurs on melting allows water molecules to be, on average, closer* in the liquid phase at 0˚C.

As ice float in water, it must have a lower density. If the density of some water is higher than that of the ice from which it was produced on melting then (as the mass will not have changed) the volume must have decreased. As the number of water molecules has not changed, then the smaller volume means the particles are on average taking up less space: that is, they seem to be closer together in the water, not further apart*.

Bill was no doubt aware that ice floats in water, but if Bill appreciated the relationship of mass and volume (i.e., density) and how relative density determines whether floatation occurs, he does not seem to have related this to his account here.

That is, he may have had the necessary elements of knowledge to appreciate that as ice floats, the particles in ice were not closer together than they were in water, but had not coordinated these discrete components to from an integrated conceptual framework.

Perhaps this is not surprising when we consider what the explanation would involve:

Coordinating concepts to understand the implication of ice floating

Not only do a range of ideas have to be coordinated, but these involve directly observable phenomena (floating), and abstract concepts (such as density), and conjectured nonobservable submicroscopic/nanoscopic level entities.

* A difficulty for teachers is that the entities being discussed as 'particles', often molecules, are not like familiar particles that have a definitive volume, and a clear surface. Rather these 'particles' (or quanticles) are fuzzy blobs of fields where the field intensity drops off gradually, and there is no surface as such.

Therefore, these quantiles do not actually have definite volumes, in the way a marble or snooker ball has a clear surface and a definite volume. These fields interact with the fields of other quanticles around them (that is, they form 'bonds' – such as dipole-dipole interactions), so in condensed phases (solids and liquids) there are really not any discrete particles with gaps between them. So, it is questionable whether we should describe the particles being further apart in a liquid, rather than just taking up a little more space.