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

or

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.

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