electrical resistance - Science-Education-Research

Higher resistance means less current for the same voltage – but how does that relate to the formula?

The higher resistance is when there is less current flowing around the circuit when you have the same voltage – but how does that relate to the formula?

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" and told me that "Resistance is current over, voltage, I think" although he did not think he could remember formulae. He thought that an ohm was the unit that resistance is measured in, which he suggested "comes from ohm's law which is the…formula that gives you resistance".

Two alternative conceptions

There were two apparent alternative conceptions there. One was that 'Resistance is current over voltage', but as Adrian believed that he was not good at remembering formulae, this would be a conception to which he did not have a high level of commitment. Indeed, on another occasion perhaps he would have offered a different relationship between R, I, and V. I felt that if Adrian had a decent feel for the concepts of electrical resistance, current and voltage then he should be able to appreciate that 'resistance is current over voltage' did not reflect the correct relationship. Adrian was not confident about formulae, but with some suitable leading questioning he might be able to think this through. I describe my attempts to offer this 'scaffolding' below.

The other alternative conception was to conflate two things that were conceptually different: the defining equation for resistance (that R=V/I, by definition so must be true) and Ohm's law that suggests for certain materials under certain conditions, V/I will be found to be constant (that is an empirical relationship that is only true in certain cases). (This is discussed in another post: When is V=IR the formula for Ohm’s law?)

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.

Linking qualitative understanding of relating concepts and the mathematical formula

As Adrian considered resistance as slowing down or stopping current I thought he might be able to rationalise how a higher resistance would lead to less current for a particular potential difference ('voltage').

Okay. Let’s say we had, erm, two circuits, and they both have resistance and you wanted to get one amp of current to flow through the circuits, and you had a variable power supply.
Okay.
And the first circuit in order to get one (amp) of current to flow through the circuit.
Yes.
You have to adjust the power supply, until you had 10 volts.
Okay.
So it took 10 volts to get one amp to flow through the circuit. And the second (unclear) the circuit, when you got up to 10 volts, (there is) still a lot less than one amp flowing. You can turn it up to 25 volts, and only when it got to 25 volts did you get one amp to flow through the circuit.
Yes, okay.

In mathematical terms, the resistance of the first circuit is (R = V/I = 10/1 =) 10Ω, and the second is (25/1 =) 25Ω, so the second – the one that requires greater potential difference to drive the same current, has more resistance.

Do you think those two circuits would have resistance?
Erm, (pause, three seconds) Probably yeah.

This was not very convincing, as it should have been clear that as an infinite current was not produced there must be some resistance. However, I continued:

Same resistance?
No because they are not the same circuit, but – it would depend what components you had in your circuit, if you had different resistors in your circuit.
Yeah, I've got different resistors in these two circuits.
Then yes each would have a different resistance.
Can you tell me which one had the bigger resistance? Or can’t you tell me?
No, I can’t do that.
You can’t do it?
No I don’t think so. No.

Adrian's first response, that the circuits would 'probably' have resistance, seemed a little lacking in conviction. His subsequent responses suggested that although he knew there was a formula he did not seem to recognise that if different p.d.s were required to give the same current, this must suggest there was different resistance. Rather he argued from a common sense position that different circuits would be likely to have different components which would lead to them having different resistances. This was a weaker argument, as in principle two different circuits could have the same resistance.

We might say Adrian was applying a reasonable heuristic principle: a rule of thumb to use when definite information was not available: if two circuits have different components, then they likely they have different resistance. But this was not a definitive argument. Here, then, Adrian seemed to be applying general practical knowledge of circuits, but he was not displaying a qualitative feel for what resistance in a circuit was about in term of p.d. and current.

I shifted my approach from discussing different voltages needed to produce the same current, to asking about circuits where the same potential difference would lead to different current flowing:

Okay, let me, let me think of doing it a different way. For the same two circuits, erm, but you got one let's say for example it’s got 10 volts across it to get an amp to flow.
Yeah. So yes okay so the power supply is 10 volts.
Yeah. And the other one also set on 10 volts,
Okay.
but we don’t get an amp flow, we only get about point 4 [0.4] of an amp, something like that, to flow.
Yeah, yeah.
Any idea which has got the high resistance now?
The second would have the higher resistance.
Why do you say that?
Because less erm – There’s less current amps flowing around the circuit erm when you have the same voltage being put into each circuit.
Okay?
Yes.

This time Adrian adopted the kind of logic one would hope a physics student would apply. It was possible that this outcome was less about the different format of the two questions, and simply that Adrian had had time to adjust to thinking about how resistance might be linked to current and voltage. [It is also possible too much information was packed close together in the first attempt, challenging Adrian's working memory capacity, whereas the second attempt fed the information in a way Adrian could better manage.]

You seem pretty sure about that, does that make sense to you?
Yes, it makes sense when you put it like that.
Right, but when I had it the other way, the same current through both, and one required 10 volts and one required 25 volts to get the same current.
Yes.
You did not seem to be too convinced about that way of looking at it.
No. I suppose I have just thought about it more.

Having made progress with the fixed p.d. example, I set Adrian another with constant current:

Yes. So if I get you a different example like that then…let’s say we have two different circuits and they both had a tenth of an amp flowing,
Okay. Yes.
and one of them had 1.5 volt power supply
Okay yes.
and the other one had a two volt power supply
Yeah.
but they have both got point one [0.1] of an amp flowing. Which one has got the high resistance?
Currents the same, I would say they have got different voltages, yeah, so erm (pause, c.6s) probably the (pause, c.2s) the second one. Yeah.
Because?
Because there is more voltage being put in, if you like, to the circuit, and you are getting less current flowing in and therefore resistance must be more to stop the rest of that.
Yes?
I think so, yes.
Does that make sense to you?
Yeah.

So this time, having successfully thought through a constant p.d. example, Adrian successfully worked out that a circuit that needed more p.d. to drive a certain level of current had greater resistance (here 2.0/0.1 = 20Ω) than one that needed a smaller p.d. (i.e. 1.5/0.1 = 15Ω). However, his language revealed a lack of fluency in using the concepts of electricity. He referred to voltage being "put in" to the circuits rather than across them. Perhaps more significantly he referred to their being "less current flowing in" where there was the same current in both hypothetical circuits. It would have been more appropriate to think of there being proportionally less current. He also referred to the greater resistance stopping "the rest" of the current, which seemed to reflect his earlier suggestion that resistance is how much something is being slowed down or is stopped going round.

My purpose in offering Adrian hypothetical examples, each a little 'thought experiment', was to see if they allowed him to reconstruct the formula he could not confidently recall. As he had now established that

greater p.d. is needed when resistance is higher (for a fixed current)

and that

less current flows when resistance is higher (for a fixed p.d.)

he might (perhaps should) have been able to recognise that his suggestion that "resistance is current over, voltage" was inconsistent with these relationships.

Okay and how does that relate to the formula you were just telling me before?
Erm, No idea.
No idea?
Erm (pause, c.2s) once you know the resistance of a circuit you can work out, or once you know any of the, two of the components you can work out, the other one, so.
Yeah, providing you know the equation, when you know which way round the equation is.
Yes providing you can remember the equation.
So can you relate the equation to the explanations you have just given me about which would have the higher resistance?
So if something has got a higher resistance, so (pause, c.2s) so the current flowing round it would be – the resistance times the voltage (pause, c.2s) Is that right? No?
Erm, so the current is resistance time voltage? Are you sure?
No.

So Adrian suggested the formula was "the current flowing round it would be the resistance times the voltage", i.e., I = R × V (rather than I = V /R ), which did not reflect the qualitative relationships he had been telling me about. I had one more attempt at leading him through the logic that might have allowed him to deduce the general form of the formula.

Go back to thinking in terms of resistance.
Okay.
So you reckoned you can work out the resistance in terms of the current and the voltage?
Yes, I think.
Okay, now if we keep, if we keep the voltage the same and we get different currents,
Yes.
Which has, Which has got the higher resistance, the one with more current or the one with less current?
Erm (Pause, c.6s) So, so, if they keep the same voltage.
That’s the way we liked it the first time so.
Okay.
Let’s say we have got the same voltage across two circuits.
Yes.
Different amounts of current.
Yes.
Which one’s got the higher resistance? The one with more current or the one with less current?
The one with less current.
So less current means it must be more resistance?
Yes.
Ok, so if we had to have an equation R=.
Yes.
What’s it going to be, do you think?
Erm
(pause, c.7s)
R=
(pause, c.3s)
I don’t know. It's too hard.

Whether it really was too hard for Adrian, or simply something he lacked confidence to do, or something he found too difficult being put 'on the spot' in an interview, is difficult to say. However it seems fair to suggest that the kind of shift between qualitative relationships and algebraic representation – that is ubiquitous in studying physics at this level – did not come readily to this advanced level physics student.

I had expected my use of leading (Socratic) questioning would provide a 'scaffold' to help Adrian appreciate he had misremembered "resistance is current over, voltage, I think", and was somewhat disappointed that I had failed.

Resistance is how much something is being slowed down

"Resistance is how much something is being slowed down or is stopped going round"

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". However, when asked about the formula, he suggested, without conviction, that "resistance is current over voltage". So, I asked him how he might go about explaining resistance to a younger student:

We will come back to the formula in a minute then, so let us say you had a younger brother or sister who hasn’t done much physics.
Yes.
And doesn’t do, doesn’t like maths, doesn’t like formulas.
Okay.
So what does it mean though? Why is it important? What’s resistance about?
Erm – I would say it was how much something is being slowed down, or erm how much it is being stopped going round. If it is in electric¬… electricity then it is in a circuit. If it’s in like the wide open range of things it's like erm how resistant is something if you push it? How much force does it give back?

So Adrian was aware of electrical resistance, and also aware of resistance in the context of mechanics.

Oh I see, so, erm if I asked you to push that table over there
Yes.
There might be resistance to that?
Yes.
And that’s different to if we were talking about meters and wires and things?
Yes.
Are they similar in some way?
They have got the same name. {laughs}
Got the same name, okay.
They probably are similar. I've never really thought about it.

So although Adrian associated electrical resistance with 'resistance' in mechanical situations, the similarity between the two types of resistance seemed primarily due to the use of the same linguistic label. This was despite him describing the two forms of resistance in similar terms – "how much something is being slowed down… how much it is being stopped going round" cf. "how resistant is something if you push it".

To a physicist, a property such as resistance should be defined precisely, and therefore preferably mathematically – and so operationally in the sense that there is no ambiguity in how it would be measured. However when students are learning, definitions and formulae may be abstract and have little meaning or connection to experience, so qualitative understanding is important. Students' initial suggestions of what technical terms mean when they first learn about them may be vague and flawed, but if this is linked to a feeling for the concept this may ultimately be a better starting point than a formula which cannot be interpreted meaningfully – as seemed to be the case with Adrian.

Arguably, understanding a relationship in qualitative terms can support later formalising the relationship in mathematical terms, whereas trying to learn a formulae by rote may lead to misremembering and algorithmic application (and so, for example, not noticing when non-feasible results are calculated).

Adrian's suggestion that resistance might be"how resistant is something if you push it? How much force does it give back?" presumably linked to his own experiences of pushing and pulling objects around. However, it seemed to confuse notions of inertia and reaction force (as well as possibly frictional forces). If Adrian were to push with a force of 100N on the wall of a building, a puck on an ice rink, or on a sledge on gravel the reaction force would be 100N in each case (cf. Newton's third law) – although the subjective experience of resistance would be very different in the different situations – as would the outcome on the object pushed.

In these situations it may be difficult for a teacher to know if a vague or confused description reflects conceptual confusion (and/)or limited expression. Yet, students need time and opportunities to be able to explore concepts in their own terms to link the abstract scientific ideas with the 'spontaneous conceptions' they have developed based on their own experiences of acting in the world.

The teacher should offer feedback, and model clear language, but needs to recognise that understanding abstract scientific ideas takes time. After all, Aristotle would be considered to have alternative conceptions of mechanics by comparison with today's science, but Aristotle was clearly highly intelligent and gave the matter a lot of thought!

After this there was extended discussion on the way resistance related to current and voltage, following Arian's comment that resistance is current over voltage. As part of this he was asked about ⚗︎ an example where different voltages were needed in different circuits to allow the same current to flow. ⚗︎ He suggested that the circuit with the higher resistance would be the one where "there is more voltage being put in, if you like, to the circuit, and you are getting less current flowing in, and therefore resistance must be more to stop the rest of that".

Adrian's way of talking about the current in the circuits did not seem to reflect a view of current as driven by a given p.d. across a circuit and limited by a certain resistance, but almost as a fixed potential flow, some of which would be permitted to pass, but some of which would be stopped by the resistance ("how much it is being stopped going round", "resistance … to stop the rest of that"). Yet, as suggested above, it can take time, and opportunities for exploration and discussion, for students' concepts and ways of talking about them to mature towards canonical science.

That Adrian could talk of "more voltage…less current…therefore resistance must be more" seemed promising, although ⚗︎ Adrian could not relate his qualitative description to the mathematical representation of the formula. ⚗︎

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

"Resistance is current over voltage, I think"

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.

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.

Current only slows down at the resistor

Current only slows down at the resistor – by analogy with water flow

Keith S. Taber

Students commonly think that resistance in a circuit has local effects, and in part that is because forming a mental model of what is going on in circuits is very difficult. Often models and analogies can be useful. However when an analogy is used in teaching there is also the potential for it to mislead.

Amy was a participant in the Understanding Science Project. Amy (when in Y10) told me she had been taught to use a water flow analogy for electric current. However, because her visualisation of what happens in water circuits was incorrect, she used the analogy to inform an alternative conception about circuits:

Do you have any kind of imagined sort of idea, any little mental models, about what (the flow of electricity round the circuit) might look like? Do you have a way of imagining that?
Erm, yeah, we've been taught the water tank and pipe running round it. … just imagine the water like flowing through a pipe, and obviously like, if the pipe becomes smaller a one point, erm, the water flow has to slow down, and that's meant to represent the resistance of something.
So, so if I had my water, er, tank and I had a series of pipes, they'd be water flowing through the pipes, and if I had a narrower pipe at one point, what happens then?
The water would have to slow down.
So would it slow down just as it goes through the narrow pipe, or would it slow down all the way round?
Erm – just through that part.

(Amy does not appreciate the implications of conservation of mass {that is, the continuity principle} here – at steady state there cannot be a greater mass flow at different points in the circuit).

…
And so how do you imagine that's got to do with resistance, how does that help you understand resistance?
…well resistance, it slows the current down, but then erm, once it passes a resistor or something it, the current is free to flow through the wire again

Analogies can be very useful teaching tools, but when using them it is important to check that the students already understand the features of the analogue that are meant to be helpful. It is also important to ensure that they understand which features are meant to be mapped onto the target system they are learning about, and which are not relevant.

Analogies are only useful when the learner has a good understand of the analogue. In this case, as Amy did not appreciate that the water flow throughout the system would be limited by the constriction, she could not use that as a useful analogy for why a resistor influences current flow at all points in a series circuit. This is an example of where a teaching model meant to support learning, which actually misleads the learner. That is, for Amy, with her flawed understanding of fluid flow, the teaching model acted as a pedagogic learning impediment – a type of grounded learning impediment.

Electrical resistance depends upon density

Keith S. Taber

Amy was a participant in the Understanding Science project.

Amy (Y10) suggested that a circuit was "a thing containing wires and components which electricity can pass through…it has to contain a battery as well". She thought that electricity could pass through "most things".

For Amy "resistance is anything which kind of provides a barrier that, which the current has to pass through, slowing down the current in a circuit", and she thought about this in terms of the analogy with water in pipes: "we've been taught the water tank and pipe running round it… just imagine the water like flowing through a pipe, and obviously like, if the pipe becomes smaller at one point, erm, the water flow has to slow down, and that's meant to represent the resistance of something".

So for Amy, charge flow was impeded by physical barriers effectively blocking its way. She made the logical association with the density of a material, on the basis that a material with densely packed particles would have limited space for the charge to flow:

So electricity would "not very easily" pass through a wooden bench "because wood is quite a dense material and the particles in it are quite closely bonded".

In air, however, the particles were "not as dense as a solid". When asked if that meant that electricity can pass through air quite easily, Amy replied: "yeah, I think so".

Amy's connection between the density of particles and the ease with which charge could flow is a logical one, but unfortunately involves a misunderstanding of how charge flows through materials, i.e., from a canonical scientific perspective, thinking about the charge flowing through gaps between particles is unhelpful here. (So this can be considered an alternative conception.) This seems to be a creative associative learning impediment, where prior learning (here, the spacing of quanticles in different materials) is applied, but in a context beyond its range of application.