The division of one dimensional quantity by another with a different dimension produces a new dimensional quantity which is interpretable as a rate of the first by the second.

It is no coincidence that dividing the voltage difference in a circuit over the resistance of a resistor in that circuit produces a unit of current - this is because the resistor impedes the flow of current across it relative to the voltage which facilitates the flow of current. So you can think of V = IR as R = V/I; the scaling of a potential difference to the current flow.

The numerical equality of R and V/I is less important than the concepts which are represented by it; upon understanding why V=IR holds one will see that current can be measured in voltage difference over resistance felt.

You can use symbolic units, equivalence classes of dimensions, which show the relationships between the variables. For example, 'length', 'speed', 'time' are inter-definable, speed always has dimensions 'length per time'. Particularly common combinations of dimensional units are given names, like the ampere (which is charge flow per time), and there are often equivalent dimensions (like potential difference over resistance) which express the same unit in different contexts.

One can multiply and divide units of inequivalent dimensions, linking to rates through proportionality, but one cannot add inequivalent dimensions together. This is literally adding apples and oranges.

One must also be careful with functions, for example if $f(t) = a + bt$ is a length, $a$ must be a length and $b$ must be a speed. Different sides of equalities and inequalities must always have the same dimension, for 'how many apples are in an orange?' is a senseless question, whereas 'how many apples per orange are there on the table?' is not.

Hope that helps. You might enjoy reading about 'dimensional analysis', which is a label given to modelling techniques and mathematical tricks based on considerations of unit dimension alone.

How is it that you can divide 8 apples among two people but not 8 volts by 2 ohms? — Alan

For the same reason that you can divide 8 apples among two people but not 8 pears by 2 peaches?

For the same reason that you can divide 8 apples among two people but not 8 pears by 2 peaches? — Frank Apisa

Thanks for answering.
My issue is that mathematical division seems to be limited by its concepts, by the things that are to be divided and not only by mathematical rules. Dividing apples among people makes sense because people can actually do stuff with apples but peaches can't.

It is no coincidence that dividing the voltage difference in a circuit over the resistance of a resistor in that circuit produces a unit of current - this is because the resistor impedes the flow of current across it relative to the voltage which facilitates the flow of current. So you can think of V = IR as R = V/I; the scaling of a potential difference to the current flow. — fdrake

Thanks a lot for your isnightful answer. I'm still going through it. I imagine Georg Simon Ohm in his lab making measurements, when he increased voltage current increased proportionally but what determined the slope of this function, if he increased voltage linearly of course, was resistance. Do you think the same scheme could be applied to more concrete concepts? like saying: the amount of apples per person increases if more apples are to be given.

olts divided by ohms do not mean pretty much anything. — Alan

The way it was described to me when I was 14 was that volts is a measure of potential difference, caused by there being more electrons on one side than the other of an obstruction to electron movement. Current is movement of the electrons, like water finding a level, electrons flow through conductors until there is an equal distribution of them. Resistance is a measure of the amount of obstruction to current flow, and amperes are a measure of the amount of current. Typically the potential difference is maintained capacatively in direct-current applications.

To avoid the need for coefficients, the three measures are scaled to provide a direct multiplicative relationship, that is, volts are the direct mathematical product of amps and ohms. Also, the power, watts, is a product of volts and amps. So you only need to know two of the quantities to deduce the other two.

So to answer your question, they are different things with a different mathematical relationship. the equivalent to apples would be if you had 8 batteries, or 8 resistors, and divided them among two people.

If the load across an 8 volt battery is two 1-ohm resistors in series, the voltage drop across each resistor will be 4 volts.

For the same reason that you can divide 8 apples among two people but not 8 pears by 2 peaches? — Frank Apisa

Clearly you can - the answer is four pears per peach.

volts divided by ohms do not mean pretty much anything — Alan

They do. The usual way to ground electrical concepts in intuition is to treat potential difference - (PD) measured in volts - and current, measured in amps, as the fundamental quantities. Current can be thought of as number of electrons passing through a surface per second. PD can be thought of as an indication of the strength of electric field.

Resistance, measured in Ohms, is a derived concept, expressed as volts per amp. It tells you how much PD you have to apply between the ends of the resistor in order to pass a current of one amp through it.

Given that, PD divided by resistance has units that are volts divided by (volts per amp), which is just amps, measuring the intuitive notion of current.

In Physics problems, intuition about units is aided by carefully choosing which quantities to take as fundamental and which to treat as derived.

Oranges are granular and volts are a continuum.

I remember being fascinated by this in infants' school and have never ceased since.

The reappearance of both the granular and the continuum keeps recurring everywhere in nature, as I have been reading in Charles Seife's books.

I have a very basic conception of math.

I think the relationship is a ratio which looks, walks and quacks like everyday division. The fundamental assumption in ratios/divisions seems to a uniformity of some kind e.g. in your example we assume that each person will consume the same number of apples. From what I know this isn't problematic because we can use ratios that are able to handle non-uniformity like one person eating more/less than the other.

A classic case of division failing to give details is when calculating speed of a car on a trip. In reality the car travels at different speeds (pedestrians, traffic lights, other vehicles, etc) but this information is lost when you divide the total distance by the time taken and get average speed.

The volt/ohm is a ratio.

If someone knows the truth about this question kindly post it here for our benefit.

Thanks.

The volt/ohm is a ratio. — TheMadFool

Only if the load has multiple elements in series. If the load has parallel elements, each element enjoys the same voltage drop.

A common example of a parallel load is houses connected to a substation. Every house sees the same voltage. This wouldn't be true if the houses were connected in series.

The question has an answer which can be googled. This isn't even philosophy.

The parallel loads produced by connection in parallel as described, are a mechanical way of transforming from a continuum of quantities to integers, like using a knife to slice a long stretch of salami, or like when an island in a river causes a separate flow each side. This paradoxical part of the philosophy of mathematics has fascinated me since infancy. Apples are integers whereas the electric current, or a long stretch of salami, are a continuum of quantities, and we have discovered ways of treating the latter as if it were the former.

Nobody seems to have mentioned Ohm's Law which states that ...

Current flowing in a conductor is DIRECTLY PRORTIONAL to potential difference across its ends. Usually this is stated with p.d. as the subject
p.d ∝ current ....... p.d.= a constant x current ....... V= R x I

(NB With current as the subject the constant of proportionality would be 'conductance', the inverse of 'resistance')


The constant of proportionality was named after Ohm, and considered to be 'resistance'. Similarly 'current' and p.d. were named after other physicists, (Ampere and Volta)

There is NO physical theory involving a constant of proportionality relating 'apples' to 'persons' and that is the only basis of an answer to your question.

Nobody seems to have mentioned Ohm's Law — fresco

The OP did.

Sorry....give me the quote ? The 'law' is not the equation.

Second sentence of the OP. The OP assumes the reader knows Ohm's Law.

No it doesn't ! It implies that people know the derived equation. That is not 'Ohm's Law' which specifies the concept 'proportionality' and naming 'the constant' as a physical property. My reply points out that no such concept applies to 'apples and people', and that reply highlights the superficiality of 'the question'.

BTW, this 'not knowing' of Ohm's Law itself, is mirrored by those who might think one of Newton's laws was Force=Mass x Acceleration . But that is merely a derivation of the actual law...'The rate of change of momentum is proportional to the applied force'.

No it doesn't ! It implies that people know the derived equation. That is not 'Ohm's Law' which specifies the concept 'proportionality' and naming 'the constant' as a physical property. My reply points out that no such concept applies to 'apples and people', and that reply highlights the superficiality of 'the question'. — fresco

The equation V=IR expresses proportionality. Resistance is mentioned.

We can divide apples among people. If you examine the voltage drops in a circuit, it does appear that we're dividing voltage among the portions of the total load. Should we think of these two dividing processes in the same way?

Seems like a damn good question to me.

You still don't get it. See my edit.

You still don't get it — fresco

I guess I don't.

The question has an answer which can be googled. This isn't even philosophy. — S

I usually don't ask in forums if I can google things myself. I'm clearly not satisfied by the answers on google and thought that this would need more than math to be explained. Hope that helps.

Nobody seems to have mentioned Ohm's Law which states that ...

Current flowing in a conductor is DIRECTLY PRORTIONAL to potential difference across its ends. Usually this is stated with p.d. as the subject
p.d ∝ current ....... p.d.= a constant x current ....... V= R x I

(NB With current as the subject the constant of proportionality would be 'conductance', the inverse of 'resistance')


The constant of proportionality was named after Ohm, and considered to be 'resistance'. Similarly 'current' and p.d. were named after other physicists, (Ampere and Volta)

There is NO physical theory involving a constant of proportionality relating 'apples' to 'persons' and that is the only basis of an answer to your question. — fresco

Thanks for your answer Fresco. It's quite clear.

The analogy I often use is with water pipes. Voltage corresponds to the water pressure in the pipe, while amps corresponds with the rate of water flow (e.g. liters per second). Ohms would be sort of like the friction from the surface of the pipe (give me some fudge room here). Therefore, you are dividing the pressure by the friction, which determines rate that water flows through the pipe. The friction on the pipe does not impact the overall pressure, as long as our pump has enough capacity to keep up with the flow. Just like with the electrical model, the resistance does not impact the overall voltage, as long as your power source can keep up with the amps.

I usually don't ask in forums if I can google things myself. I'm clearly not satisfied by the answers on google and thought that this would need more than math to be explained. Hope that helps. — Alan

The answer I found online is that you can't convert volts to ohms since volt and ohm units do not measure the same quantity.

Maybe I don't understand why E=mc2, and am not satisfied with the answer, but is that philosophy? Maybe if I was more of a genius than Einstein it would come closer to being philosophy, but it would probably just mean that I need to learn more.

Why you should think what you said above is 'an answer' beats me ! :smile:

But you are correct about this 'not being philosophy'. Its about specific mathematical modelling.
There is a philosophical angle which could be developed (elsewhere) about the epistemological status of mathematical models in general, and the role played by 'analogy' with respect to that status.
However, this (contrived analogy) question hardly lends itself to such extrapolition.

Why you should think what you said above is 'an answer' beats me! :smile: — fresco

Why not?

You can't divide 8 volts by 2 ohms because you can't convert volts to ohms since volt and ohm units do not measure the same quantity.

You can divide 8 apples amongst 2 people by giving them each some of the 8 apples.

Faulty logic I'm afraid, and nothing to do with the answer. If I define the 'nerd' as the 'physical property' of apples per person, then you will get 2 nerds. Please refer to my 'constant of proportionality point ' above which defines a physical property called 'resistance' measured in 'ohms'.

A seemingly irrelevant reference to Ohm's law somehow answers the question. Okay then.

Many resistors are made of compacted carbon. If you could probe inside one, you'd see the resistance is divisible. 1/2 way through a 1.5 Kohm resistor is probably around 750 ohms.

You get that, right?

Ohm's Law is a model. It's the same model used in gas laws. We use the model and simultaneously see the world through the lens of the model. Voltage could also be considered a "constant." It's potential energy.

The answer to the OP is that we can divide voltage across 2 ohms.

A cook can turn 2 small meat loaves into one bigger one and reroute several currents of smaller power into one bigger one, but not turn two apples into one big one.

Your answer is like answering the question, "How long is a piece of string?", by answering that it's twice as long as half of its length. It tries to be clever by looking for a loophole, but just misses the point.