If amps and volts can vary independently, how does Ohm's law actually work?

This is a very basic question as I'm just starting to get into things, but I've hit a roadblock with comprehending the definition of voltage as it relates to the relationship defined in Ohm's law.

It might be that I'm misunderstanding the definition of voltage. As it says "voltage is the work needed per unit of charge to move a test charge between two points." My confusion centers around the idea of 'per unit of charge', which suggests to me that amps and volts can vary independently (that is one 'unit' of charge may require different quantities of work to move between two points not related to resistance).

If that's the case, how can I = V/R be a thing? That suggests that current is precisely and deterministically derivable from voltage and does not vary independently. I guess I'm just not really understanding how current actually varies - whether or not it's a property, say, of the conductive medium (copper has 'more available charge' to move, or some such) vs. current actually being directly related to voltage and does not vary independently and I've totally misunderstood the definition of voltage.

I'm hoping I can understand this and move on to more interesting things, but I'm just kept awake at night not really understanding this. My tl;dr question is really summarized in the title.

• I haven't checked their hypothesis, but it would appear to me that this implies that if you divide a joule (work) by a coulomb (unit of charge) you get a volt, which is a useful little factoid. If you were to think of voltage as performing similarly to pressure in pipes that are already completely full, current as being the rate of flow of the fluid(electrons) and resistance representing things that get in the way of the flow (constrictions for example), you would not be too far wrong, and that formula (Ohm's Law) will make more sense. – K H Dec 26 '18 at 7:31
• It's the 'that are already completely full' thing that I guess clarifies my question in the context of that metaphor. That I guess implies that there is nothing that varies current in a conductor that isn't resistance and voltage. That helps, thanks. – Sam Dec 26 '18 at 7:50
• The resistor does not care if I or V is the independent variable. It just makes sure that the equation is true: V=I*R. Some sources, like solar panels, are neither fixed voltage nor fixed current. If you plot the V vs I curve for a solar panel and plot the V vs I curve for a resistor, the place where they cross is the voltage and current (operating point) that you will observe when you connect them together. – mkeith Dec 26 '18 at 7:50
• See my answer for more, but once you've learned the basics, you have to move on to impedance, which is a more complicated model for resistance that takes into effect things like capacitors and inductors that impede or encourage the flow of electricity by temporarily storing energy, rather than burning it off as heat the way that resistors do. – K H Dec 26 '18 at 7:52
• Are you looking for a deeper understanding from physics? Or just an electric/electronic view? For example, one volt between two plates in outer space will impart one Joule of energy to one coulomb of charge regardless of the distance between the plates or the mass of the charge. That is basic physics but less often taught as electronics. Bringing up Ohms's law makes me unsure where you want to go. (Note I didn't mention a resistor.) – jonk Dec 26 '18 at 8:51

The short answer is that, if you are varying I and V independently, you must have some means of varying R.

For a fixed R, you understand fine : as you increase V, I will increase dependent on V, at a rate which is exactly the resistance R. Lock in that understanding, it is important.

But that's a special case.

You often want to vary the power level from a fixed voltage - that means varying the current, which generally means varying the resistance. And there are various ways of doing that.

A crude way is to vary the number of loads on a circuit- switch on additional lamps on a lighting circuit, or heating elements in a heater.

Another way is to switch the resistor on or off relatively fast, so its resistance alternates between R, and infinity, to reduce the average current. Examples are thermostatically controlled heaters (or the simmerstat controls in ovens) and PWM "pulse width modulation" used for dimming lights and some motor speed controls. These can only reduce power by increasing the effective "R" - once R is connected all the time ("100% duty cycle") you are at full power.

(Motors are a bit more complicated than that. because motors are also generators. Motor control will keep for another day)

There are also resistances which are variable in themselves - either as a function of temperature (which is a function of the power dissipated in them) or as a function of the voltage across them, or some other effect (like light falling on them). Ohm's Law still applies ( V = I * R ) but R is no longer a constant, and the equation may not be a linear one.

Some of these devices are semiconductors; but consider a simple incandescent lamp bulb first. As its filament wire heats up, its resistance increases, and is more than 10x as high at full power than it is when cold.

Let's talk about ideal resistors only. Real resistors are reasonably close to ideal, but I don't want to get into that. It is the resistor that enforces Ohm's law (V=IR). This law is the essence of being a resistor. Voltage or current sources can set the voltage or current, but the resistor always enforces Ohm's law.

Using the framework you have outlined, you could say that the resistor decides how much work is required to move a charge through it, and that amount of work depends on how many charges are already moving through it. If you can do the work (raise the voltage) you can move more charges (increase the current). It is similar to pumping a fluid through a restrictor plate (if that doesn't help you just ignore it).

So real world power supplies (such as solar panels) may not readily conform to an ideal voltage source or an ideal current source (they are a little of both). But if they are connected to a resistor, the voltage and current will always obey Ohm's law. If the voltage goes up, the current must go up also. And vice-versa.

Mathematically, a resistor is nothing more and nothing less than a two terminal element that enforces Ohm's law. Mathematically this all works out just fine.

In the real world, resistors may have a temperature coefficient. As they get hotter or colder, the resistance may change a bit. They also have power limits. If you dissipate too much power in a resistor, it may fail (melt, crack, blow up, whatever). Likewise, some resistors may have a bit of inductance as well as resistance. For DC or low frequencies that might not matter, but at high frequencies it can. There are other non-idealities, but I don't think that is what you are asking about so I will just leave it at that.

• So it's the case that resistance itself determines current given a certain voltage? IE it is not dependent on the 'charge quantity' determined by the current source, thus 'resistance' is as you said the enforcer of Ohm's law? If that's right then I think I'm starting to get it, hopefully. I think I'm failing to correctly articulate my confusion because I don't have a well developed vocabulary for this stuff, but it seems like I was confused by the idea of a current source 'quantity' having some real impact on the values in Ohm's law. – Sam Dec 26 '18 at 10:09
• Another way to look at it. All the elements in an electric circuit, the voltages, the current sources, resistors etc, all of them have a say in what the final currents and voltages will be. A power supply cannot set the voltage and current simultaneously into a resistor (unless, by chance it is exactly the right voltage and current to satisfy Ohm's law) because the resistor will not allow it. – mkeith Dec 26 '18 at 18:35

George Ohm was in the Cavendish Lab (in Britain) about 1800 (true story) and had permission to use the Lab's Volta Piles.

Using these Piles (relatively constant voltage sources), he experimented with various lengths of various conductors, using a current meter of some sort.

Given the Piles were producing constant voltage, guess what Mr Ohm measured?

The current was inverse to the resistance, the length of the various wires.

Mr Ohm concluded V = I * R

It might be that I'm misunderstanding the definition of voltage. As it says "voltage is the work needed per unit of charge to move a test charge between two points."

I haven't checked their hypothesis, but it would appear to me that this implies that if you divide a joule (work) by a coulomb (unit of charge) you get a volt, which is a useful little factoid.

My confusion centers around the idea of 'per unit of charge', which suggests to me that amps and volts can vary independently (that is one 'unit' of charge may require different quantities of work to move between two points not related to resistance).

The per unit of charge just defines the relationship between the voltage(pressure) and the amount of material(units of charge) being moved in a test.

If that's the case, how can I = V/R be a thing? That suggests that current is precisely and deterministically derivable from voltage and does not vary independently.

Ohm's Law describes the interaction between voltage, resistance and current. You may think of voltage as being pressure, current the rate of flow, and resistance a measurement of constriction to the flow. If you apply the same pressure to a large pipe and a small pipe, more fluid will flow through the large pipe. The same applies to a large wire (low resistance) and a small wire (high resistance). This is resistance. If you apply a voltage to a large and a small resistance, more current will flow through the small resistance than the large one, though it will flow through both. If you know any two of the three values, Ohm's law will allow you to calculate the other. When you learn about RLC circuits and AC power, things become a bit more complicated, but when simply choosing and matching devices or finding replacements, Ohm's law comes in handy a lot.

Resistance of pure resistors varies by material, temperature, cross sectional size, and in the case of AC skin effect, by even more complicated geometry than size.

Silver is the best elemental conductor but some people think it's too expensive for power transmission, so a lot of copper and aluminum get used instead. Many other conductors are used for different applications (temperature, corrosion resistance, etc) and the resistance of each material affects its viability for each application. Copper is most commonly used in housing to minimise maintenance, but aluminum sees some use there and proceedingly more as conductor size increases, due to lower weight and cost. Many metals and conductive plastics are used as resistors for anything from the tiny circuits on small electronics to the large heating elements on a traditional electric stove. Gold sees use in platings (often extremely thin) for corrosion resistance and I believe also sees use for microscopic wires due to its ductility.

As a general rule, for reasonable DC voltages, if you double the cross sectional area of a wire, you halve the resistance and double current flow as a result.

Corrections by @G36, @Solar Mike =)

• Gold is not the best conductor, the best conductors are silver, copper, gold and aluminum. – G36 Dec 26 '18 at 7:54
• Gold as in gokd plating is used because it does not tarnish or corrode unlike other metals... – Solar Mike Dec 26 '18 at 8:07