# What are the dependent and independent variables in Ohm's law?

I can't quite wrap my head around Ohm's law. The relationship itself is quite intuitive to me. What I don't understand is when a system has dynamic voltages, currents, and resistances. I don't quite understand which variables are dependent and which are independent. For example, one could take multiple 9V batteries, connect them, and get a larger potential difference between the positive and negative terminal. If we assume the resistance didn't change between the terminals, that would imply there must be a higher current between the terminals. This example leads me to believe that voltage and resistances and independent variables, and current is dependent on those two variables. But, if we had a circuit with multiple resisters in parallel, then there would be voltage drops between them, making voltage dependent on resistance. I appreciate all help.

• Are you familiar with Kirchoff's laws? – Marius Ladegård Meyer Jun 26 '20 at 12:20
• I suspect that your problem is not with choosing the dependent variable. You seem to be confused regarding how to apply Ohm's Law to series vs. parallel circuits. Is that correct? – David White Jun 26 '20 at 20:25

First off, Ohm's law is not the equation $$V = IR$$ alone. Instead, $$V = IR$$ is significant in at least two different ways, only one of which is properly called as "Ohm's law":

• One of these is that it is a definition of "resistance" as a physical quantity. In that case, it would perhaps better be written as $$R := \frac{V}{I}$$ . In this sense, the equation is analogous to the definition of capacitance: $$C := \frac{Q}{V}$$ The reason this is not a "law" is because a "law" in scientific parlance means a rule that describes an observed relationship between certain quantities or effects - basically, it's a . A definition, on the other hand, synthesizes a new quantity, so that the relationship is effectively trivial because it's created by fiat.
• The other, however, is what is properly called "Ohm's law", and it refers to a property of materials, the "law" being that they generally follow it: a material that behaves in accordance with Ohm's law (often only approximately) is called an "ohmic" material, and Ohm's law here says that the voltage-current relationship looks like $$V = IR$$ for a constant value of $$R$$. Note that in the definition sense, there is no reason at all that $$R$$ needs to be a constant. In this sense, though, Ohm's law should be understood perhaps as analogous to the idea of modelling friction in elementary mechanics by $$F_\mathrm{fric} = \mu F_N$$ giving a linear dependence between the friction and the normal force $$F_N$$ through the coefficient of friction $$\mu$$. (Once more, though, you can also take this as a definition of a CoF - the "law" part is in that $$\mu$$ is constant so the linear relationship holds.)

And so I presume that your question is asking about the first sense: if we consider $$V = IR$$ just a defining relation between three quantities, which one is the "dependent" and which is the "independent" quantity? The answer is that this is not a really good question given the parameters. The terms "dependent" and "independent" quantities are kind of an old-fashioned terminology from the less-rigorous earlier days of maths that keeps getting knocked around in not-so-great school texts, and relate to functions: if we have a function $$f$$ with one variable $$x$$, which in a fully modern understanding would be called the function's argument or input, then in the specific case where we bind (i.e. mandate it has the same value as) another variable $$y$$, to have the value of the function in question, so that $$y = f(x)$$ following the binding, then $$y$$ is called as the dependent variable, and $$x$$ the independent variable.

To see why that doesn't work so well in this case, note the logical structure of the above statement: the givens, argument, and conclusions. We are given a function $$f$$, then we create a binding between a variable $$y$$ and the value $$f(x)$$ of the function, then finally, we name the two. But in the case of "$$V = IR$$", we are simply giving this relationship; there is no "function" here of any type, much less being employed in this very specific manner.

(What do I mean by "binding"? Well, that's what the symbol $$:=$$ earlier means: to bind variable $$y$$ to some expression means that we are to declare that $$y$$ now can only be substituted for the expression given, and not something else, at least within a particular context. Writing $$y := \mathrm{(expr)}$$ means $$y$$ is bound to expression $$\mathrm{(expr)}$$.)

And this is also why I say it is "old-fashioned" from a modern point of view - in modern usage functions are far more general and flexible than they used to be, and a modern point of view is that an expression like

$$x + y > \cos(xy)$$

is in fact entirely built from functions: not only $$\cos$$ but the multiplication $$\cdot$$ (here suppressed in favor of juxtaposition) and addition $$+$$ but also interestingly, the symbol $$>$$ itself: that is a special kind of function called a "Boolean function" or a relation, which asserts that something is true or false about the arguments you put into it. When you say that an "equation holds", you mean the Boolean function $$=$$ evaluates to "True".

Likewise, in modern usage, the terminology of "dependent" and "independent" variables really is more at home in a scientific/empirical context: in conducting an experiment, the independent variable is the one we modify, while the dependent variable is the one we seek to analyze with regard to if and how it responds to changes in the independent variable. In the case of an experiment involving electric circuits, any of the three variables here may serve those roles (yes, even $$R$$ - think about swapping resistors, or using a variable resistor, and for $$R$$ as dependent variable, think about heating up a resistor with suitably high current, causing its resistance to change [i.e. behave non-ohmically]).

That said, if we are going to really insist on sticking to this regardless, I'd say that in most cases, we would want to say that the current is the dependent variable, the other two are independent variables. This is because we can typically control voltage and resistance much more easily, and we think of voltage as the "causative" element in the situation. Hence, in light of our previous discussion, we take $$I$$ to be a function of $$V$$ and $$R$$:

$$I(V, R) := \frac{V}{R}$$

and note that $$V = IR$$ then holds.

• Your suggestion that the terminology of "dependent" and "independent" variables really is more at home in a scientific/empirical context seems to be historically incorrect. The distinction seems to have originated in mathematical textbooks, such as "A course of mathematics" (1837): "In an equation of two variables, that which may be determined at pleasure is called the independent of primitive variable, and the other, whose value must afterwards be found in the given equation is called the dependent variable." link – jkien Jun 27 '20 at 12:23
• @jkien : Reworked it now. – The_Sympathizer Jun 27 '20 at 12:57

In school I "learnt" that Ohm's law consists of three equations \begin{align} U &= R \cdot I \tag1 \\ R &= U / I \tag2 \\ I &= U / R \tag3 \end{align} In the eq.(1) the independent variables are $$(R, I)$$, in eq.(2) the independent variables are $$(U, I)$$, and in eq.(3) the independent variables are $$(U, R)$$.

Once we learn how to manipulate relationships Ohm's law reduces to a single relationship -- each of the three equations will do. Each equation has two input variables (=know values, which are also called independent variables) and only one output variable (=unknown value, which is also called dependent variable). There exist no unique way to define dependent/independent variables, because these "names" depend on the used equation.

I think it's a matter of personal preference, or the situation at hand.

We usually think of resistances as fixed values for a device. Like a resistor. And voltage sources are more common than current sources. So in my head I tend to think of current as the dependent variable. But current sources do exist, and resistances can vary, so in some other situation I might think of voltage as dependent.

tl;dr Probably best to forget about independent-vs.-dependent variables in Ohm's law; they're all just variables. Generally, independent-vs.-dependent is a conceptual framework for dealing with incomplete models in experimental contexts before reference-classes are worked out; you don't need to worry about it when it comes to textbook physical relationships like Ohm's law.

# Forget independent-vs.-dependent variables.

The variables are just variables. That's it.

Independent-vs.-dependent is more of an experimental concept: independent variables are controlled and dependent variables are observed. Once the abstract-model, e.g. Ohm's law, is worked out, then we don't talk about independent-vs.-dependent variables anymore.

### Thought experiment: Sale at the grocery store.

A grocery store wonders how many new customers come into the store when they have a sale on apples. So, they do an experiment, then create a correlation that describes their results.

Turns out 10 more people come in when there's a sale on apples, $${ \left[\text{number of customers}\right] ~=~ \underbrace{100}_{\begin{array}{c}\text{normal} \\[-25px] \text{number}\end{array}} + \underbrace{10 \, \delta_{\text{sale}}}_{\begin{array}{c}\text{extra if} \\[-25px] \text{there's a sale}\end{array}} }_{\large{,}}$$ so we can increase the $$\left[\text{number of customers}\right]$$ by modifying if there's a sale, $$\delta_{\text{sale}} .$$

But, say that we found 10 more people and shoved them into the store.. would that cause apples to go on sale?

Since the model doesn't work in reverse, one might describe $$\left[\text{number of customers}\right]$$ as a "dependent variable" since we can't change it without breaking the model.

This is sort of an intellectual retreat, though. The fuller issue can be addressed by appropriately working out the reference classes. For example, we can be more precise and say that a sale-on-apples doesn't directly increase the number of people in the store, but rather increases the number of people in the store who wouldn't have otherwise come if not for the sale on apples. Once the various reference-classes are appropriately modeled, then we can drop the whole independent-vs.-dependent thing since all of the variables ought to be sufficiently well-defined that the model'll just generally hold, e.g. as Ohm's law is meant to.

### Reference-class issues can happen with electric-resistance, too.

When there's resistance that obeys Ohm's law, we can describe it "Ohmic resistance". However, a circuit might have non-Ohmic resistances, too.

When a situation gets more involved like that, it's important to not confuse the variables. For example, you don't want to just use $$R$$ for everything, including the Ohmic-contribution, non-Ohmic-contribution, and total-resistance, as, obviously, it can get all distorted if those variables are mixed up.

But as long as you keep everything straight in a known-abstract-model context, then there's no need to worry about independent-vs.-dependent variables.

The decision of dependent versus independent variables depends on the situation.

In most circuits you will face there will be components which are better at fixing one of the variables. For example, a AA battery is very good at fixing the voltage between its ends at 1.5V. Other elements are usually designed to have some flexibility. A resistor, for example, is designed to have a very fixed resistance but is happy to have the voltage depend on current.

As a general principle, "sources" will fix a voltage or current, so you can rely on them to start the process of selecting which variables are dependent or independent.

Later, this will get more murky. For example, a AA battery outputs 1.5V, until I bridge it with a very small resistor (like a 10ohm resistor). Then we start to get into the limits of the chemical reactions in the battery and voltages get messy fast. However, in introductory work, you will typically not be given these cases. The problems should always tell you which things should be fixed, and which should be solved for.

Physical equations are always created in order to describe a class of "systems" that have certain properties. Specifically, Ohm's law is used to describe systems which are called ohmic resistances.

Regarding electric circuits, the two main quantities that are observed, which means measured, are voltage and current. In contrast to these two, the resistance is not observed directly, instead, it is always measured by measuring both voltage and current at the same time. That said, one may think of the resistance as a quantity that is a property of an ohmic resistance. Which means it is independent in the scope of Ohm's law.

In practice, that means that the resistance is assumed to be constant (in the ohmic case), and it describes the response of a system to an applied current or voltage. So, one can calculate either

• the voltage $$U$$ that drops across the resistance if a current $$I$$ flows through it: $$U = R \cdot I$$. In this case, $$U$$ is the dependent variable one may measure, while $$R$$ is determined by the characteristics of the system, i.e. the resistance and $$I$$ is given by the corresponding experimental setup, i.e. the source of current.
• the current $$I$$ that flows through the resistance if a voltage $$U$$ (which is a difference in potential) is applied across the resistance: $$I = U/R$$. Now it is the other way round: $$I$$ is the dependent variable, and $$U$$ is given by the voltage source.

But this is only one half of the story. As it is the case with all physical theories (which equations like Ohm's law are, in the end), they have a limited scope which one has to be aware of.

In other cases than the ohmic case, the resistance can be dependent on other parameters, for example:

• the temperature $$\theta$$ of the resistor (using a thermistor in a resistance thermometer)
• the frequency $$f$$ of an applied alternating voltage (using a capacitor)
• the voltage itself, which is the case in semiconductors or even for the resistance of the human body, for example (it is also highly dependent on other factors, see here)

In these cases the resistance is no longer an independent quantity, so the system you are confronted with is not very well described using the resistance. Quantities that are more suitable for describing the system are for the examples mentioned are

• the temperature coefficient $$\alpha$$ of the thermistor, in many of those cases the resistance can be approximated by $$R(\theta) = \alpha\cdot\theta$$
• the capacity $$C$$ of the capacitor. More things happen when applying an alternating current to a capacitor, since one needs to talk about impedance and reactance instead of resistance here. Have a look at the corresponding Wikipedia articles for further information on this topic, since it is out of the scope of this question. For our concerns, it is enough to realize that Ohm's law can not simply be applied here.
• in the case of the resistance being dependent on the voltage, one needs to look at the relation $$R(U)$$ for finding independent quantities. If it were possible to approximate things with $$R(U) = a_1 \cdot U + a_2 \cdot U^2$$, then $$a_1$$ and $$a_2$$ would be the independent variables you are looking for.

Still, resistance can be a valid quantity to do calculations with, since its the nature of physical theories that theories like Ohm's law are limit cases of "larger" theories. Here, "Larger" means for example that the theory may describe things happening in other orders of magnitude (like in the case of a thermistor if the linear approximation made above is no longer valid for very high or low temperatures) or when things are getting time-dependent (like in the case of an alternating voltage).

Naturally, more mathematics are required to describe these more complex cases, but if you are looking at the limit cases, "smaller" theories still have to hold. In fact, when physicists are looking for new theories to describe new phenomena they observed, one crucial criterion for their ideas is that old, well-proven theories (like Ohm's law) are contained in limit cases.

Here's what it comes down to. You have three quantities: voltage $$U$$, current $$I$$, and resistance $$R$$. Without any more information or physical laws, all are free to be any value (and you perhaps want to pick each one).

Ohm's law, however, gives you more information, and restricts the possible values $$(U,I,R)$$ in an ohmic circuit that way. It says: you can vary each one of $$U$$, $$I$$, and $$R$$, but at the end of the day, the circuit will always satisfy $$RI=U$$. That means you can pick two of the three in an experiment without Ohm stopping you, and he will dictate the third.

When you have a fixed-resistance resistor, or a fixed-voltage battery, what you are doing is exactly that: picking $$R$$ and/or picking $$U$$, and if you've picked two, Ohm's law allows you to calculate the third, because it holds.

When first introduced to Ohm's law, the canonical example is a battery with a resistor across it. In this case, one builds up an intuition that voltages are somehow "fixed" or "independent", from which currents are to be inferred. However, this is a misconception.

V=IR is a (mostly) true expression relating the voltage drop across a resistor to the current passing through it, but one must know what the resistor is a connected to in order to determine actual values for these variables. An ideal voltage source defines the voltage across it to be fixed at some $$V$$, and is taken to instantaneously adjust the current it supplies to maintain a fixed $$V$$ across its terminals. An equally reasonable object is the ideal current source, which can produce any voltage to maintain a fixed $$I$$ through it.

For low currents, a battery may be modeled as an ideal voltage source of voltage $$V_\text{Batt}$$. In this case, for a battery connected to a resistor of resistance $$R$$, the current in the loop is $$I=V_\text{Batt}/R$$. If the resistor were instead connected to a constant current $$I_{CC}$$, it would remain to determine the voltage drop across the resistor, $$V_R = I_{CC} R$$.