# Why is the unit for capacitance so large?

Most capacitors are in the µF, nF and pF range. I know there are some special ones that go that high, but at the time faraday was around, and the unit was named after him, they didn't have such a thing. Why is the unit so large if we rarely use caps with that high of a value?

• For elementary particle physicists, the meter and the second are enormous units. It's all a matter of context. For electronic engineers, mA and uA are common. For electrical engineers, kA and MA are common. – Alfred Centauri Jun 10 '13 at 23:16
• The unit you're talking about wasn't defined as we currently know it until more than a decade past Farady's death. (Source) Units named after people are typically assigned posthumously. – Warren Young Jun 10 '13 at 23:58
• He was giant in his time. We can only hope to posses uF's worth today ;-) Like fE (femto einsteins). – user6972 Jun 11 '13 at 7:49
• ...And you need a unit for those bigger capacitors. If I'm right, they are trying to use "supercapacitors" in electric cars. – Anonymous Penguin Jun 19 '13 at 19:14

As others have mentioned, 1 farad is 1 coulomb per 1 volt. But the rabbit hole goes deeper -- the question then becomes why is 1 coulomb what it is, and why is 1 volt what it is?

Following this rabbit hole to the bottom will eventually lead us to the 7 base SI units, which are units of measure for the 7 physical attributes of our world: distance, mass, time, electric current, thermodynamic temperature, amount of a substance, and luminar intensity. They're like axioms in mathematics. From here, other units are defined in terms of these. So volt = (kilogram meter meter) / (ampere second second second). Meanwhile coulomb = ampere * second. You'll notice that 1 of a derived unit is expressed in terms of 1's of a base units.

So ultimately, 1 farad is so large because the base units are so large, at least relative to the sizes of electronic components nowadays where we fit billions of transistors onto several square millimeters.

• I wonder how many Farads you could fit in a capacitor that takes up a cubic metre. – user253751 Jan 2 at 16:36
• @user253751 That's an invalid question, because for any given technology capacitor volume is roughly equal to the amount of energy the thing can store -- i.e., its voltage rating squared times the capacitance. Life is further complicated by the fact that there are different technologies, each with its own advantages and disadvantages -- so there is no "canonical" capacitor to wedge into a 1-meter cube. – TimWescott Jan 2 at 19:03
• @TimWescott It's completely valid. Physicists can't answer it, but engineers can. – user253751 Jan 2 at 19:11

Because it fits in with all the other (SI) units we have. 1 farad is 1 coulomb per volt. It just so happens that 1 coulomb is... a lot of charge.

• Let's put it another way; it allows the formula $f = \frac{1}{2\pi RC}$ to work with any mysterious conversion factors. – Kaz Jun 10 '13 at 23:01
• It'd be nice to hear why the other SI units (i.e. coloumb) are so big then. Is it the definition of ampere, charge or voltage? – Macke Jun 11 '13 at 7:14
• @Macke 1 coulomb is 1 ampere × 1 second. – Random832 Jun 11 '13 at 12:44
• @Macke: A second is a nice unit for a human-perceivable timescale, but it's huge relative to the amount of time a typical capacitor can supply what would have been a reasonably-measurable amount of current. – supercat Jun 19 '13 at 22:07

Because 1 Ampere is a unit so large compared to the amount of current we normally use. Because 1 second is a unit so large compared to the audio and rf frequencies we normally use.

If you normally use currents much smaller than 1A, for periods much shorter than 1sec, and don't have a lot of money to waste or a lot of space to waste, you can use capacitors much smaller than 1F.

On the other hand, if you wanted to do electrical power, instead of radio electronics, 1F isn't very big. Here is a recent press release on a 400F capacitor. http://www.engineering.com/ElectronicsDesign/ElectronicsDesignArticles/ArticleID/5290/Is-it-a-battery-No-its-a-Supercap.aspx -- and note that the special feature is that it is no larger than a deck of cards.

• 400F with the size of a deck of cards is by no means a large capacitance in a small package. There are capacitors in the kiloFarad range and above, which are much smaller. They, however, operate on very small voltages. – vsz Jun 11 '13 at 15:19
• @vsc Energy stored is proportional to voltage squared, so that is no surprise. – starblue Jun 11 '13 at 19:40
• 1kWh, 300kW, 477kg, 1900x950x455: bombardier.com/en/transportation/sustainability/technology/… – starblue Jun 11 '13 at 19:45

The SI units for electricity fit in with the SI units for everything else. The relationship becomes clear if you look at the definition of a joule:

$$J = N\cdot m = W\cdot s$$

Notice that it has both mechanical units you'd naturally consider mechanical (newtons, meters) and electrical units (watts). We can break it down into more basic units:

$$J = \frac{kg\cdot m^2}{s^2}$$

Or we can expand watts to more basic, but still electrical units:

$$J = V \cdot A \cdot s$$

And now you have volts and amps, by which the farad can be defined:

$$F = \frac{A\cdot s}{V}$$

If you analyze this carefully, you will notice that a joule is a watt-second, and a watt is some ratio of current and voltage, but that ratio is undefined. This is why the ampere is an SI base unit, defined as

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length.

So if you want to blame something for the farad being so large, blame the ampere. Or, blame the other SI base units referenced by its definition, the second, meter, or kilogram (indirectly, by the newton).

It has nothing to do with Faraday. It is a definition.

From Wikipedia:

$F = \dfrac{A\times s}{V}$

Manipulated Algebraicly:

$A=\dfrac{F\times V}{s}$

And in terms of $i_c(t)=C\dfrac{\mathrm{d}v}{\mathrm{d}t}$.

Expressed Algebraicly:

$I=C\dfrac{\Delta V}{\Delta t}$

The Farad is a derived unit, it is an Ampere second per Volt.

This is not an exhaustive explanation since the SI unit of mass is the kilogram, so there is precedence for SI base units already carrying a multiplier prefix.

One could have defined the Megafarad as an Ampere second per Volt. The resulting comparatively huge inconvenience of doing so may appear like an attractive tradeoff for units that bear considerable likeness to measures in historical or everyday use, and a whole lot of goods are traded in the likeness of pounds and kilograms.

The Farad had no preexisting meanings or uses assigned to it. It is, however, interesting that as opposed to capacitor capacities, battery capacities are not given in something like Megafarad but rather mA h or, for lead batteries, A h (and for multi-cell Li ion compounds commonly used with DC-DC converters, in the more significant mW h ). This again reflects some hands-on measure more desirable for quick estimates for people that try solving problems in their head rather than reverting to sliderules.

The reason for this is because the system of units upon which it is based - the International System (SI) - is designed for mathematical coherency, and it is "built" from other units which happen to be more reasonably sized. The requirement of mathematical coherency basically means that you try to eliminate as many "physical constants" as you can when measuring in your system by suitably choosing the sizes of your various units so their values equal 1 when measured in those units.

To better understand this, consider the basic law of mechanics, Newton's second law:

$$F = ma$$

In fact, this is not the most general form. Newton's law, in fact as I believe he originally stated it, is really a proportionality between mass, acceleration, and force, and it should most "properly" be

$$F = k_N \cdot ma$$

for some constant $k_N$, where "N" stands for "Newton". In SI - by design, which is what we're getting to - $k_N$ numerically has the value 1, so we can elide it. In other systems, however, this might not be the case: for example in the American system of units, if we measure mass in pounds of mass (lb or just "lb"), force in pounds of force (lbf), and acceleration in miles per hour per second ("0 to 60 in 6 seconds" = 10 MPH/sec, and so forth). then the constant $k_N$ is very much not 1, but rather

$$k_N \approx \frac{1}{21.936}\ \mathrm{\frac{lbf}{lbm \cdot \left(\frac{MPH}{sec}\right)}}$$

(to understand where this value comes from, a hint is to note that 21.936 MPH/sec equals nearly 32.174 ft/sec^2... and remember what the conceptual relation between lbm and lbf is ...)

So if I want to make my 2000 lbm vehicle accelerate at 10 MPH/sec, then the engine must supply a force of

$$F = \left(\frac{1}{21.936}\ \mathrm{\frac{lbf}{lbm \cdot \left(\frac{MPH}{sec}\right)}}\right) \left(2000\ \mathrm{lbm}\right) \left(10\ \mathrm{\frac{MPH}{sec}}\right)$$

which is roughly 911 lbf, or half a ton of force.

The way that coherent units work is that we, in effect, specify some of the units and then allow this requirement that these constants should be 1 to "derive" the others. Hence, if we have already defined units for, say, mass and acceleration, we can use $F = ma$ to define a coherent unit for force. Or we could have units for mass and force and then define a unit for acceleration from them. In particular if we use lbm and lbf, this unit is the gee - the "standardized Earth-surface gravitational acceleration" or about 32.174 ft/sec^2. So in the SI, we set our scales by measuring mass in kilograms (kg) and acceleration in meters per second per second (m/s^2), and then ask for the unit of force needed to make Newton's second law look like its common textbook form, and you get the unit of force called the Newton (N) as the result, which is the coherent SI derived unit of force.

The base units of the SI - here the kilogram, the meter, the second, and the ampere - were derived because they are all reasonably-sized from a human perspective: a kilogram is about the mass of a small jug of water, the meter about a human's waist off the ground, the second about the time of a human's heart pulse at rest, and the ampere an electric current level in the right order of range for many common electrical devices. Of course, compromises were made in the actual definitions of various kinds to make them easier to work with and also with an aim to reliable reproduction (e.g. originally, the meter was technically defined by fractionating the circumference of the Earth, and the second as a suitable fraction of the day [one rotation period of Earth], which were for their time the best standards), but these can be considered the ultimate reasons for choosing those more technical standards - to aim roughly to this human scale.

But when you start deriving other and more esoteric things from them under this demand of mathematical coherency, things tend to not like to always stay in human scale and we end up with some often "phEEEpy" surprises - and the Farad is a good example of that. So what we do is we stick in all those good prefixes to scale those coherent units up or down to make bigger or smaller non-coherent units of more convenient sizes from them. In effect, we make those "k"s into simple decimal powers (and not garbage like the American system factors above), even if not 1.

On a deeper level, you could think of this as that "the Universe's natural scales are 'fighting' to assert themselves against our anthropocentrism", because we are in effect ceding some ground to let the mathematical form of the laws of the Universe dictate our other units. Going the farthest extreme the other way - "don't fight it, let the Universe do all the talking" - gets you a system similar to the Planck units: a system of units that is ludicrously at variance with human scale (e.g. one meter is roughly 100 billion trillion trillion Planck units of length), but is extremely useful to simply state the fundamental governing equations of the most profound physical phenomena.