# How much is a Coulomb, really?

I've heard it said in my physics class that a Coulomb "is a lot of charge". And I believe it; most of the problems I've done in the class so far involve charges on the order of micro-Coulombs (or, occasionally, nano or milli). But I still don't understand the sense of scale I'm working with.

So, let me ask this:

• What single object would contain about one Coulomb of charge (positive or negative)?
• And if I touch it, will I die?
• A one Farad capacitor at one volt has one Coulomb on it. Now, one Farad is a big capacitor for most electronics applications, but you can go to Digikey and find 6000F capacitors in stock. And, at 1 volt nothing will happen if you touch it. Yet, a 10mA current through you can kill you. It isn't simple. – Jon Custer Feb 25 '16 at 18:28
• The exercise I ask my class to do is compute the electrostatic force between two $1\,\mathrm{C}$ point charges at a distance of $1\,\mathrm{m}$. After you do that consider the mechanical engineering constraints on the capacitor that @Jon describes... – dmckee Feb 25 '16 at 18:46

Avogadro's number is $6.02\cdot 10^{23}$; a single electron has a charge of $1.6\cdot 10^{-19}$ C, so $1.04\cdot 10^{-5}$ moles of single-ionized material carries a net charge of 1 C.

To carry that much charge, you need a large capacitance or a large voltage, since $Q=CV$. An object with a 1 mF capacitance and a voltage of 1000 V would be sufficient, or a 1 F capacitance with 1 V. The difference between these is the amount of energy stored, which goes as $E=\frac12 CV^2 = \frac{Q^2}{2C}$ - so for the same amount of charge, a larger capacitor will have less stored energy.

And that is the hint to the "will it kill me" part of the question: you are not killed by charge, but by current flowing. If you have a charged object with a low potential, the flow of current through your body will be slow - and you will survive. But if the voltage is high, it will easily overcome the resistance of your skin and give you an almighty jolt - possibly enough to kill you.

So what is the size of a sphere with a capacitance of 1 F? Capacitance of a sphere is $4\pi\epsilon_0 r$, so you would need a radius of about $9\cdot 10^9$ m - quite a bit bigger than the Earth. A sphere with a 1 m radius, with a 1 C charge on it, would have a potential of about 1 GV. That's a very large voltage - if you could even maintain that potential (not in ordinary atmosphere), touching it would kill you.

Supercapacitors can be created in which two conductors are brought in very close proximity, while having a dielectric layer in between that produces a very high capacitance in a small package. Such a device can easily be charged with a Coulomb - although that isn't a net charge (one plate will be positive, the other negative). And whether such a capacitor could give you a lethal shock will again depend on the capacitance.

• I think saying that when the voltage is "high" it will overcome the resistance of the skin could use a little more specificity, as folks who learn from this Q&A are probably precisely those who don't know the relevant numbers. – DanielSank Feb 26 '16 at 7:51

A lot of charge is relative. Here's a chemistry perspective. Faraday's number gives the charge held by 1 mole of electrons: 96,485 C. Therefore, 1 Coulomb is 1/96485 moles of electrons, or around $10^{-5}$ moles. Based on this, you can deduce that any corroded object of a reasonable size formed by passing at least 1 Coulomb of charge.

Another point of comparison would be thinking about how much charge is in a small battery. According to Google, typical AA batteries have 400-900 mAh of capacity. mAh is a strange unit of charge, but it's very useful for converting between current, capacity and time. Essentially, 400 mAh is the amount of charge passed by a current of 400 mA over 1 hour (or 200 mA over 2 hours, 800 mA over 0.5 hours, etc, hence its utility). Since 1 A is defined as 1 C/s, passing a current of 400 mA for 1 h requires $0.4 \frac{C}{s} \times 3600s = 1440 C$. So, next time you want to imagine 1 C of charge, think of a AA battery and divide by 1000.

• I think you dropped the Coulomb unit with the 1440 number. – DanielSank Feb 26 '16 at 7:31
• I like the analogy of the AA battery - although they come in significantly higher capacity these days. – Floris Feb 2 '17 at 1:26