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I have become confused as to the meaning of what a C rating of a battery means. My understanding is that because a C rating is measured in milli-ampere hours the units should cancel to be a unit of charge. However, sources such as A Guide to Understanding Battery Specifications (Paragraph 3), and This physics stack exchange post seem to indicate that a $C$ rating is a unit of capacitance which is of course measured in Coulombs per Volt.

I my question is what a $C$ rating really is and if there is a more intuitive interpretation.

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$1~\textrm{Ampere-hour}$ equals $3600~\textrm{Coulomb}$ of charge.

The amount of charge a battery can hold is determined by the amount of chemicals that are in it and it's fixed by the design of the battery. Usually it only decreases because of loss of electrolyte or changes in the chemical and physical structure of the electrodes.

The voltage on a battery is also almost constant (if it's too high or too low, the electrochemical system in it will usually degrade very quickly). We therefore measure the "capacity" of a battery with a single amount of charge, with the SI units of Coulomb, however, technically the easier to handle quantity is $1\rm mAh=3.6~ Coulomb$ or $\rm Ah=3600~Coulomb$. This will immediately tell us the necessary charge current, if we are, for instance, charging at $\rm C/10$.

For example: if the battery has $12~\rm Ah$ of capacity, then the charging current has to be $1.2~\rm A$ at $\rm C/10$. For many batteries the specified charging time is $14~\rm h$ to make up for the inefficiency of the charging process because not all charge that we send trough the battery is actually used to perform a chemical conversion of one ion species into another.

In comparison the capacitance of a capacitor is constant and the amount of charge it can hold is proportional to the voltage between its terminals, which can be anything between 0 and the max. operating voltage, which is usually limited by dielectric breakdown of its dielectric. Because of this proportionality we specify the capacitance of a capacitor in $\rm [Farad]=[Coulomb/V]$, a unit that doesn't make sense for batteries, because the voltage on a battery can't be changed.

In practice batteries usually hold a lot more charge than capacitors, but they suffer from the constant voltage problem, which makes them less versatile than capacitors for electronics. OTOH, capacitors, with very few exceptions, are not useful for long term energy storage.

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