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When trying to compare the energy in a battery to the energy in a capacitor, the units don't match up. How can one compare a battery whose Ah are 10 and Voltage is 3 (for a total of 30 Wh) to a capacitor whose Farads is X and voltage is Y?

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To begin, let's call things with their names (no offense). What is measured in Farads is capacitance $C$. What is measured in $Ah$ is the charge that can be stored in a battery or a capacitor.

From the definition of capacitance, the charge on the walls of a capacitor with capacitance $C$ and potential difference $V$ is $$ q = CV $$ so you obtain the value you're interested in. However, if $C$ is in Farad and $V$ is in volts, $q$ will be measured in Coulombs. $$ \text{Ampere} = \frac{\text{Coulomb}}{\text{Second}} $$ so

$$ 1\text{ Ampere}\cdot\text{hour} = \text{1 Coulomb}\cdot\frac{\text{hour}}{\text{second}} = 3600 \text{ Coulombs}$$ So, in your notation, the stored charge is $XY/3600$.

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Thanks for the explanation. –  Jack Dagmy Jan 31 '13 at 20:30
Can you tell me why in this data sheet:maxwell.com/products/ultracapacitors/docs/… –  Jack Dagmy Jan 31 '13 at 20:31
Your calculations doesn't work. –  Jack Dagmy Jan 31 '13 at 20:31
I don't know, here we are talking about a physical device, which can be imperfect, have leakage currents etc, I don't know how they calculate it. You should ask an experimental physicist or someone expert in electronics. –  Bzazz Jan 31 '13 at 20:37
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Energy in a capacitor is $CV^2/2 = QV/2$ because its voltage starts at 0 when uncharged (unlike a battery, where the voltage is more or less constant). See note 8 in the Maxwell doc you referenced, which uses this formula and then converts from joules to watt-hours in the same fashion correctly described by Bzazz's answer.

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