If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Energy of a capacitor

## Video transcript

check out this capacitor look at what happens if I hook it up to this light bulb yeah nothing happened because the capacitor is not charged up but if we hook it up to a battery first to charge up the capacitor and then hook it up to the light bulb the light bulb lights up the reason this happens is because when a capacitor is charged up it not only stores charge but it stores energy as well when we hooked up the capacitor to the battery the charges got separated these separated charges want to come back together when given the chance because opposites attract so if you complete the circuit with some wires and a light bulb current is going to flow and the energy that was stored in the capacitor turns into light and heat that comes out of the light bulb once the capacitor discharges itself and there's no more charges left to transfer the process stops and the light goes out the type of energy that stored in capacitors is electrical potential energy so if we want to figure out how much energy is stored in a capacitor we need to remind ourselves what the formula is for electrical potential energy if a charge Q moves through a voltage V the change in electrical potential energy of that charge is just Q times V looking at this formula what do you think the energy would be of a capacitor that's been charged up to a charge Q and a voltage V yeah that's what I thought it would have been too but it turns out the energy of a capacitor is one-half Q times V where does this 1/2 come from how come the Energy's not just Q times V well the energy of a capacitor would be Q times V if during discharge all of the charges were to drop through the total initial voltage V but during discharge all of the charges won't drop through the total voltage V in fact only the first charge that gets transferred is going to drop through the total initial voltage V all of the charges that get transferred after that are going to drop through less and less voltage the reason for this is that each time a charge gets transferred it decreases the total amount of charge stored on the capacitor and as the charge on the capacitor keeps decreasing the voltage of the capacitor keeps decreasing remember that the capacitance is defined to be the charge stored on a capacitor divided by the voltage across that capacitor so as the charge goes down the voltage goes down as more and more charge gets transferred there'll be a point where a charge only drops through 3/4 of the initial voltage wait longer and they'll come a time when a charge gets transferred through only a half of the initial voltage wait even longer and a charge will only get transferred through 1/4 of the initial voltage and the last charge to get transferred drops through almost no voltage at all because there's basically no charge left that's stored on the capacitor if you were to add up all of these drops and electrical potential energy you'd find that the total drop in energy of the capacitor is just Q the total charge that was initially on the capacitor times 1/2 the initial voltage of the capacitor so basically that 1/2 is there because not all the charge dropped through the total initial voltage V on average the charges dropped through only 1/2 the initial voltage so if you take the charge stored on a capacitor at any moment and multiply by the voltage across the capacitor at that same moment divided by 2 you'll have the energy stored on the capacitor at that particular moment there's another form of this equation that can be useful since capacitance is defined to be charge over voltage we can rewrite this as charge equals capacitance times voltage if we substitute the capacitance times voltage in for the charge we see that the energy of a capacitor can also be written as 1/2 times the capacitance times the voltage across the capacitor squared but now we have a problem in one of these formulas the V is squared and in one of these formulas the V is not squared I used to have trouble remembering which is which but here's how I remember now if you use the formula with the C in it then you can see the V squared and if you use the formula that doesn't have the C in it then you can't see the V squared so these are the two formulas for the energy stored in a capacitor but you have to be careful the voltage V and these formulas refers to the voltage across the capacitor it's not necessarily the voltage of the battery in the problem if you're just looking at the simplest case of one battery that has fully charged up a single capacitor than in that case the voltage across the capacitor will be the same as the Volt yup the battery so if a 9-volt battery has charged up a capacitor to a maximum charge of four coulombs then the energy stored by the capacitor is going to be 18 joules because the voltage across the capacitor is going to be the same as the voltage of the battery but if you're looking at a case where multiple batteries are hooked up to multiple capacitors then in order to find the energy of a single capacitor you've got to use the voltage across that particular capacitor in other words if you were given this circuit with these values you could determine the energy stored in the middle capacitor by using one-half QV you would just have to be careful to use the voltage of that capacitor and not the voltage of the battery plugging in five coulombs for the charge lets you figure out that the energy is 7.5 joules you