Energy Stored by Capacitor

Hey guys. When we first introduced capacitors, we said that by separating the charge in a capacitor it stores some potential energy. Now, we want to look at is exactly, how much potential energy a capacitor stores, let's get to it. Remember, capacitors separate charges like I said and that separation leads to a potential energy stored but, how much energy is the question that we want to answer, the energy stored by a capacitor is typically given in three forms I'll start with one-half C, V squared, all of these three forms are completely equivalent and they're just related to each other by using different forms of this equation, the other forms are one-half q times V and one-half q squared over C, all three of these are completely equivalent. Alright, now we want to talk about the energy density stored in a capacitor, the amount of energy per unit volume, okay? And it's given by a little lower case u. Alright, this is just going to be the potential energy divided by the volume and I'll use Vol for the volume instead of V because we're using V for potential, the volume of a parallel plate capacitor that has an area A, on each plate and the separation distance of d, right? That volume is just a times d the area times the distance between them. So, u is going to be one-half C, V squared over a times d and if you use a whole bunch of algebra, what you end up with is one half epsilon naught times the electric field squared, this form and this form are the two useful forms of the energy density, okay? Now, we want to do a couple of examples dealing with the energy stored by a parallel plate capacitor.

Two parallel plates of area 50 square centimeters with a separation of 10 millimeters have a voltage across them of 20 volts, what is the energy stored, and what is the energy density, okay? Pretty straightforward, we're given an area, a separation and a voltage, okay? We have to choose one of the three forms of the energy to use, okay? The thing is, each of them we're going to require two of the three things capacitance charge or voltage, we definitely have voltage. So, we want to choose one that has voltage but we're not told charge here. So, we can't choose the one with voltage and charge, we are however given enough information to find the capacitance. So, we can use the one that involves capacitance and voltage and all we have to do is find the capacitance, the capacitance of a parallel plate capacitor is epsilon naught, A over d, this is going to be 8.85 times 10 to the negative 12 the area is 50 square centimeters or 50 times 10 to the negative 4 square meters, the separation distance is 10 millimeters or 0.01 meters and this ends up being 4.4 times 10 to the negative 12 farads, okay? Now, we know the capacitance. Now, we can find the potential energy stored. So, big U, the potential energy is one-half times the capacitance times the potential difference, twenty volts squared, and this ends up being 8.8 times 10 to the negative 10 joules, that's not and epsilon. Alright, that's question one down, what about the next question, the energy density? Well, Remember, I told you there are two useful forms of the energy density 1/2 epsilon naught times the electric field squared or just the potential energy divided by the volume, since we don't know the electric field that would just be another thing, we have to calculate, let's not worry about that, let's say that the little u, the energy density is the energy divided by the volume, right? The volume is just going to be a times d, so this is going to be 8.8 times 10 to the negative 10 divided by 50 times 10 to the negative 4, which is the area in square meters, times that distance 0.01 and this whole thing is going to be 1.76 times 10 to the negative 6 joules per cubic meter, right? It's a density. So, it's just the thing, which is energy in this case, divided by volume, right? So, joules per meters cubed.

Alright, the next example, what electric field strength would store 2.5 millijoules of energy per cubic centimeter, if we want to relate energy density, which is what this is, this is some amount of energy per some amount of volume, that's an energy density, if we want to relate back to the electric field we'll use this equation, one of our two equations that relates the energy density to the electric field, pretty simple, solve for the electric field, the problem is that the units of energy density are not the SI units. So, we do need to convert that first, this is 2.5 milli is 10 to the negative 3 joules per cubic centimeter, each cubic centimeter has one, sorry, each cubic meter has one million cubic centimeters. So, we are going to convert this by saying 1 million cubic centimeters goes into 1 to the meter, okay? So, this 10 to the negative 3 becomes a 10 to the positive 3, when we multiply it by 10 to the 6. So, that's 2.5 times 10 to this, sorry, 10 to the positive 3 joules per cubic meter, okay? So, that's our potential energy density in SI units. Now, we can take this equation and we can rearrange it, we're going to take the 2 and multiply it up, the epsilon naught and divide it over, that's going to isolate E squared, then we're going to take the square root of both sides and that's going to isolate E. So, our electric field looks like 2 times 2.5 times 10 to the 3, just our energy density, divided by epsilon naught, which is 8.5 times 10 to the negative 12, all of that square rooted which is just 2.38 times 10 to the 7 Newtons per Coulomb and that is the electric field. That wraps up our discussion on the potential energy stored by a capacitor, thanks for watching guys.