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# Electric power

## Video transcript

so if you've ever run current through a resistor you might have noticed that that resistor warms up and in fact if you run too much current through the resistor a resistor can get so hot it can burn you when you touch it so you have to be careful so what I'm saying is when you have current flowing through a resistor it warms up and we want to explain why conceptually why does current moving through a resistor heat it up and is there a way to calculate exactly how much that current would heat up that resistor over a given amount of time there is a way to calculate it will derive that in this video but first we should just explain conceptually why is it that current moving through a resistor heats up the resistor and so we'll explain that with this current now the way I've got it drawn here notice that I've got these positive charges positive charges don't actually move through a wire but physicists always pretend like they do because positives moving one way through a wire is equivalent to negatives moving the other way through the wire and if you use the positive description even though it's technically incorrect you don't have to deal with all those negative signs it's easier to deal with and it's equivalent so we may as well use it but you can go through this whole description with negative charges moving the other way and it works just as well you just have to be very careful with the negatives and it kind of obscures the conceptual meaning because it's hiding behind a bunch of negative signs so we'll just use these positive charges but know that it's really negatives going the other way so why do positive charges flowing through a resistor cause that resistor to heat up well here's why so we know that when current flows through a resistor there's a voltage across that resistor in other words between this point here and this point here there's a difference in electrical potential so there's a voltage across that resistor technically I'm gonna call it Delta V because it's a difference in electric potential in other words V on this side the electric potential on this first side of the resistor is going to have a different value from the electric potential on this second side of the resistor so why does this matter it matters because this final side where the charges end up is going to have a lower electrical potential than the beginning so these positive charges are going to be moving from a high potential region to a low potential region and that means they're going to be changing their potential energy so these charges have electric potential energy and if they go from a region of higher electric potential to a region of lower electric potential they've started with more potential energy electrical potential energy than they end with so they're decreasing their electric potential energy and in case that's confusing remember that the definition of electric potential is the amount of electric potential energy per charge so to just put a number in here so it's not so abstract let's say v2 was 2 joules per Coulomb that would mean for every Coulomb of charge at that point v2 there'd be 2 joules of electric potential energy and if v1 is at a higher electric potential maybe this is at 6 joules per Coulomb that would mean over here at this position v1 for every Coulomb of charge there'd be 6 joules of electric potential energy so as these charges move through the resistor they're going to be decreasing their electric potential energy and so the obvious question is well where does that energy go if these charges are decreasing their electric potential energy where's that potential energy going my first guess is that they'd increase their kinetic energy because I'd remember that on earth if you drop a ball and it decreases its potential energy its gravitational potential energy we know that when it decreases its gravitational potential energy and falls down it increases its kinetic energy it just speeds up so the decrease in gravitational potential energy just corresponds to an increase in kinetic energy of that object and so maybe that's happening over here maybe as these charges lose potential energy they speed up but that can't happen remember the current on one side of a resistor has to be the same as the current on the other side these charges don't speed up they're losing potential energy but they don't speed up this is a little counterintuitive we're used to things speeding up when they lose potential energy but these charges aren't going to speed up what they do is they just heat up the resistor so as these charges fly through this resistor they strike the atoms and molecules in this lattice structure of the solid so this resistor is made out of atoms and molecules and as these charges flow through here and again it's really electrons flowing the other way but as the charges flow through same idea they strike the atoms and molecules they transfer energy into them and as they pass through in their wake they leave a resistor that's hotter at a higher temperature which means these atoms are jiggling around more than they were before and since they're oscillating more than they were before they're jiggling they've got more energy the temperature of this resistor increases so these charges rather than keeping all the energy for themselves they actually just spread it out over that resistor as they pass through and they spread it out in the form of heat or thermal energy and they emerge with basically the same kinetic energy that they started with so this change in potential energy electrical potential energy corresponds to an increase in thermal energy of this resistor and that's why the resistors heat up but is there a way to calculate exactly how much this resistor will heat up how much energy it's going to gain per time there is we just have to use the definition of power so we know the definition of power is the work per time or since work is the change in energy or the energy transferred we can just write this as the amount of energy this resistor is gaining per time so what we want is a formula that tells us how much energy are these charges depositing in the resistor per time well this energy gained by the resistor is coming from the loss of potential energy of these charges so these charges are losing potential energy they're losing electric potential energy and that electric potential energy is turning into thermal energy so the thermal energy this resistor gains is just equal to the amount of electric potential energy that these charges lose so I can just rewrite this I can just say that the power is going to be equal to the change in electrical potential energy of these charges per time and so I'll just continue down here power is going to be equal to how do we find the change in electric potential energy we'll remember potential is defined to be the potential energy per charge so that means the electric potential energy is just the charge times the electric potential so if I want to find Delta U I can just say that that's going to be u when they emerge u2 minus u1 and this is a way we can find the U values so the U at two since it's Q times V is just going to be the charge at two times V two and then the U at 1 so we'll do minus the U at 1 is the charge at 1 but that's the same charge whatever charge enters this resistor has to exit it so it'd be the charge at 1 times the viet 1 this is the change in electric potential energy so I could rewrite this I pull out a common factor of Q in this expression right here and we get that the power is going to be equal to this common factor of Q times V 2 minus V 1 so that's Delta U and that's what we're plugging in right here for Delta u Delta U is just the difference in these Q times V values and then we still have to divide by x since we're talking about a power but what is V 2 minus V 1 that's simply the voltage across this resistor Delta V is V 2 minus V 1 so I can rewrite this I can just say that this is Q times Delta V the voltage across that resistor divided by the time it took for the charge to pass through that resistor and now something magical happens check this out so we got power equals I've got charged the passed through the resistor divided by the time that it took for that charge to pass through the resistor but charge per time is just the definition of current so we get this beautiful formula if I just factor out this Q over T I get Q over T times Delta V the voltage across the resistor but Q over T is just the current so I get that the power is going to equal the current through that resistor times the voltage across that resistor and this is the formula for electrical power this tells you how many joules of thermal energy are being created in that resistor per time so the units are joules per second or in other words watts because what this is telling you is the amount of thermal energy generated per second if the power value came out to be 20 watts that would mean there'd be 20 joules of thermal energy generated every second which is a really useful thing to know this formula is extremely useful when you want to figure out how much energy is going to be used by a light bulb or a toaster or a TV or whatever electronic device you want to use this tells you how much energy that it's going to turn into either thermal energy or light or sound or whatever other kind of energy that it's converting that electric potential energy into because notice we never really assumed this was thermal energy we just called it e and then we said that whatever energy got transformed came from the change in potential energy of those charges and that's always going to be true it's always going to be electric potential energy converting into something whether it's heat or light or sound so this doesn't just work for resistors works for almost all electrical opponents that turn electric potential energy into some other kind of energy and you can rewrite this in different forms sometimes you'll see it a different way so there's this form and then I'm just going to copy this I'll show you there's a couple other forms you can see this in so let me clean this up we'll put this formula down here this gives us the power but we know from Ohm's law that Delta V is equal to IR so if both of these formulas are true I can plug Delta V as IR so I can take this IR I can plug it in for Delta V and I get an alternate expression for the power used by electrical component I get that the power is going to be I times I times R which is just I squared times R so this one might be more useful if you've got a situation where you don't know the voltage but you happen to know the current and the resistance but there's one more I could have solved this Ohm's law formula for I and I get that I equals Delta V over R and then if I plug Delta V over R in for I up here I'd get an alternate expression for the power I get that the power equals Delta V over R times Delta V is just going to be Delta V squared the voltage across that resistor squared divided by the resistance of that resistor and this formula might be more useful if you know the voltages and the resistance but you don't know the current so depending on what you know these can get you the power used by a resistor they're all equivalent they will all give you the correct and the same power it's just a matter of what's more convenient for the actual problem that you're dealing with so recapping when current passes through a resistor it converts electrical potential energy into thermal energy and you can calculate the amount of electrical potential energy converted per second using current times voltage current squared times the resistance or voltage squared divided by the resistance