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Solved example - Calculating power & heat dissipated

Video transcript

the heating element of a toaster has a resistance of hundred ohms if it's connected across two hundred volt supply find the heat produced in ten seconds so let's try and write on what's asked of us we are asked to calculate the heat produced in that heating element in ten seconds so let's write that down h4 Heat we need to calculate heat produced in ten seconds this is what we want to calculate and what's given to us that's always the next step for we what's given to us are let's see in heating element of a toaster has a resistance of hundred ohms so we are given the resistance of the element so the resistance of the material is even is to 100 ohms and we're also given it is connected across two hundred volt supply so which means voltage is given to us so V is 200 volts so what's going on over here well just to give some context in a previous video you seen that when you have a current passing through any material due to the resistance of that material they will be heat generated and we also derived a formula to calculate that heat generated in that material and we found out that the heat generated in any material due to its resistance due to the electric current that can be calculated as also write the down here heat generated heat generated per second that can be calculated as the product of voltage voltage and current and if you are not clear about where this comes from we need more clarity on this we have derived this in previous video so it'll be a great idea to go back watch that video and then come back over here but this is the basic formula to calculate the heat generated in any resistor for a second due to the current but wait a second you may be wondering well there is no current given in our data so how do we calculate it Oh remember Ohm's law Ohm's law connects voltage resistance and and Ohm's law was V let me write that down over here V equals I I times R so if you look at this carefully one way we could solve this is we can calculate what the current is in this heating element by using Ohm's law and then we can substitute that over here or we can directly substitute this in our formula so let's do that so if we substitute Ohm's law in our formula there are two things we can do one we can substitute V as IR so if it's absolute IR over here we'll get I squared times R so that's another formula for us and one more thing which we can do is we can get rid of I as V over R so if we subsidy over R here we'll get V squared over r v square or and now because we're given voltage and resistance all we can do is directly go ahead and use this formula to calculate the heat produced every second and then from that we can go ahead and calculate how much heat is produced in ten seconds so it'll be a great idea to pause the video right now and see if you can try this yourself all right let's do this so heat produced per second I'll go with this one because we know the voltage and the resistance so voltage is 200 so that's 200 square right that's 200 times 200 divided by R divided by our I know we can cancel some 0s that's why I write it as 200 times 200 instead of 200 square and so I can cancel two 0s over here and that gives me 400 400 watt well you look at it you have heat heat is energy so it's joules for a second so for hundred joules per second so what we have calculated is that in this heating element there were four hundred joules of heat produced every second but we are asked to calculate the heat produced in ten seconds how do we do that well we know in one second there's 400 joules of heat in two seconds it would be two times that in ten seconds it'll be ten times that so last step heat in ten seconds that's going to be 400 joules per second times ten seconds and that's going to be four thousand joules so the total heat produced in ten seconds in our heating element is going to be four thousand joules all right let's solve another one a boat produces 240 joules of heat every minute when the voltage across it is 2 volts find the resistance of the bulb and the current through it so the concept is very similar it's about heat generated for a second but this time we are given the heat heat generated every minute and we have to find the resistance and the current through the bulb so let's again go ahead and write down what's asked and what's given to us what's asked is we need to calculate the resistance R and we need to calculate the current let's use the same color yellow eye and what we are given is heat produced every minute so let's write that down heat for a minute that's given as - 40 joules per minute and we're also given the world is across it is tours voltage is towards the concept is similar so let me write down the same thing that we wrote earlier we know Heat produce every second is V times I that's our basic formula calculate electric power here the power is dissipated as heat energy per second and then we can also use Ohm's law and we can write this as I squared R I squared R which is also the same as we squared over R V squared over R and if you look at it carefully we we are given heat produced per minute from this we have to calculate what is the heat produced for a second that's going to be our first step because the formula over here deals with heat breeze per second but once we know that then we know this and we know the voltage so from that can we calculate the current yes we can we can use the same formula as before and calculate the resistance and then maybe we can go ahead and calculate the current so again pause the video and see if you can try this yourself all right let's do that the first step as said as mentioned would be to calculate heat produce per second instead of per minute so how do we do that well what we can do is we can convert minutes to second because I want second in the denominator right so this can be written as so let's write this let's write that down somewhere over here let's continue this one minute is how many seconds one minute is 60 seconds so I would just write this out joules per 60 seconds and the zeros and the zero cancel and so I end up with 24 by 6 that's 4 joules per second so that's how much heat is produced in our boat and now that we know this we can plug it in over here so I'm going to use this formula heat produced per second is going to be V squared over R so let's take some more space over here all right so heat produced per second that's 4 joules per second I'm gonna I'm not going to put in the unit's because I already know the unit of resistance that's homes right so I'm not gonna put in units over here so for that's going to be V squared over R V is 2 so V squared is 2 times 2 divided by R and then we can just do the algebra we can multiply both sides our and we'll divide by 4 on both sides so we'll get R on the left hand side and that'll give me 22 that's 4 divided by this for this cancels and it gives me 1 so I get one oh and that is the resistance so the resistance of the bulb must be 1 oh all right what else do we want to find out we want to find out what's the current well now there are a couple of ways in which we can do this either we can go ahead and use Ohm's law V equals IR because we know V and we know R yeah we know R and so we can go ahead and calculate I or you can also use this and calculate you should get the same answer so I'm just going to go ahead and use Ohm's law I'm gonna go ahead V equals IR so I would be V over let's write that down over here I equals V over R V over R and that's going to be V which is 2 divided by r which is 1 and that's going to be 2 amperes 2 amperes and so that's the current through our bulb and that is this is the resistance through our bulb there we have it