Area expansion coefficient

when you heat or cool things not only with the length change but their area and volume changes as well in this video we're gonna find out how to calculate changes in area when you heat things up or when you cool them down imagine you have a gold plate sitting on your table with all your other gold stuff and let's imagine it's a perfect square so it has a length L so it's length is L and the temperature of that plate right now let's call it as T well you heat it up that's what you do you heat up that plate and you see the whole thing expanding so here's what you see the whole thing expands and as it does carefully see that it's length has expanded over here and we'll share as well in both sides the length has expanded and therefore there's an extra area the area has expanded the area has increased let's keep that over here so let's say right now the new temperature is T plus delta-t T plus delta-t and it's new length well that would be the old length L plus some extra length Delta L some extra length Delta L and the same story on this side will be L plus Delta L over here and now the big question is how much has the area change that's what we need to figure out so how do we do this well we already know how to take care of chile's in length we have seen previously that the change in the length during thermal expansion or contraction can be written as Delta L equals our foil our foil is the linear expansion coefficient times L times delta T we've talked a lot about this expression in previous videos and so if you're not familiar with this or maybe you need a refresher or something it'd be a great idea to go back watch those videos first and then come back over here so we know how to take care of lengths and changes in length so the question now is how how to calculate age in well all we need to do is ask ourselves how do you get area from linked hey that's easy for square the area is just the side square right let's let's write that down so the area initially that would be L square site squared is L square this is the initial area what would be the final area well the final area let's call it as a - that would be the final side squared right so final side length squared so what's the side length now well it is L plus Delta L so it would be L plus Delta L the whole square and now to calculate the change in the length we just have to subtract these two so the change in the area we write that over here that's going to be the final area which is over here L plus Delta L plus Delta L the whole squared minus the initial area and that is just l squared and at this point I want you to pause the video and see where the algebra takes you just go with the flow and see if you can do something just see what expression you end up with all right let's do it so what can we do next well we have an A plus B whole square form let's expand that it'll be a square plus B square plus 2 a B so if you expand this you'll get a squared plus B squared plus 2 to a B - l squared and notice this cancellous and so we end up with we end up with - L Delta L plus Delta L whole square that is our change in area and at this point you may say okay there's nothing more to do right but there is once one more step we can do over here well we need to remind ourselves that Delta L is an extremely small quantity check out over here we've seen this before also that when things expand Delta L is much smaller than L so what can you say about L Delta L and Delta L square let's compare them can you see that L Delta L is going to be a much larger quantity compared to Delta l squared let me give you an example let's say L is something like 100 then Delta L would be something like point 1 point 1 so in that case 100 times 0.1 well this this term forget about the - forget about the - I'm just kidding about this term that would be just 10 well this would be point 1 squared that would be point 0 1 now 10 plus point zero 1 can we approximately say it's almost 10 we can do that right if if you don't want exact result we just want approximations so in this case just to simplify this we can just forget about this so we could forget about this and we can just forget about this right so we can neglect Delta L square because it's much smaller compared to L Delta all right let's do that ok so we will neglect this term so this term is neglected so neglect this and so what we now end up with is Delta a equals to L times Delta L but what is Delta L we already know that and you might be wondering well country isn't this only applicable for wires and rods no it's applicable for any object 2d 3d any kind of object even always here we can apply this so we could just say well we already know how to calculate Delta L Delta would be alpha L our file times L times delta T oh this is getting interesting now let's see what we end up with we end up with and just put a division over here all right so what we end up with is we get to alpha L let's write that first to alpha l times l squared or what's L squared oh that's the area that's the initial area so that's just a times delta T let's go ahead and box this because I think we have simplified this as much as possible and now now just see what we have done we have now figured out the change in the area so if we know what the initial area is and we know what the change in temperature is and if we know the Alpha L for material then we can calculate Delta a we don't need anything more so just by knowing this we can figure this out but can you see the similarity between this expression and this expression there are very analogous to each other just like how you have change in length equals some number times length original length times delta T you have change in area equals some new number some new constant times area times delta T so just like how alpha L is the linear expansion coefficient tells you how much length has expanded this to alpha L we can now call that as the area expansion coefficient and we often write this as alpha a and we call this as the area expansion getting a little crowded over here Co efficient so just like how alpha L helps us calculate changes in length alpha a helps us calculate changes in area and if it any numerical they ask you to calculate change in area and alpha is not given to you and only alpha L is given to you you can still do it because alpha is just 2 times alpha but one thing we need to remember though is that we arrive at this expression by a zooming dealt by L by neglecting Delta L squared and we can only do that when Delta L is very small compared to L in other words this whole thing is an approximate relationship which works when the expansions are very tiny and one last thing which you can check for yourself is that alpha a should have exactly the same units as alpha L because it's just 2 times alpha L right so 2 has no units so it has the same units Kelvin inverse would be the units of alpha now you may wonder that we derive this expression and we got this result for a perfect square plate but would it work for any other shapes the answer is yes and you can try and do it yourself for shapes like say a rectangle or a circular plate try them it would be a great exercise but it turns out that in general if you take any shape even 3d for that matter and you calculate the changes in the area to do due to changes in temperature you end up with the same expression you end up with exactly the same result as long as we're dealing with very tiny changes