Poisson's Equation for Beginners: LET THERE BE GRAVITY and How It's Used in Physics | Parth G

big thanks to skillshare for sponsoring this video check out a free trial of skillshare premium by clicking the link in the description below hey everyone parth here and in this video we'll be looking at the meaning behind poisson's equation thank you very much for voting for this topic in the poll over on my community tab as always we'll be trying to deal with this rather complicated mathematical equation without going into too much heavy mathematics so if you enjoyed this video then please do hit the thumbs up button and subscribe for more fun physics content let's get into it now the general form of the poisson equation can be written like this we'll be looking at the meaning of each one of these symbols in a moment but before we do you may have seen the poisson equation written in a few different forms for example like this as well this is the version of poisson's equation used when studying gravitation we'll be taking a look at that too but first let's focus on the more general form we'll start by understanding the mathematical meaning of each one of these symbols and then applying some physical intuition to it the first thing we see in our equation is this downward pointing triangle squared this symbol has a very particular meaning in vector calculus the first thing we need to know is that the downward pointing triangle is known as a nabla or a del and we can think of it as a vector containing partial derivatives d by d x d by d y and d by d set each one of these measures how quickly a certain quantity changes over a small distance in the x direction the y direction and the z direction so for example if we have a packet of flour and we open that packet and then we squish it so the flour goes everywhere and then we plot how much flour is found at every point along the x direction for example let's say our flour distribution looks like this lots of flour near the origin and then less and less as we get further away from the origin well if our flour distribution is labeled f let's say we want to find df by dx this simply measures how quickly the flour distribution changes as we move along the x direction we can think of this as measuring the gradient or slope of our flower distribution over here for example the gradient is quite steep so our flower distribution drops off quite quickly whereas over here the gradient is not so steep so the flower distribution isn't changing a huge amount in this region now we could apply what's known as the nabla operator to our flower distribution function and the first term essentially would just give us df by dx as we've seen except we've got curly d's here we don't have normal d's luckily though there's a simple way to think about the curly d's if we realize that the flower distribution doesn't just change over the x-direction it also changes over other directions and the change in other directions might be different to the change in the x direction then we can understand that the curly d's are telling us that we're only measuring the change of our flower distribution over the x direction whilst assuming that the flare distribution remains constant in other directions similarly curly df by d y measures how quickly the flower distribution changes in the y direction whilst keeping everything else constant and so we can think of the curly d's as essentially isolating the change that we're trying to look at in a particular direction without worrying about any other direction if you're unfamiliar with partial derivatives by the way i'll leave some resources in the description box below anyway so this is what the downward pointing triangle nabla or del represents as always we've just sort of scratched the surface there's obviously a lot more to it than that but for the purposes of this video we've covered what we need to know the thing is though poisson's equation actually has del squared rather than just del now if we're thinking of squaring a vector how how does that work well del squared is simply a notation that is actually used to represent del dot del that's the dot product between two del vectors for those of you familiar with gradient divergence and curl what we're actually looking at here is the divergence of the gradient of some quantity but that's not what's important here what is important is that the dot product or the scalar product between two vectors is given by multiplying the corresponding components of those two vectors and then adding up all these little products and we do a similar thing when finding the dot product between these two dell vectors except we don't multiply together the components what we're technically doing is applying a partial derivative on a partial derivative in all three cases and then we add them together the same way we would for a normal vector again for those of you familiar with calculus we can write these as the second derivative in x y and z of the function f whatever f may be and on the other side of our poisson equation we have some other function we haven't really specified what this function is but all we're saying is that in this particular case the function f has to be such that the second derivatives as we've seen on the left hand side have to all add up to give us this other random function phi now before we go any further i'd like to thank this video's sponsor skillshare skillshare is an online learning community where you can find a large number of inspiring classes focusing on topics such as productivity and lifestyle to building a business to learning creative skills many of you may know that one of my hobbies is creating music check out my music channel linked below and i've taken some classes on skillshare that have taught me some really cool skills for example i took a class called audio mixing on the go professional sound without the studio by king arthur which gave me lots of tips for improving my mixes without lots of fancy equipment and that's the key here skillshare has a large number of classes to choose from and it's all about learning so there are no adverts and skillshare costs less than 10 a month with an annual subscription but the first 1000 of you to click the first link in the description box below will get a free trial of skillshare premium please do go check it out and big thanks to skillshare once again for sponsoring this video now so far we've had a very generic look at poisson's equation nothing specific nothing intuitive because we've just said that the functions found in poisson's equation are just some random functions they've not really represented anything physical yet so let's have a look at that let's look at how poisson's equation is used in the study of classical or newtonian gravitation we'll start by considering a law in classical gravitation known as gauss's law gauss's law deserves a video of its own and i'll leave some resources in the description below if you're unfamiliar with it gauss's law looks a little bit like this and to understand it let's imagine that we're considering the gravitational field of the earth we may know already that earth's gravitational field is meant to look like this radially inward pointing gravitational field lines meaning that any little mass placed at any one of these points will accelerate towards the center of the earth but how do we know that earth's gravitational field looks like this well this is where gauss's law comes in we begin by considering a random surface a closed surface that entirely encapsulates the earth for simplicity's sake we will consider this spherical closed surface then we break up this arbitrarily chosen surface into lots of little area chunks we'll say that each one of these has a little area d a and what we can also do is to represent each one of these surfaces with a vector that vector is exactly perpendicular to each one of these surfaces so instead of a surface we now just think about a vector and the size of that vector corresponds to the area of the surface this by the way is a really clever way to represent surfaces in general because you can simply think of one single vector rather than a whole flat surface we get all the information we need from the vector because the direction in which the vector is pointing we know is directly perpendicular to the area and the size of the vector gives us the size of the area and so this is how we mathematically encode something about those little area surfaces into our working next we see that gauss's law tells us that we want to find the dot product between the gravitational field at every single point in space which is exactly what we're trying to find here and the little area vectors d a the dot product by the way is a measure of how well two vectors align with each other so if they're both pointing in the same direction then the value of the dot product is as large as it can be and if they're both exactly perpendicular to each other then the value is zero and in this case then we're simply measuring how well the gravitational field line aligns with our little area vector or another way to think about it is how much of our gravitational field lines pass through those little area elements so we find the dot product g dot d a for each one of these little area elements and then we add up all the contributions to find what's going on over our entire closed surface and that's exactly what the integral sign is about it's telling us to add up all of those little contributions and the sum over the entire closed surface must be equal to minus 4 pi g m where g is the universal gravitational constant and m is the mass enclosed within our closed surface in this case the mass of the earth now finding a solution to this is quite tricky but we do find that the gravitational field along this entire surface must look like this and when we choose other arbitrary surfaces for which to do this same process again we find that the full gravitational field of the earth looks like this just as an intuition thing by the way there's a negative sign here because the little area elements had vectors that pointed out of them whereas the gravitational field points in the exact opposite direction it points inward so gauss's law can help us find the gravitational field of anybody we happen to be considering and we've been looking at gauss's law in integral form but it can be written in differential form which looks a little bit like this what we see here is that the divergence of g is equal to negative four pi g rho where all the symbols have the same meanings apart from rho which is actually the mass density that's the mass per unit volume in our enclosed surface now again we don't need to worry about what divergence actually means but we can see that this equation that we've got looks pretty similar to poisson's equation from earlier we've got some function on the right hand side minus 4 pi g rho and the only difference is that we've got the divergence of some vector quantity rather than del squared of some scalar function luckily for us the gravitational force is what's known as a conservative force what this means is that if an object is placed in a gravitational field and then move to some other point in that gravitational field it doesn't matter how it got from point a to point b the work done in order to take the object from point a to point b is the same regardless of how it got from a to b now mathematically this translates to the curl of the gravitational field being equal to zero how that's equivalent we'll discuss some other time but the important thing here is that there's a mathematical identity so not anything to do with physics but it's all purely mathematical that says that the curl of the gradient of a scalar field is always equal to zero and because we've seen that the curl of our gravitational field must be equal to zero what this tells us is that we can write our gravitational field as the gradient of a scalar field we can write that g is equal to grad v where v is a scalar field at which point we can go back to gauss's law and substitute in grad v for the gravitational field g and so now what we have on the left hand side is the divergence of the gradient of some scalar field v but also remember that del dot del is equal to del squared and so we've arrived at the poisson equation for gravitation this is a specific use of the general poisson's equation we saw earlier in the video we arrived here by starting with gauss's law and then remembering that gravity is a conservative force field by the way the scalar field that we've substituted in here v is known as the gravitational potential many of you may have heard of this and it's quite similar to a gravitational potential energy but it's not quite the same thing what exactly the differences are between those two well i'll leave that for you to dig into at this point i'm gonna finish up because this is a long-ass explanation already thank you so much for watching if you enjoyed it please do hit the thumbs up button subscribe for more fun physics content and hit that bell button to be notified when i upload check out my patreon page if you'd like to support me on there and thank you so much to skillshare for sponsoring this video once again check out the first link in the description box below for a free trial of skillshare premium i really appreciate all your support as always thank you so much for watching and i'll see you very soon you

Channel: Parth G

Views: 78,412

Keywords: poisson equation, poisson's equation explained, gauss law of gravitation, parth g, physics, poisson equation physics, laplace operator, nabla operator, del operator, classical physics, conservative force

Id: k91KDItxif0

Length: 12min 11sec (731 seconds)

Published: Tue Mar 09 2021