Maxwell's Equations: Gauss' Law Explained (ft. @Higgsino physics ) | Physics for Beginners

hey what's up you lot path here coming at you this week with one of my most requested videos yet yeah that's right it's the third in the Maxwell equation series this one's a super interesting one the equation that we'll be looking at is this one here and we've actually got some help with us today that's right oh boy mr. Higgs II know physics is going to be here helping us to understand this equation if you enjoyed this video then please do hit the thumbs up button and subscribe to my channel if you want some more physics e content and if you haven't seen my two Maxwell equation videos that I've made before this then definitely go check them out up here as well well that further ado let's get into it so like I said in today's video we will be studying this equation here this is one of Maxwell's equations of electromagnetism we're going to try and understand what this equation is telling us in as intuitive and visual away as possible so as always you don't need to be studying like university-level maths or physics to understand what's gonna go on here so this particular equation here deals with the behavior of electric fields these electric fields are generated by charged particles or charged surfaces or charged objects in general we're looking a bit more detail in a moment what we mean by electric fields but let me start out by saying that an electric field is a very specific kind of example of something known as a vector field basically a field of vectors let's say for whatever reason we happen to be studying this region of space and we're studying a vector field in that region of space well we can think of a vector field as basically this region of space and every single point in that region of space can be assigned a particular vector vectors of course being represented by an arrow with a certain length and pointing in a certain direction we're not talking about electric field specifically just yet we're just thinking of a general vector field in a general vector field these vectors can represent one of many things for example this could be a vector field representing the speed and direction of wind at each point above the United Kingdom or we could have a vector field that describes the direction and speed of little sneeze particles that come out every time somebody sneezes topical right or we could have a vector field that describes the size and direction of the gravitational force exerted by a planet on a mass placed at a particular point in that vector field if you want a more in-depth discussion of vector fields than I've talked to these in my first Maxwell equation video check it out up here but now let's look at electric fields electric fields being a particular example of a vector field here is an electric field this field represents something very closely related to the force exerted on a charged particle that we place at any point in their field basically if we take let's say a positively charged particle and this charged particle we can say has a charge of one Coulomb and we place it at this particular point in the electric field then this diagram tells us that this one Coulomb charged particle will experience a force in this direction the magnitude or size of that particular force is directly related to both the size of the vector starting at that point as well as the magnitude or size of the charge specifically the charge of the particle multiplied by the length of the vector or the size of the arrow gives us the magnitude or size of the force exerted on that charged particle as a clarifying example we could take that same charged particle and place it now here in the field this diagram tells us that the force exerted on a charged particle when placed here will be in this direction and will have a smaller magnitude in this case because even though the charge on the charged particle is still one Coulomb the original vector had a smaller magnitude and so the force exerted on the one Coulomb charge is going to be smaller at this location and just to clarify things a little bit more if we took a negative 1 Coulomb charge and placed it in this electric field let's say at this point once again then the force exerted on this negatively charged particle would be equal to the magnitude or size of the vector multiplied by the charge but this time the charge is negative so the force exerted on a negatively charged particle is going to be in the opposite direction to the force exerted on the positive particle because both the positive and the negative charges we considered had the same magnitude the forces will be of the same magnitude just in opposite directions so this teaches us something important when we take a positive charge and place it in an electric field the electric field diagrams are drawn in a very specific way such that the force experienced by a positive particle is in the same direction as the electric field diagram and the force experienced by a negatively charged particle is in the opposite direction to the arrows or the vectors drawn in our diagram this is by convention scientists sat down a long time ago decided that the vectors used to represent electric fields will be pointing in the direction of forces experienced by positively charged particles is this a choice basically it doesn't really matter but the majority of the physics world goes by this convention but more interestingly we also learn that the force experienced by a particular particle in an electric field is equal to the charge on that particle multiplied by the magnitude of the vector at that particular point in the electric field so the electric field is a representation of forces felt by charged particles whilst also removing the dependence on the charges of those charged particles so coming back to our equation from earlier that's what the e stands for the e represents the whole of the electric field and you'll notice that there's an arrow above the e this tells us that it's a vector or vector field but anyway that's not so important next what we can do is to look at what this downward pointing triangle is for those of you that have seen my first Maxwell equation video again I've talked about this in a bit of detail and you'll recall that the downward pointing triangle when specifically paired with the dot in between the triangle and the e this is known as an operator called divergence let's think about what that means first of all like I said we're dealing with an operator it does something mathematically to our electric field representation and secondly this particular mathematical operator is known as the divergence operator and so just out of interest if we want to read or interpret the left hand side of this equation we can say that this is taking the divergence of the electric field e will come back to the right-hand side of this equation in a moment but first let's look in a bit more detail at the divergence mathematical operator again this divergence operator can be applied to any generic vector field not just specifically the electric field but what do we mean by finding the divergence of a vector field well to answer this question I am bringing out the big guns that's right mr. Higgs Eno is going to be answering this question for us he's going to show us a very intuitive and visual way of understanding what it means to take the divergence of a vector field as opposed to just a mathematical understanding take it away thank you very much for inviting me past talking about one of the best equation is truly an honor anyway so I'll be talking about the divergence operator here so this part of the equation of Gauss's law and all explanation of this step versions over is that it kind of misha's flow around a vector fields so for example if you have a positively charged particle sitting here that we can see around the circle here we have an outlet flow of lexus and so we say the divergence is positive or in this example we have a negatively charged particle and around the same region the basis will point inwards as if this particle was a sink and so we say the divergence around this field here is negative that's also a different scenario which is just pointing in one direction but the same size it could be when moving around with no turbulence and wherever we put our circle there are as many vectors going in as going out and so the divergence is zero everywhere so that's a fine a good explanation I think but we can do better it will make things easier to understand the exam we looked at we only had a couple of weights are strong but of course this is just a representation so in reality there will be infinitely many vectors just like between 1 & 0 there's infinitely many numbers and the reason there is an infinite amount of vectors is of course because the vector field is given by an equation so for example this equation could render this vector field and again we only draw some basis or another equation 1 over R squared will result in this field here now ting the divergence means that everywhere in this field we measure how much is flowing in out not in a big area but in an infinitely small sphere or actually point I guess and we do that everywhere on the fields so now before seeing the divergence operator every point on space had a vector pointing in some direction but after taking the divergence operator every point of space has a number associated measuring how much of this vector is flowing in out let's try and examine with our previous equations so let's go back to this one if we take the divergence of this field it turns out if we get this equation and so when you plug in the numbers this is what the divergence look like you're also try with your other example and here comes something special because we have a 1 over R to the power of 2 the divergence is zero everywhere I'm telling you this because that's how charges behaves so this equation represented an electric field of a charge and this is what this equation means everywhere we don't have a charge the divergence is zero so here is zero here zero here zero but here we have a charge and so the divergence equals to the charge density epsilon0 I know it might not seem as interesting but it is extremely important for calculations where symmetry is special for example if we have an infinitely big charged plane so here we have charges in recreant each fill going out like this now we can construct what we call a Gaussian surface in the e field it's just whatever surface we want and so the idea is to measure how much is flowing in and out because we know the divergence is zero everywhere we can construct whatever surface you want so somewhere potato a sphere a box or we can construct something that matches the symmetry such that the calculation will be easier to do so here it is a cylinder the electric vector field flow will be zero if we don't have any charges in capsules because the divergence is zero everywhere and so we can place this Gaussian sphere so it in capsules sum of this charge plane and now we can calculate the electric fields classic so huge thanks to mr. Higgs II know for that explanation if you haven't heard about his youtube channel or haven't seen any of his videos then i highly recommend heading over to his channel and giving him a subscribe as you've seen he makes some very intuitive and visual videos helping you understand different complicated concepts and physics so head over there and tell him I sent you but anyway we've just seen what it means to take the divergence of a vector field essentially we're just measuring how much of that vector field flows in or out of a particular region of space that we happen to be considering an interesting example we saw in mr. Higgs his explanation was the one over R squared a vector field again we're not too worried about the actual maths affair we're more worried about what the vector field looks like what the arrows look like and what direction they're pointing in well as it turns out a negatively charged particle placed at the origin in our original vector field will produce an electric field that looks identical to this what this means is we could take another particle let's say our positively charged particle from earlier and place it at some point in this electric field when we do we see that the force on the positively charged particle in this particular case is pointing toward the negatively charged particle and this hopefully makes intuitive sense we've learned maybe all the way from high school that positively charged particles and negatively charged particles attract each other remember the convention that we use for electric fields is that the arrows point in the direction of force experienced by a positive particle play at that particular point so we could take a positively-charged passcode and place it anywhere in this electric field created by the negative particle and the positive particle would always experience a force pointing toward the negatively charged particle in other words they're both attracting each other and the other interesting thing is that the vectors closer to the origin the charged particle which created the field are larger West farther away they're smaller which also makes intuitive sense in that let's say we took a positively charged particle and placed it here well that particle is going to be more strongly attracted to the negative particle because it's closer whereas if we place it much further away it's going to be much more weakly attracted to the negatively charged particle they still attract they're just much further away so they don't attract each other as strongly let's return to our Maxwell equation now we've seen that the divergence of our electric field is a measure of how much of that field is flowing in or out of any particular region of space that we want to be considering and this particular equation is telling us that that divergence is equal to Rho divided by epsilon not so what are these two quantities Rho and epsilon not well Rho is the electric charge density inside the region of space that we happen to be taking the divergence for basically this means the total charge inside that region divided by the volume of that region it's the charge per unit volume inside that region we're looking a bit more detail at this in a second but let me just quickly explain more epsilon naught is epsilon naught is a fundamental property of the vacuum the empty space of our universe this particular constant of nature is known as the permittivity of free space or vacuum permittivity and this is a constant that comes up a lot when discussing electromagnetism I'll talk about this more in a separate video but for today it just represents a constant and this constant is a property of vacuum of empty space because what we're considering when we're talking about electric fields is empty space aside from where we place charges let's now look at linking the two sides of our Maxwell equation the divergence of the electric field in a particular region of the electric field that we want to consider and the charge density inside that particular region of space let's say we want to take the divergence of the electric field in this particular region here I'm not going to draw 3d just imagine that this is a sphere rather than a circle and we can see that in this particular region there are vectors pointing into the sphere and on the other side there are vectors pointing out so as much of the vector field flows into a sphere as it does flows out this means that the flow in is exactly balanced by the flow out and the divergence of our electric field in this region is zero and this exactly works with our Maxwell equation because the charge density in that particular region for this particular extra field is zero there is no charge inside this particular region of space and so the charge per unit volume is also zero so for this particular region of space left hand side is zero right hand side is zero equation makes sense let's now consider this region of space this gets a bit more interesting we can see that here all of the vectors are pointing in to the sphere none are pointing out and from earlier we said that this conventionally was and negative divergence so over all the left-hand side of our equation is equal to some negative value let's call it negative a it doesn't matter what the exact value is but the interesting thing is that here it's not equal to zero and so this means that the right-hand side must also be some negative value which it is because remember on the right hand side we've got the charge density in that particular region of space which is equal to the charge in that particular region of space divided by the volume of that region and of course we've got the scaling factor epsilon naught which like I said for this video is just a constant and so we see that the charge negative Q divided by the volume let's call that V divided by epsilon naught as well must be equal to negative a again particular values don't matter but what we see here is that on the left hand side we've got some negative value and on the right hand side we've got some negative value as well the equation again makes sense but this equation that we're thinking about one of Maxwell's equations technically Gauss's law because gas came up with us first does a lot more than just mathematically relate the left-hand side of this equation to the right-hand side in fact it's a prescription as to how electric fields must behave what this is telling us is that if we have a particular kind of charge maybe a charged sphere where the charge is distributed all over the surface of that sphere as opposed to just a charged particle now the electric field generated by that charge sphere must behave in a very specific way and that specific way is the following the charge density in any volume that we want to consider any particular volume that we want to consider divided by epsilon naught must be equal to the divergence of the electric field and this isn't just true for a charge sphere or a charge particle this is true for any distribution of charges so this particular equation Gauss's law is very prescriptive in a way it tells us the electric field generated has to behave in a very specific way so that's an overview of this particular equation here I've simplified many things down and I know that I haven't gone into like the intricacies and the particular details about this equation but hopefully I've explained it well enough that it makes sense let me know in the comments below if that's not the case I do have one more thing to talk about though I keep talking about the first Maxwell equation video that I made a while ago now and the reason for that is that the equation I talked about in that video is very very similar to the equation that we talked about in this video that equation deals with the divergence of magnetic fields and it tells us that the divergence of magnetic fields is always zero and this shows a fundamental difference between classical electric fields and classical magnetic fields while electric fields can have a nonzero divergence they're dependent on the charge density of any particular volume that we want to be considering magnetic fields cannot have a non 0 divergence they must always have zero divergence which can be boiled down to one very specific thing magnets can only exist as both a north and a South Pole together at least that's what these particular classical equations tell us I've gone into a bit more detail about this about whether or not that's true at the moment it seems like it is in that previous video but for the moment if these Maxwell equations are a good description of our universe then magnets cannot exist solely as north poles separate from South Poles because if they did we could have regions of space where the divergence of the magnetic field was nonzero but it turns out that magnets if they follow Maxwell's equations cannot exist like this they must exist as North and South Poles together and with all of that being said I'm going to end this video here guys thank you so much for watching as always thank you so much for the support that you give me and thank you to mr. Higgs Eno for providing such a brilliant explanation as part of this video head over to this channel add subscribe to him if you haven't already he makes some really really cool stuff and if you enjoyed this video please leave a thumbs up and subscribe to my channel for more physics e content hit the bell button if you want to be notified every time I upload new videos and if by any chance you're interested in listening to some music that I produce or if you're a metalhead if you like metal music then head over to my second channel party shenanigans I'll leave a link in the description below last bit of self plug if you're interested in following me on instagram and head over to act path blogs and follow me there and as you can probably tell by this point I've spoken way too long so I'm gonna end this video thank you so much for watching I'll see you really soon buh buh buh buh bye

Channel: Parth G

Views: 61,309

Rating: 4.9729657 out of 5

Keywords: physics, maxwell's equations, gauss's law, maxwell's equations in differential form, electromagnetism, divergence, curl, gradient, vector calculus, electric field

Id: pTMh1yyqVC8

Length: 18min 13sec (1093 seconds)

Published: Tue Apr 21 2020