 I'm Professor Stephen Secula, and I'd like to welcome you to Modern Physics at SMU. Modern Physics is the first course past the two-semester introductory sequence, where we begin to get closer to the physical principles that are at play in the modern technological and scientific world. Now, like all good sciences, physics builds on the past discoveries that have driven the field forward, to continue to move through the frontier of human knowledge. Modern Physics is going to challenge you to step beyond the comfortable confines of introductory physics and into the modern view of the universe, particularly space, time, energy, and matter, which are the four subjects at the heart of the science known as physics. Well, introductory physics tends to leave off at the end of electricity and magnetism, and that period coincides with the end of the 19th century, the late 1800s, which was a feverish time of experimentation. The foundations of space and time had been laid in the centuries before by people like Isaac Newton, and the laws of electricity and magnetism, which were relatively the new kids on the block as regards scientific law, had been established firmly in the mid-1800s. It was believed at the time that, for the most part, everything that needed to be known about the universe had been established, and all that was really left was to sort out some lingering puzzles that hadn't quite yet been fit into the framework of Newton's mechanics and the laws of electricity and magnetism. Now, one of those phenomena is light. It's fairly straightforward to make a light source in the modern world. All we have to do, for instance, is take a sealed tube filled with gas, for instance, and expose it to a strong electric field, a large electric potential. And we can coax it into emitting light. Now, light is, of course, all around us. It's what's literally illuminating the scene here. What light was was only thinly established by the end of the 1800s. We'll look at the foundations of light and electricity and magnetism and mechanics in this video. But one of the puzzles that was left over at the end of the 19th century was why certain elements emitted certain kinds of light, but not others. So for instance, this sealed tube here is containing hydrogen gas. And you'll notice that it gives off a fairly strong reddish color. On the other hand, if I place it with another tube of gas, this one of mercury vapor, mercury being a metal, the only elemental metal that's liquid at room temperature, in this case, sealed as a vapor in the tube, you'll notice that in this case, we get a very strong blue light from this particular element. If you study the fingerprints of these light emissions very closely, you'll observe that they have strong colors in some places, but not in others in what is known as their atomic spectrum. Why was that? The mystery of atomic spectra would only be fully understood in the early part of the 20th century with the advent of what we now call modern physics, specifically quantum physics. Now another interesting phenomenon that had been observed in the 1800s, but which was not fully understood, had to do with electric currents. So what I'm going to do here is using the triboelectric effect, using friction to build up an electric charge on a piece of plastic. And then placing that plastic in contact with a conductor so that it can soak up that excess charge. You'll see that I've now caused a net electric charge to sit on this mylar material attached to this aluminum soda can. Nothing dramatic here so far. The charges can freely move on the conductor. And in this case, they don't like to be near each other because they're electrons and they all have the same electric charge. And so they rush as far apart as they can get while remaining on the conductor. So they're trapped on this conductor, they can't escape, but they've done everything in their power to get away from each other. And in the process, they've exerted an electric force that mechanically causes the mylar to spread out in space. Now, while I've been talking, the light in this room, which comes from something like a dozen light fixtures in the ceiling, has been blasting this metal. And yet, while there's a breeze in the room that moves the mylar sheets around, nothing's really draining the charge off of this. We don't see these sheets appreciably falling down. So we have many watts of natural light coming in from light sources here. Nothing happens. But it was observed in the 1800s that if you expose certain metals to certain colors of light, or even colors of light that are beyond human perception like ultraviolet, they'll begin to allow an electric current to flow. This is the so-called photoelectric effect. And it was unexplainable by Newton's mechanics and the laws of electricity and magnetism as they were established in the 1800s. So all I have to do, according to the photoelectric effect idea, is take something that emits ultraviolet radiation. In this case, a sanitizing wand for a sink or a counter or a toilet or something like that. Just go ahead and switch it on. And it will begin to emit ultraviolet C radiation or light down toward the surface of the table. And if I move it over the aluminum can, it will begin to almost instantaneously drain the electric charge off the mylar. And if I sit here long enough, it will eventually pull almost all of that electric charge off the mylar, leaving it hanging back down in a more vertical position. This is four watts of UV light, compared with tens of watts of natural light coming from the light fixtures in the ceiling. Why? Why does the color of the light matter to this effect more than the intensity of the light? That was a mystery left over at the end of the 1800s. Now another mystery which would ultimately combine to lead to a firmer understanding of matter, energy, space and time had to do with heat energy. Now heat energy is something that we will explore in this class. If I light this burner on fire so that it emits a nice blue flame, I can take a bimetallic strip. This is a strip of two metals bonded together back to back. And I can place it in the flame. Now we'll take a look at some of mechanically what's going on here later in the course when we establish the foundations of heat energy in about a month or so. But if I leave this metal in the flame, not only does it begin to bend, but after enough time, it will also begin to glow on its own. Originally it had a silvery color at room temperature. But as I continue to expose it to the source of heat energy, this open flame, and heat energy is transferred into the metal, eventually the metal begins to glow of its own accord with its own light. Now this wasn't such a mystery to physicists and chemists of the 1800s, but what was a mystery had to do with the degree of absorption of energy and the degree of re-emission of energy at other frequencies and wavelengths, other kinds of light that you can see with your eye. And the exact relationship between heat energy and temperature and the kinds of radiation that should be emitted from a heated body proved a real challenge to mechanics, electricity, and magnetism, and the laws of heat energy transfer or thermodynamics that were also established in the 1800s. So modern physics is your gateway into a world that's more consistent with the kind of world we live in today, not the world of the 17 and 1800s, but the world of the 1900s, the 20th century, and now the 21st century. The foundations that we will establish in this course will lay the groundwork for a variety of important technological revolutions, non-invasive imaging of the human body, the harnessing of the energy at the heart of every atom, the construction of semiconductor devices which revolutionized our ability to do computations quickly and efficiently, and a host of other advancements whose roots were all laid down in a revolutionary period in the transition between the 1800s and the 1900s that led into the era of modern physics. Welcome to this course. For the rest of the video, we'll do a foundation's review of introductory physics to refresh your memory about the most salient things from the past two semesters of material, and then we'll move on to the foundations of modern physics. In this lecture, we'll re-explore the foundations of introductory physics, the basic concepts that should have been communicated to you in the first two semesters of introductory physics. Physics builds on the past, like all sciences, the discoveries of an earlier era influence our understanding of new discoveries and how to adapt our mathematical descriptions of nature in order to describe what we know from the past but include new observations that don't quite fit into the original framework that we had developed. The big picture that I want you to take away from this foundation's lecture can be broken into four large parts. First of all, a foundation of the physics that you have learned so far is Newton's mechanics. These are laws of motion. They link forces that act on objects to changes in the states of motion of those objects, and a state of motion is characterized by the velocity of an object. The laws of mechanics were first established by Isaac Newton in his foundational publication, Philosophia naturalis Principia Mathematica, or the Principia, published in 1687. This set of laws illuminates how velocity, the state of motion of an object, can be influenced by external forces and codifies mathematically using geometry, algebra, and the newly invented calculus, the way in which you can describe the interaction of these things in order to understand the natural world. What would also be developed over the following centuries were a series of what we now call conservation laws. These are principles that establish that certain quantities appear to be conserved that is left unchanged even by complex phenomena in nature. These include things like the total energy of a system, including internal forms of energy like chemical energy, the total linear momentum of a system, and the total angular momentum of a system, and foreclosed in isolated systems where no external forces, especially of the non-conservative variety, those that can't store and release energy in some kind of potential. In those systems, conservation laws will absolutely hold, and they were established through careful chemical and physical work up through the 1700s, and they continued to be built on in work on heat energy in the 1800s. Heat energy and the laws that govern its transfer from the mechanical form to the thermal form will be revisited in later lectures, the second part of the foundations of modern physics. The third key idea is Newton's law of gravitation. That is the law that relates the distance between material bodies and the force between them, a force that requires no actual physical contact, no medium to be present between two things in order for them to exert a force on each other, and this was also established in Newton's Principia. And finally, the last set of laws of physics that we have to accept as a foundation for what's going to happen in this course are the laws of electromagnetism. These are the rules of electricity and magnetism, describing them as forces in the same way that gravity is a force, that can induce changes in states of motion, again without physical contact between material bodies, electric forces and magnetic forces can operate even if there's no medium between the two bodies that are interacting with each other via these forces. They were established in the 17 to 1800s, and they were finally codified formally in four equations known as Maxwell's equations in 1862. One of the mathematical foundations of describing nature in physics is a kind of number known as a vector. These are essential to describing any multi-dimensional quantity, and they have a well-defined algebra which you should have exercised in previous physics courses. You probably have also exercised these in a dedicated math or engineering or both course. Vectors are numbers that can be built from scalars. Scalars are numbers that have no directional information. So for instance, a good example of a scaler would be if you asked for directions to somebody's house and they told you, go 10 miles. Well, that might eventually, by going 10 miles, get you to their house, but without some crucial directional information, how far east, south, north or west should I go to add up to those 10 miles, you're probably not going to make the journey successfully. Scalars, however, can be assembled using, for instance, component notation into a vector. So here, for instance, is demonstrated a vector, denoted A with a little arrow over its head. And it's broken into components. It has a component that lies entirely along the X coordinate axis in a Cartesian coordinate system with length A with a subscript X. And similarly, it has a component along the Y axis in a Cartesian coordinate system, A with a subscript Y. And these little vectors here, I with a little triangular hat over it, J with a little triangular hat over it, we'll come back to those in a moment, but they're essential in indicating a dedicated direction, either only along the X or Y or Z axis. Vectors, even though they carry both length information and direction information, can be summarized as having a singular length that characterizes the full straight line distance that you would have to go to get from the beginning of the vector to the end of the vector. And this is known as its length or its magnitude. This can be denoted in one of several ways, either just drawing the vector with no arrow over it, so A in this case, or putting absolute value signs around the vector, that's another common notation for length or magnitude of a vector. And this can be computed using the sums of the squares of the components, and then you take the square root of that total sum. In two dimensions, this will recall the familiar Pythagorean theorem, which, given the lengths of the sides of a right triangle, will tell you the length of the hypotenuse. Then there are unit vectors. This is a subspecies of vector, and they're special because they are vectors whose length is always exactly one in whatever unit system you choose to use. Unit vectors are denoted with that little triangular hat symbol, so for instance, i hat, j hat, and k hat, as they would be denoted in spoken terms, are special, and they're unit vectors that point only along the x, y, and z axes, respectively, of a Cartesian coordinate system. This also means that because the angles between the x and y, y and z, and z and x axes are 90 degrees, the angles between these unit vectors are also always 90 degrees for any pair. You can add vectors, so for instance, if I have a vector a and a vector b, and I wanna know what the resulting vector, for instance, c with a little vector arrow over its head looks like, all I have to do is take the x components and add them together, noting that they point along the i hat direction, take the y components and add them together, noting that they point along the y direction, et cetera, and this will give you the resulting sum of two vectors. You can replace the sum with a minus sign to get the difference of two vectors, but the math is the same. There are two kinds of multiplicative products of vectors, the dot product, which gives you a number, and the cross product, which returns a vector. The dot product is given by the following notation. C can be represented as the dot product of two vectors, a and b, with a little dot between them, and it's a number, it's a pure scalar, whose size is the magnitude of a times the magnitude of b times the cosine of the angle between a and b. In component notation, you can calculate this by taking the x components and multiplying them together, taking the y components and multiplying them together, et cetera, and then adding all of those products together. And again, this yields a pure scalar, a pure number with no direction. On the other hand, the cross product, the other multiplicative operation between two vectors, yields a vector. So in this case, the cross product of two vectors, a and b, would yield a third vector, c. The cross product is denoted by putting a cross multiplicative sign in between the two vectors, a and b. This one's a little bit more complicated, and you have to be a bit more careful with this. I like component notation because you can essentially distribute the multiplication algebraically between the two vectors, a and b. And you wind up with terms that look like the x component of a and the y component of b with this cross product of unit vectors next to it, and then the y component of a and the x component of b with the reverse cross product of i and j hat next to it, and then a bunch of other terms that look similar to this, depending on how many dimensions this thing has. And in the end, this yields a pure vector with a length given by the magnitude of a times the magnitude of b times the sign of the angle between the two vectors, a and b. Also, the vectors c will always point at exactly a right angle to both a vector and b vector. That's one of the natural consequences of the cross product. Now the cross products of coordinate axis unit vectors like i hat, j hat, and k hat obey the following rules. The cross product of any unit vector with itself is zero because there is no vector that's perpendicular at the same time to both i and itself. There's an infinite number of those vectors and the cross product yields a result of zero for this. Similarly with j cross j and k cross k. Now, the rule of thumb for computing all of the other cross products is that i cross j is k, and then if you kind of conveyor belt k to the beginning of this operation, move i to where j is and move j to where k is, you get one of the other cross products, k cross i is j. And then similarly doing this conveyor belt permutation one more time, so-called cyclic permutation, you get j cross k is i. Now, what about j cross i, i cross k, or k cross j? Well, if you swap the order on the left side of these equations, then the right side changes by a minus sign. So j cross i would be negative k hat, i cross k would be negative j hat, and so forth. Vectors are an essential building block of everything that happens in mechanics. But the real laws of nature that we encounter in a course on introductory mechanics are Newton's famous three laws of motion. The first law states that the state of motion, that is the velocity of an object, remains constant unless the object is acted upon by an external force. Absent external influences, the natural state of an object is to maintain whatever velocity it presently has. This can be summarized in an equation as follows. The sum of all forces with subscript i, and there can be from one, two, three, all the way up to capital N forces acting on an object, if all of those add up and cancel each other out so that there is zero net force acting on an object, then the resulting acceleration, that is the change of velocity with respect to time, or the change of the state of motion with respect to time, given by the second derivative of a position vector of the object is zero. No net force, no change in state of motion. The more general form of this equation is given by Newton's second law, which relates the net unbalanced force acting on an object to any resulting acceleration or change in the state of motion of that object. The change in Newton's second law is proportional to something. Force and acceleration can be related to each other by a simple equation and the constant of proportionality between force, F, and acceleration A is given by M, the so-called inertial mass of an object. Because you can write the acceleration as the second derivative with respect to time of the position vector of an object, I've put here the calculus notation for the acceleration in three dimensions where r vector is a position vector x, y, and z that not only can change with time, but whose change with time can be further altered by having an external force act on it. That is an acceleration. And then finally there's Newton's third law that in every interaction of two material objects, let's call them A and B, two forces are in action. The direction of the force exerted by object A on object B is the opposite of the force of object B on object A, but they are otherwise equal in magnitude. So if I take my hand and push on the surface of a table, the table pushes back against my hand with an equal magnitude, but opposite direction force, that's why my hand doesn't go through the table. Now usually after learning about Newton's laws of motion, we then learn about quantities that are associated with motion. These are known as energy and momentum. What is common between these quantities is that they vary in some proportion to the degree of motion. So for example, the quantity of energy associated with a moving object, so-called kinetic energy, is proportional to mass and to the square of the velocity of an object. It is a scalar because you square the velocity, you lose all directional information about it, and the exact equation for kinetic energy is determined to be one half times the mass times the velocity squared or the speed squared of an object. There is a direction full quantity of motion and that is known as linear momentum. It is proportional to mass and directly to the velocity of a body, at least in this classical physics and this introductory mechanics we learn about, this is observed to be the thing that appears to also be conserved in nature like energy. Linear momentum is denoted by the letter P that the vector had over it and it's the product of inertial mass and the velocity of the object. We can write this in calculus notation as the mass times the first derivative of the position vector with respect to time. Now there's another momentum quantity that's associated with a body that can rotate as well. So the degree of its rotation around some axis imparts some angular momentum to the system and we also learn that in closed and isolated systems, this quantity can be conserved. It's proportional not to the mass of the body but to the distribution of mass around the axis of rotation, the so-called moment of inertia and to the rotational velocity of that body. All points on a rigid body that can rotate about an axis will have the same rotational velocity regardless of their distance from the axis of rotation. In the moment of inertia, it describes using an integral which is shown here. I is the integral of r squared dm where r is the distance from the axis of rotation for the little bit of dm mass that you're considering at the time. The product of these two things yields the angular momentum and this is observed to be conserved in systems that are closed and isolated. Now if an external conservative force acts, one where the work done by the force in moving an object from point A to point B is the negative of the work in moving from point B to point A by any path that you can take, then there is an associated potential energy as well which we denote you. This is another kind of energy. So there's kinetic energy and then for conservative forces where those things like gravity for instance can act on a system, you have an associated potential energy, you can lose kinetic energy and store it in potential energy and you can lose potential energy and gain it in kinetic energy. There's an interplay in these kinds of energy in systems and the total energy can be conserved. On the other hand, for external non-conservative forces such as friction or air drag, there is no associated potential energy but other forms of energy such as heat which is the motion of atoms in a material object can result from losses of kinetic energy through the action of those forces. Now as I've hinted at before, energy and momentum can be conserved for a system that is acted upon only by conservative forces which have an associated potential energy and is otherwise closed to and isolated from all other kinds of forces. In that specific case, what is known as mechanical energy is completely conserved. Mechanical energy is the sum of all kinetic and all potential energy in the system at any moment. So for instance, there might be some initial moment of time where there's a total kinetic energy Ki and a total potential energy Ui. And if the system obeys the constraints I've listed above then I can look at any other time, say some time final later denoted with an F and I can see that although kinetic and potential energies may have morphed one into the other, the sum of these two things across all objects in the system is the same sum as I had at the earlier time. Now for a non-closed and non-isolated system and especially where non-conservative forces can act, total energy will be conserved but not just mechanical energy. And total energy is the sum of kinetic, potential and all other forms of internal energy like heat due to friction or drag or even chemical energy if for instance mechanical energy has been converted into stored chemical energy through some chemical and mechanical and electrical process then you can retain the energy in that form and you may be able to get it back later in the form of either potential or kinetic energy depending on what kinds of non-conservative forces are acting in the system. But if you can figure out all the energy buckets where energy can go in a system even one where non-conservative forces can act then you can still see that the total energy in all of those buckets added up remains constant over time even if you can't recover mechanical energy when it's lost into forms like heat or chemical energy. And for a closed and isolated system of objects total momentum, both kinds, a linear and angular is conserved. So if I sum up all the linear and all the angular momentum at one time initial, type Ti, I will find later on that the sum of all momentum and all angular momentum, all linear momentum and all angular momentum is the same even if it's been interchanged between objects maybe they've collided with each other, things like that. Now if only elastic collisions of these objects are possible that is the number and mass of the objects never changes then the total momentum and kinetic energy are conserved in that case. But if in elastic collisions are possible where objects can stick together for appreciable periods of time or if they can lose mass or gain mass then only momentum will be conserved. But again you have to be very careful with how closed and isolated the system is. Now another law that we encounter in introductory physics which seems a strange beast compared to the other kinds of mechanical phenomena that we encounter in these courses is the law of gravitation which governs the gravitational force between any two bodies with mass. It acts without physical contact and it does so even across empty space and I've illustrated that here by showing you the planet Jupiter which is the heaviest planet in our solar system and four of its moons, the ones that were first spotted by Galileo when he turned his telescope to the night sky to see what he could see. These are the so-called Galilean moons. They're the biggest moons of Jupiter. Jupiter has many more moons than this but these are the four most visible, the most easily visible even with a modest aid to the eye. And those are Io, Europa, Ganymede and Callisto. And these four moons do an orbital dance around Jupiter. They don't orbit the Earth, they orbit this planet and this was a remarkable observation in the days of Galileo that you had objects in the night sky that didn't go around the Earth. And they do this under the influence of gravity, the same force that holds our moon in orbit around our planet and our planet and all the other planets of the solar system in orbit around the central star, our sun. It's gravity, gravity explains all of this stuff. Now the gravitational force that an object A exerts on an object B is proportional to the masses of both objects and inversely proportional to the square of the distance between them and this is codified in the law of gravitation. That is the gravitational force between any two bodies. So for instance the force on A that's exerted by B is proportional to the product of their masses divided by the distance squared between them. The constant of proportionality G I'll get to in a moment but the force points from the object that's acted upon A toward the object that's doing the acting B. So it's an attractive force. Now again, this is the force that A experiences exerted by B. Now G is this universal constant of proportionality. It must be determined by experimental methods and it's currently known to be about 6.67 times 10 to the negative 11 Newton meters squared per kilogram squared. Not a very big number. Gravity may seem like a strong force but that's because we're being pulled on for instance by all the atoms of the planet Earth and that's why when we try to jump off the surface of the planet Earth we get pulled back down to the surface. So all the atoms of the Earth below us are pulling back on us as we attempt to accelerate away and it re-accelerates us back to the center of the Earth. But of course we don't go through the surface of the Earth when we hit it. Why is that? That's because another set of forces, electromagnetism, governs the interactions between atoms and atoms tend to repel each other because they have clouds of electrons around them and the electrons have the same electric charge and in the laws of electromagnetism this causes a repulsive force to occur. And so while gravity may seem strong the truth is because we don't get pulled through the surface of the Earth and down to the core of the planet is because of the strength of electromagnetism which overcomes an entire planet's worth of atoms pulling on you. Now what's worse, gravity seems like a strong force but it's not. And also this force law doesn't really tell us its origin, it has something to do with mass and it weakens or strengthens depending on your distance squared between two objects. It tells you what direction it points but it doesn't explain what the origin of gravity actually is. What is this force? Where does it come from? So one of the unsatisfying things about the law of gravitation is that it's very descriptive but it is by no means explanatory. And this was something that even Isaac Newton recognized and because he could provide no evidence to explain the origin of the force known as gravity he preferred not to speculate on it and left it open for the people that would come after him to try to figure out but it was certainly one of those puzzles he never managed to resolve in his lifetime and its resolution would be left until the modern era of physics. Now speaking of the laws of electricity and magnetism let's take a look at those and I'm gonna do so in a form that may not be very familiar to you but it will be beneficial to you later even if you don't completely understand notation. Now electric and magnetic forces have something in common with gravitation. They can act without physical contact across stretches of empty space. However it's pretty much right about there that they part ways from gravity. Their strength is proportional to a completely different physical property of nature electric charge which various bits of matter like the electron for instance appear to carry as a fundamental property. Now like gravity the strength of say the electric or magnetic force appears to vary inversely with the square distance between charges or flows of charges depending on the situation we're talking about here but I can wave my hands sort of make that rough approximation. A density of electric charge however is the source of the electric field of force. Mass has nothing to do with the electric field of force. It has something to do with the gravitational field of force but again this is roughly where gravity and electricity and magnetism all part ways. Now an electric current density that is a flow of electric charge is the source of a magnetic field of force. So a static electric charge just sitting there in space will exert an electric force on another charge somewhere nearby but in order to get a magnetic interaction to occur one of those charges has to be moving relative to the other. Now I'm going to define a symbol it's this funny triangular symbol known as nabla because it resembles an ancient harp of the same name. It's got a little vector sign over which immediately tells you that whatever this thing is it has directional information and it's funny because it's not made of numbers it's made of derivatives and specifically it's made with either the full or partial derivatives with respect to space. So for instance the derivative of something with respect to x the derivative of something with respect to y and the derivative of something with respect to z. This exposed triplet of derivatives is known as an operator. It doesn't itself return a number but when used on another thing like another vector it can return a number. So you can think of it as a function that when finally given something on the right hand side to act on will give you some information back but on its own it doesn't really give you information it's just prepared to tell you how something changes in space. Now you may not have seen this symbol before and that's okay but by defining it it allows me to write the laws of electricity and magnetism so-called Maxwell's equations in four compact mathematical lines. Now the laws governing these electric and magnetic fields are four in number. The first one is known as Gauss's law for electric fields and believe it or not from this compact little equation here you can under special conditions derive Coulomb's law which is probably what you really learned was the law of the electric force in introductory physics. There is a simple exercise one can go through to show that this reduces to Coulomb's law but this is the most preferred in general form of this particular law of electricity and magnetism and in English what it tells me is it tells me that a charge density that is a charge per unit volume row is the source of an electric field on the left hand side. We have this operator I defined above which is just a triplet of space derivatives acting on an electric field via the action of the dot product. So this thing returns a number and that number is equal to the charge density divided by epsilon naught which is a constant of nature. The second law is Gauss's law for magnetic fields and this one is probably the simplest of the four. It's that same operator action the Nabla symbol with a dot product with the magnetic field but on the right hand side you get zero and what this equation tells you is that so far as we know there are no such thing as a magnetic charge in order to create a magnetic field you have to have moving electric charge and so far as we know and many experiments have tried and many experiments have failed there is no such thing as a magnetic charge. That's what this equation codifies. Then there's the Faraday Maxwell law. The Faraday Maxwell law tells me that if I have a time-changing magnetic field this can generate an electric field. Now I have a different vector operation on the left hand side. I have this Nabla symbol, the vector cross product with the electric field which returns a vector and indeed I have a vector on the right hand side as well the time derivative of a vector field is also a vector and then finally there's the ampere Maxwell law and this tells me something a little bit similar to the Faraday Maxwell law and that is that if there's a time-changing electric field or if there's a current density of electric charge a flow of electric charge or both then this results in a magnetic field. So the left hand side tells me that there's a magnetic field that exists the right hand side tells me where those magnetic fields might come from either from a charge current density or from a time-varying electric field and mu not here is another fundamental constant of nature epsilon not and mu not you should have encountered in introductory physics and you can go ahead and look up their values. Now what's amazing about the laws of electricity and magnetism, Maxwell's equations is that when you consider them in a particular situation it finally clarifies what the heck the nature of light is light is an amazing phenomenon it carries information from one place to another and it does so at a seemingly immense speed and it turns out that by solving Maxwell's equations in a certain regime you find out what light is it's a very rewarding exercise one that you would presumably go through in a more advanced course than this one but I'll tease it here. So for instance if you consider empty space where there are no electric charges no row no charge densities and where there are no electric currents no J's with the vector hat over the top of it nonetheless Maxwell's equations are not just simply all zero. So let's take a look at those equations under those conditions. I've rewritten the four equations with no electric charges and no current densities. So I have this nabla dot e vector is zero nabla dot b vector is zero I have nabla cross e vector is just negative db dt and nabla cross b is something proportional to the time derivative of e. So there is a trivial solution to this e and b can be zero that works out just fine but there's another solution to this that isn't the so-called trivial solution and the non-trivial solutions are vector functions of space and time and this is what they look like. The electric field and the magnetic field as a function of space and time that also satisfy these four equations are these time and space varying functions over here they're co-synosoidal and they can all be written in terms of the electric field. They describe some kind of oscillatory phenomenon oscillatory phenomena like waves are things you should have learned about in an introductory mechanics class. k hat here simply indicates a unit vector that's in the direction of travel of the phenomenon and this number c with a zero subscript that turns out to be the speed of the phenomenon in empty space because that's the kind of space we're considering here empty no matter no charges no currents. And it turns out that you can solve for that speed and you find out that it's equal to one over the square root of those fundamental constants of nature mu knot times epsilon knot and if you plug those numbers in you get an amazing fact out of this that whatever this phenomenon is it travels at 2.998 times 10 to the eighth meters per second and for the astute among you this is the speed of light. So what Maxwell's equations in empty space tell us is that when solved they describe a phenomenon that can travel from point A to point B seemingly through empty space and it does so at precisely the speed at which light was known to travel in the days when this was solved. So light is what is known as an electromagnetic wave and like a mechanical wave which was the only analogy that physicists had at the time it was originally assumed that it must travel in a medium sound travels in air water waves travel in water they are distortions of a medium and so it was presumed that light must too be some kind of mechanical wave and that means that seemingly empty space couldn't really be empty something's got to be there that distorts to allow this wave to travel that was the assumption based on mechanics. Now finally I want to go into the subject of relativity which would have been introduced to you probably under the phrase relative motion. In introductory physics you get some exposure to relative motion that is a person standing on a train the train is moving relative to somebody on the ground the person on the train throws a ball up in the air what is the person on the ground see that's usually the way in which this is couched the person on the train for instance who throws the ball straight up in the air will see it go up gravity will accelerate it and eventually it will come straight back down into their hand so it just goes up slows to a stop and then accelerates down back to their hand all along a straight vertical line that's what the person on the train sees a person on the ground watching this sees the ball follow a parabolic trajectory because the ball and the person have a horizontal velocity because they're standing on the train so the ball goes up and comes down yes but it doesn't land at the same coordinate along the horizontal that it started at it appears to follow a parabola and so the two observers will disagree on the motion of the ball the person on the train says no, no, no it goes straight up and then comes back down to my hand and the person on the ground says well no it didn't go straight up it followed a parabolic trajectory but your hand moved too and so it was there to catch it when it came back down and it's possible to use mathematics to relate these differing observations of space and time and to do this you assume that time passes the same for all observers the person on the train and the person on the ground all experience time the same way when you make that assumption you get out of this something known as the Galilean transformation that allows you to relate spatial coordinates and velocities of objects from a frame you consider to be at rest to a frame that you consider to be moving so in our case you might consider the platform or the ground next to the train to be the rest frame you might consider the train to be the moving frame and these equations shown down here will relate coordinates, velocities and times in the moving frame with the primes next to them to things in the rest frame the numbers without the primes attached to them okay so that's not so bad it's actually one of the more complicated things that most students encounter in introductory physics because it forces you to think in two different frames of reference and this is not always as straightforward as it seems but the math itself is not that bad it's more the conceptual issues that go along with this that pose a particular challenge for most people who see this the first time so that is basically a summary of what we now call classical physics introductory mechanics and the laws of electricity and magnetism for semester one and semester two physics and even though classical physics is challenging there are many difficult things that you have to do there's new math you haven't seen before you're often learning calculus at the same time you're expected to use calculus in introductory physics nonetheless at the end of the day if you stop and look at all of this stuff you'll often say okay the mathematical or some conceptual difficulties aside all of the stuff feels to me very intuitive I can throw a ball up in the air I can catch it I can watch somebody do that in a train and see it moves in a parabolic arc okay yeah we disagree on what's happening but we can explain to each other why we see what we see it's all very normal day to day human scale stuff really this is intuitive it just had to be described by mathematics and that often is the difficult part but you have to be very careful about intuition intuition is largely based on experience with events that involve the following things speeds that turn out to be very close to zero you know driving at 70 miles an hour may seem really fast to you as a human being or getting on a rocket ship that goes into earth orbit might seem really extreme and they are for human beings but compared to the fastest known phenomenon in the universe which is light 2.998 times 10 to the eighth meters per second 70 miles an hour seems pretty pathetically slow and in fact is so close to zero that from the perspective of light it might as well be nearly at rest not very impressive to light so you have to be careful one because the speeds that you're used to encountering are really close to it turns out zero and so your intuition is built on a very narrow spectrum of experience in the universe the other thing that you may take for granted is that the sizes of things that we usually think about in classical physics with the exception of electrons and protons and electricity and magnetism the sizes of the things tend to be very large by comparison to what are known to be the building blocks of the material universe and for the stuff around us that's mostly going to be atoms that's the day-to-day stuff that we are interacting with but when you interact with a table that table has like Avogadro's number worth of atoms in it that is a huge number of atoms and the scale of the structure built from those atoms is vast by comparison to the atoms themselves and so as a result as we begin to encounter phenomena and this was true of physicists at the end of the 1800s as you begin to encounter phenomena that are very fast or very small so objects moving very close to the speed of light or objects that are really more at the atomic or even the sub-atomic scale or things that make up the atoms you begin to find the classical physics needs to be modified to describe the universe more completely it works for slow things at large scales like human scales or planet-sized scales or even bigger but it breaks down in regimes where it was never designed to operate the very fast and the very small so as a result you're often going to find as you go into modern physics that what you think to be true about the universe is based on intuition from a limited set of experiences in the cosmos and as a result your intuition is actually fundamentally wrong but the good news is is that this only means that you are finally, finally experiencing the breadth of the universe all it has to offer at all of its scales and speed and size rather than that limited scale of phenomena closer to human experience so let's use classical physics and let's make some predictions to set ourselves up for where people started to go really wrong with these ideas in roughly the late 1800s now the tenets of classical physics which I can summarize based on the earlier part of this lecture are encoded largely in Newton's laws and Maxwell's equations and they should if this is all there is to the universe apply to all phenomena in the natural world after all if this was really the complete set of all the laws of nature that had been discovered in the 16 and 1700s then it must be true that they describe everything otherwise they're not a complete set of laws so let's take a look at light what would the framework of classical physics then insist be true about light? well from Maxwell's equations we know that light is some kind of oscillatory phenomenon like a wave and so our experience with waves in the 1800s was that they must be mechanical in nature they must represent the distortion of a medium so they gave it a name they named it before they ever discovered it and they called it the ether and it was believed to be the thing that actually fills empty space empty space isn't empty it's made of this substance called the ether that we normally can't experience but light experiences it and the distortion of the ether is what we call light that was the hypothesis based on the mechanical understanding of wave phenomena so the speed of light in so-called empty space the number that we got from Maxwell's equations that isn't really the speed of light in empty space it's the speed of light measured relative to an observer at rest with respect to the ether the ether is the universal reference frame for light and if you can be at rest with respect to the ether then you will observe that light moves at 2.998 times 10 to the 8th meters per second it's a big number, okay? but this would then make ether the universal rest frame that is the frame that you could define to always and absolutely be at rest and then everything else is in motion relative to it that would be awesome the Galilean relativistic and Newtonian mechanical view of the universe would have allowed something like this to exist now the problem was that sort of the new kid on the block Maxwell's equations which really only emerged in the second half of the 19th century they were silent on the topic of the ether they described no substance that required this electromagnetic wave called light to propagate so it was assumed that they must be incomplete that the new kid on the block they're probably not complete they need to be completed and the ether would complete them so it was assumed that Newton's mechanical view of the universe the laws of motion and all that stuff that that was correct but that Maxwell's equations was just incomplete and needed to be completed with this mechanical substance, the ether so if we then apply this thinking to a problem involving light and travel and time what would we predict? let's put ourselves in the role of sort of late 19th century physicists we've learned all this stuff it's been solid for 200 years so what are we going to predict? so let's do a thought experiment a thought experiment is a kind of experiment that you can carry out entirely inside of your head what you do is you imagine a scenario you analyze the scenario using the understood principles of nature or laws of physics and you look to see if the conclusions of running this imaginary experiment would in any way violate logical or physical consistency and if you determine that that's the case you may have hit upon a useful inconsistency in our understanding of nature that could then be used to figure out what the correct description of nature might be so to do our thought experiment let's imagine that we are in a space that is filled with ether the medium in which light traveling as a wave disturbs the medium and propagates at 2.998 times 10 to the 8th meters per second now imagine into this volume of ether we place two cars one car at the left, one car at the right and the car at the right has its headlights aimed at the car on the left so that an observer in the car on the left could look back out the window and if the headlights of the car behind them were on they should be able to see the light but let's put a 30 kilometer gap between the front of the right car and the back of the left car so that light if it wants to go from the car on the right to the car on the left has to cross this gap of 30 kilometers okay fine so we've placed the cars in the ether the cars are at rest with respect to the ether so they're in the frame of reference of the ether and the car on the right switches on its headlights how long does it take for an observer in the car on the left, the second car to see the light reach them? well this seems pretty straight forward right? you know the distance it's 30 kilometers from where the light leaves the right hand car and arrives at the left hand car and Maxwell's equations tells us that light travels at a fixed speed it doesn't say anything about the ether but we've invented the ether to help us to have electromagnetic waves comport with all prior knowledge of mechanical waves so it's a medium with mechanical properties that can stretch and squash and those stretchings and squashings are electromagnetic waves and in that medium light travels at 2.998 times 10 to the 8th meters per second okay everything's at rest with respect to the ether light travels at the speed of light in ether so we just run the numbers we take the distance, we divide by the speed and we get the time that is required to make this journey and we find that that time comes out to be about 0.1 milliseconds 1 times 10 to the minus 4 seconds okay nothing hugely revelatory here but let's take our thought experiment one more level forward now let's imagine that both cars have been plopped into this ether volume and they accelerate at the same time up to a constant velocity of half the speed of light that's a speed of 1.5 times 10 to the 8th meters per second and let's imagine that the cars are both moving together at the same velocity from right to left so they're traveling from the right to the left in the ether at all times they maintain a fixed distance between the front end of the right car and the observer at the back end of the left car of exactly 30 kilometers the car on the right turns on its headlights now how long does it take the light to reach the observer in the other car well let's review what we think we know about light speed and this so-called ether that distorts to allow electromagnetic waves to propagate light travels at sea the number given by Maxwell's equations 2.998 times 10 to the 8th meters per second in the rest frame of the ether but now from the perspective of the cars the ether is a wind that's rushing past them still air on a calm day leaves no sensation on your body but if you were to start running forward you would perceive a wind hitting you in the face and that's sort of the equivalent situation here both of these cars are now traveling through the ether they're doing so at half the speed of light and so from their perspective the ether is rushing past them as a wind and its speed is also half the speed of light it's as if they perceive themselves to be at rest and the ether to be rushing past them at half the speed of light so the velocity of this wind is the negative of their velocity with respect to the ether now Galilean relativity and Newtonian mechanics demand that from the perspective of observers in the car that the light that leaves the car on the right while it's traveling at 2.998 times 10 to the 8th meters per second in the rest frame of the ether is encountering this wind of ether that has the apparent effect of slowing it down this is sort of like sound waves or water waves in their respective media if the medium is moving then the medium's speed can add or subtract from the velocity of the wave in that medium and so Galilean relativity and Newtonian mechanics are going to demand that the observed speed of light in the frame of the cars is the speed of light in the rest frame of the ether minus the velocity of the cars and so you would actually see the light leaving from the right hand car and traveling the gap between the right hand car and the left hand car at what seems like a slowed speed as if it's encountering resistance as it moves forward it's not moving at 2.998 meters per second anymore it's moving at about half that and so you would answer that well the distance between the cars is still the same it's 30 kilometers and the speed of light has been reduced by the ether wind and so you would predict based on all knowledge at this stage that the time it takes for the light to get to the other car is greater than it was before it's about 0.2 milliseconds now twice the time that was required when the cars were at rest with respect to the ether now that's a prediction and it comports with all prior experience in the pre-20th century world it comports with ideas about how velocities add in relative motion it comports with the idea that waves can only travel because they're distortions in some kind of medium a mechanical explanation for waves that's consistent with Newton's mechanics all of this seems to be perfectly acceptable from the perspective of the bare bones introductory physics to which you would have been exposed but a fair question to ask is this is the outcome of a thought experiment what would be observed in a real experiment in the real world and we'll take a look at that so let's review the basic ideas that are the foundations for modern physics the groundwork for modern physics are newton's mechanics the concepts of energy and momentum quantities associated with motion that can be conserved under certain conditions the law of gravitation and the laws of electromagnetism however these were largely built to describe phenomena that comport with typical human experiences phenomena at our size scales or slightly larger or smaller essentially within our ability to see the world around us including with a microscope or a telescope that would all be within the human scale the exception however is Maxwell's equations they were developed by studying electric charges which are very small and they are really beyond the scale of everyday experience except in their large scale macroscopic effects like electric and magnetic forces electric currents lightning strikes refrigerator magnets things like that they have these big macroscopic effects that feel familiar to us but at the individual level of an electron let's say things are not typical compared to the human world by the end of the eighteen hundreds chemists and physicists were beginning to directly interact with scales that really were beyond human experience so for example the electron is discovered in eighteen ninety seven and it turns out to be the first sub-atomic particle although that really wouldn't be fully understood for several more decades in addition an invisible radiation like for instance what we now call x-rays this was discovered at the end of the eighteen hundreds and eighteen ninety five in the case of x-rays and these phenomena and other phenomena at the same scale even atoms themselves or other general forms of light they turn out to be way beyond human experience and so trying to adapt our intuition in the form of newton's mechanics for instance to these phenomena would lead to spectacular fails now not only were such new phenomena small they also turned out to be capable of moving extremely fast x-rays move at the speed of light electrons with minimal effort can be compelled to move at almost the speed of light such speeds are also very much beyond human day-to-day experience although you might lead yourself foolishly to think that you understand them really well so this concludes a foundational lecture a review of the material you should have been exposed to already in semester one and semester two physics i know that i've couched this in some ways that are unfamiliar but i'm trying to rattle you out of any complacency you might be in after having had a couple of introductory semesters of physics and we're going to begin to explore the consequences of these classical physics predictions on phenomena like light in class and then we will build on what we conclude from those explorations into the first steps of modern physics