 Charge that changes its location in space with respect to time or electric current generates magnetic field. But what happens when magnetic field penetrating something that can conduct electricity suddenly changes its strength? This very basic question is the foundation of all electric power generation and all electric motors that currently exist in human societies. There's a deep symmetry between electric charge magnetic field and the interplay between the two. Let's begin to explore that now as we see the first glimpses of the interconnectedness of the electromagnetic world. So, so far this semester we've looked at magnets and magnetism, the force exerted between a pair of magnetic objects, for instance the compass needle here and the self-pole or north pole of an actual bar magnet like the one shown here. We've looked at the underlying cause of magnetism which appears to be the motion of electric charge. So, for instance in this device I ran a current through it, maybe about one or two amps from a power supply and I was able to deflect the compass needle in the same way that this magnet is able to deflect the compass needle by overcoming the Earth's magnetic field and the way that it influences this little handheld magnet here that you can use for navigation. We're going to explore one more phenomenon for this course and then I'm going to wrap up the subject of magnetism and discuss its context in the larger role of modern physics, especially in our understanding of the phenomenon which we take for granted called light, but before we get to that I need to demonstrate something called magnetic induction and it's easiest to just describe that phenomenon by showing you a few demonstrations. So what I have here is just a simple copper wire solenoid. You can see it's just a hollow tube and it's got all of these copper wire windings around it. You see it's empty inside and that's just an insulating material around which the wire has been wrapped and it has wires at each end where I can connect for instance an electric potential difference from some external source so I can hook a voltmeter up to this and look for an electric potential difference to be created. So here I have a trusty voltmeter and right now it's not hooked up to anything so you're just seeing noise in the voltmeter itself but what I can do is I can take the two leads from the voltmeter and I can connect one of them to one end of the solenoid and I can connect the other wire to the other end of the solenoid. Now right now there are no batteries in this system so effectively I'm measuring the electric potential difference across nothing. It's a conductor but there's no voltage on it whatsoever. You can see here it just reads zero. Now what I'm going to do is take a bar magnet pair and I've got the north ends and the south ends lined up so that they're pointing in the same direction so I've made this magnet nice and strong. So again keep your eye on the voltmeter. You'll see it says zero and if I put the bar magnet inside suddenly the readings on the screen change from zero to non-zero numbers and it happens again when I take the bar magnet out. Again focus on the voltmeter. I'm going to put the magnet down inside of the solenoid and you see a voltage occurs with a negative sign and then a voltage occurs as I withdraw the magnet from the solenoid as well. And again. So what I've just demonstrated to you is an incredible phenomenon known as magnetic induction and it's basically what you just observed. You have a voltmeter hooked up to this solenoid this long winding of copper wire that's been coated in an insulator so that the wires themselves don't actually touch each other. We know from the B.O.S.S. art law that each of these loops is capable of producing a magnetic field if one runs a current through this but I'm not running a current through this. Rather I'm taking the bar magnet just a simple north and south pole archetypal bar magnet and I am merely inserting it into the tube and the response is something really astounding and that is the induction of a voltage. So let me go ahead and do this. So if I move this in we get a voltage of one sign and if I take it out we get a voltage of another sign. This is really an incredible phenomenon and it's one that's been known about since the 1800s basically what we're observing is that this is the symmetric partner, the twin if you will of the B.O.S.S. art law. The B.O.S.S. art law says that moving electric charge creates magnetic field but this experiment tells us that there's a beautiful flip side to that statement moving magnetic field can create moving charge and I'm creating an electric potential difference in this wire with no physical contact between the bar magnet and the wire I'm just moving in the plane of the solenoid and I can create a voltage down here. It's fleeting, it only occurs while the magnet is in motion while the magnetic field lines if you imagine them going into the south pole or coming out of the north pole and then being moved down into this solenoid they're penetrating the plane of the solenoid and they are as a result of this causing a voltage to occur in the wire and as we know a voltage drives a current Ohn's law is in effect in this device it's copper, it's got a low resistivity but nonetheless it has some resistance and so V equals IR works and if I induce a V by moving this magnetic field inside and out of the solenoid, if I make a V there's an R in here and I'm making an I, I'm making a current in response to that. So it's a beautiful symmetry moving electric charge, current makes magnetic fields but changing magnetic fields moving magnetic fields can also make a current flow in this case in a conductor so we're going to explore this phenomenon a little bit more with some other demonstrations but this is the phenomenon simply known as magnetic induction the ability of a changing magnetic field to induce a voltage and in a conductor something with some resistance in it a corresponding flow of electric charge the current. We can explore this phenomenon of magnetic induction a bit more closely and in a slightly more controlled way using one of these FETs demonstrators so I have here a simple simulated laboratory containing a bar magnet much like the one that I showed you in the live demonstration video and I have here a simple solenoid a two loop winding of copper wire and it's attached to a light bulb which is a resistor like any resistor if a current flows through it there will be resistance to the flow of current this will establish an electropotential difference across the resistor and that will allow electric charges for instance to do work we would expect the light bulb to light if a voltage is present well we can repeat in the live video simply by moving the magnet into the coil of wire so let me go ahead and do that we see as I move the magnet through the coil of wire the light bulb glows briefly now interestingly it glows as the end of the magnet enters less as the center of the magnet passes through the coil and then it glows again as the other end of the magnet exits the the loop so let's actually examine the magnetic field from this magnetic dipole here for just a moment we observe that at the ends of the magnet the north end here and the south end here the magnetic field is changing very rapidly the magnetic field lines are diverging as you go further from the poles of the magnet and that means that the field is weakening further from the end of the bar magnet as we get closer to the end of the bar magnet the field lines grow in density they clump together there are many more of them per unit volume or area near the end of the magnet and then as the end of the magnet passes through when we look at the center we see that while there is a changing magnetic field here in terms of the density of lines the change is not as great as it was at the ends where these lines diverge very rapidly over a distance of say half the length of the bar magnet but over here half the length of the bar magnet and spacing doesn't really change all that much and then as we get to the south pole we see the same end of magnet phenomenon again the magnetic field line density in space changes very rapidly from right here at the end of the magnet to further away from the pole of the magnet so let's observe this dragging of the bar magnet through the loop one more time and think very carefully about what is changing as we pass the bar magnet through the loop it was this kind of investigation that a young scientist named Michael Faraday would have been doing to examine the exact cause of magnetic fields inducing currents and voltages in conductors for instance so let's do our own investigation now watch very closely I'm going to bring the north pole of the bar magnet into the loop and I'm going to do this at a constant rate so I'm going to try to drag this magnet at a constant speed through the loop then the middle of the bar magnet will enter, watch very carefully what happens to the light bulb as the north end passes through and then the middle of the bar magnet traverses the central area of the coil and then finally watch the light bulb again as the south end of the of the bar magnet passes through the coil watch what did you observe well if you said that you observed that as the north end of the magnet the bulb glowed brightly for a second but then as the middle of the bar magnet passed through the glowing ceased and almost stopped altogether before the south end of the magnet passed through the area of the coil and then the glowing started again then you made a pretty good observation let me do this again but going from right to left and again I'll try to maintain a constant speed for the bar magnet as I go through the coil south middle north middle south south middle north but what is it that's actually changing as we move the bar magnet through this loop of wire that's causing a voltage and a current to flow through the resistor the light bulb well we can bring in a little device to measure magnetic field and what I'm going to do is I'm going to just drag this over here and just kind of show you that as I move this device with the little mouse hairs through the magnetic field of the bar magnet we see that as I bring it closer to the north pole that magnetic field gets very strong and this is merely a reflection of the graphical representation that we as humans use to represent magnetic fields the magnetic field lines are growing much much closer together as you approach the poles of the magnet and so you would expect the strength of the magnetic field to go up because after all it's the closeness the density of those lines that represents the strength of the magnetic field in a picture in that way if I take the magnet measuring device and I move it closer to the center of the magnet we see that here the magnetic field is relatively constant in strength it's about three or so gauss it's not changing very much over the center of the magnet and again contrast that with moving back here toward the pole whereas I sweep around the end we see we've gone up a factor of about five in magnetic field strength from five, three, four, five now almost up to thirty so actually almost a factor of ten increase in magnetic field ok so let me sweep this back along the middle we see we hit sort of our minimum strength magnetic field and then as I bring the magnet measuring device around the ends we get close to where the magnetic field lines are clumping very strongly again on the south and we get a big magnetic field so let's put our magnet measuring device right here and I'm going to move the bar magnet through the coil again I'm going to try to avoid having the magnet touch the cross hairs because after all the field is super strong inside the magnet itself and that's going to skew the measurement so let me move the magnet through the loop again and watch very closely what happens to the B average the average magnetic field in gauss you see right now it's about half a gauss and as I come in at my constant rate you see it goes up as the poles pass through and it decreases as we go toward the middle of the bar magnet it sort of levels out so let me do this again see we're up to about three gauss here and it decreases again a little bit and then it increases to three gauss again as we come through all the way and go through one more time and try to do a nice constant velocity with this it's a little hard to do with these little track pads Michael Faraday realized as he studied this phenomenon that the key thing that's changing is the density, the number of the magnetic field lines penetrating the area of the coil in this case so we have the density of magnetic field lines represented in general the strength of the magnetic field is represented by B, the magnitude of B vector so our magnetic field strength in Tesla, gauss in this case the area is some fixed thing for the coil, it's not changing the system where the area changes and the magnet stays in the same place and we would see the same effect so we have a situation where B is changing through A and B, the vector B dotted into the vector A is a quantity known as flux and this is the quantity which Michael Faraday recognized was the key quantity that when changed, whether A is changing or whether B is changing when flux is changed when B vector dot A vector is changed that's when a voltage is created and in fact he figured out the relationship between those two things which we now will explore a bit it's called Faraday's law combined with the B.O. Savart law and Coulomb's law written in various forms this is one of the four fundamental laws of electromagnetism that is the merging of electricity and magnetism and by exploring the magnetic force we've seen that electricity and magnetism are deeply connected in some way magnetic fields can tell electric charges in motion how to bend their paths we know that moving electric charge can generate magnetic fields and we also know that changing magnetic fields now magnetic induction can cause charges to move in response and thus set up an electric potential difference and a current flow for instance in a conductor so what I'd like to do now is explore a little bit Faraday's law of induction you can largely credit two key researchers with the discovery of magnetic flux and how its changes induce voltages and currents in conductors Michael Faraday is somebody I've mentioned before he was an incredible young physicist he came from poverty most at the time it was customary in England that somebody who didn't have personal resources and wealth and connections would have an extremely hard time breaking into the scientific elite of the day but Faraday managed through the fact that he was just brilliant to rise up and first start working with other well known scientists but then establish his own career through his own discoveries and to this day is still one of the finest minds that our species has ever seen he's the person who's credited discovering the fact that a changing magnetic field can induce an electric current an electric potential difference but also Joseph Henry as shown here in 1870 the two of these together can share a lot of the credit for coming up with this key idea that changing magnetic flux induces electric current in a conductor so this is the principle that we're now exploring in more detail and we'll use some mathematics to start exploring this in even more detail so we can explore this idea of magnetic flux mathematically and the way in which we in the physics community represent this in which now you will represent it as well is simply to construct the following dot product of two vectors so let's imagine we have a loop of wire and that loop of wire has some radius r that means it has an area we can calculate that area we know that it's just pi r squared now if you think back a few lectures you can remind yourself that any area like this for instance a loop of current carrying wire with some area a its area can be represented by a vector that is perpendicular to the surface of that area at all locations the easiest thing to do is to say well I'm going to just pick the center of the area to be my representative location and I'm going to draw a vector that's exactly perpendicular for instance to the radius that I've drawn here and I'm going to label that vector a vector and that can be written as the area times a unit vector a hat and that unit vector points perpendicular to the surface of the loop in this case ok so the nice thing about doing this in fact this is a standard trick for instance in mathematics and in computer graphics and animation you represent an area using a single vector that's perpendicular to the surface of that area why? well because it's much easier to describe the orientation of a tilting surface using this vector it's much easier to describe the orientation of the surface as I for instance did it like this using a single vector perpendicular to the surface and that vector will always indicate the orientation of the surface that way I don't have to worry about how all the things on the surface are pointing I can just have one vector that sums up the whole thing so if we imagine that this dry eraser here is an area and I define a vector perpendicular to it at all points as I tilt the dry eraser we see that the vector tilts very nicely in space we want to know the orientation of the surface at any point we only need to know the orientation of a single vector perpendicular to its surface we know everything about where the rest of that flat surface is located in space so it's very typical to characterize the area with a vector perpendicular to its surface like this now the magnitude of that vector is just the area in the case of a circle, pi r squared and then of course here we might say well we have an x-axis here we have some y-axis here and then this here is a z-axis pointing up out of the plane so the circle lies in the x-y plane and then we have a z-axis perpendicular to that and in this case I've just chosen for convenience sake that the area vector points along the z-axis ok well that's all well and good this lets me characterize an area using a vector why don't we actually use this in practice well imagine if you will a solenoid much like the one we saw in the FET demonstration with two loops of conductor like this and these might be hooked up to some resistor up here and we could pass a bar magnet through this if that resistor was a light bulb we could get the light bulb to light great well these are loops and they have some area a so for instance if we knew the radius of this was one centimeter we could calculate the area using pi r squared great well I can characterize the orientation of the areas of these loops because they are parallel to one another with a single vector a vector which is has a magnitude of pi r squared and a direction that's perpendicular to that surface of the loop now imagine if you will that I have some magnetic field that I can now have penetrate the area of this loop so for instance I might imagine that I could set up some external magnetic field b that points parallel to a more maybe at some angle to a I now want to characterize this sort of penetration of the magnetic field through the area and the quantity to do this is known as Greek letter phi capital phi with a subscript b and this is magnetic flux and this word just means the flow of magnetic field in this case through an area and mathematically the way that we write this is we just take the dot product of b and a the area vector the vector that characterizes the orientation of the area we can see already what this is going to be this is just the dot product of two vectors so it's the magnitude of the resulting product and that's going to simply be b a cosine of theta where theta is the angle between the magnetic field vector at a given point and the area vector at a given point now in this case I've chosen it very conveniently we have a uniform magnetic field between a and b is the same and in fact it's zero the cosine of zero is one so in this picture we just have b times a that's it, no other modifications required now on the other hand I could have chosen something a little bit more complicated I could have oriented my surface at some funny angle like this characterized again by the vector a and again I could have had my b field pass through this area still horizontal, my for instance along the x axis but now we see that there's an angle there's an angle between the orientation of the plane the flat surface, the loop and it's vector a and the magnetic field vector b and now I have a non-zero angle so now the flux would be changed and in fact we see that the minimum flux that one can obtain is when the magnetic field is completely parallel to the surface of the loop in this case so if this is my magnetic field vector this is my surface there is no penetration of the magnetic field lines through the surface if the magnetic field lines are parallel to that surface at all locations they never pass through the plane of the surface unless they tilt at a slight angle like this and so in this case when you have parallel to the surface we now have a situation where a and b are perpendicular they make a 90 degree angle of pi over 2 radians and the cosine of pi over 2 is 0 so this has all the properties of what we're looking for for characterizing the penetration of the surface by magnetic field when there is no penetration of the surface that is the lines never pass at an angle through the surface this quantity is 0 when the a vector and b vector are perfectly aligned so that all the magnetic field penetrates directly perpendicularly through the surface then we get the maximum flux that we can ever have so maximum flux occurs at 0 angle between a and b minimum flux 0 occurs when a and b are at a right angle to one another so we'll explore this quantity a little bit more and look at Faraday's law of induction which is based on this quantity Faraday's law of induction is actually quite simple I'll just summarize substitute this briefly as Faraday's law what it says is if I have for instance some kind of coil of wire and I change the magnetic flux penetrating the areas of these loops pick any one of them so any one of these will have some area a if I have a magnetic field that starts off weak but then I increase its strength over time thus increasing b.a more b gives you more b.a gives you more phi sub b more flux if I change that over time what will happen in here if I were to pick two points little a and little b measure the electric potential difference the voltage across these two points what I would find is that I would get a voltage here which is non-zero as the flux changes the voltage increases from zero to something else and so Faraday recognized that emf electromotive force which we've written as a sort of calligraphy e in the past which is related to it's proportional to so that's what that symbol means it looks like a little fish or something like that it's proportional to the change in magnetic flux with time if magnetic flux changes over some period of time if b grows stronger if a gets bigger or smaller if the angle between b changes at all that any of those things would change magnetic flux remembering again that this is b a cosine theta if any of these things changes theta, a or b any of them changes with time there will be a voltage that's established across this coil of wire now the key relationship here the constant of proportionality is actually quite simple and it's this constant of proportionality which is often known as Lenz's law it's a simple minus sign so this thing here is often referred to as Lenz's law that is that the voltage that is established is established to oppose the change in the flux with time this is the key feature of magnetic inductance the electric potential difference that's established for instance in this coil of wire is set up in such a way that the voltage attempts to oppose the change in magnetic flux so for instance if you set up a voltage in here what happens a current begins to flow well as a current begins to flow a magnetic field appears from the wire what Faraday and Lenz and others recognized is that the flow of current due to this electric potential difference is such that the magnetic field created by the change in current will attempt to cancel out the changes in magnetic field from the external field so to sort of summarize this when you have magnetic induction you get a voltage which gives you a current which establishes a B for instance in the wire which tries to cancel the change in the external okay the external magnetic field B at EXT so magnetic induction operates in a way such that the voltage that's created by the induction in this coil of wire generates a current and we know the current flowing through a wire makes a magnetic field we can call that B wire and the direction of B wire is such that it attempts to cancel out any changes in magnetic field so if you're increasing magnetic field through the coil the current flows in such a way that the magnetic field due to the wire tries to decrease the magnetic field so it tries to negate it if you decrease the magnetic field through the wire the magnetic field here to begin with and then I decrease it maybe I take a bar magnet and I move it away like I did in the FET demonstrator again picture the south pole of that magnet being pulled away from the coil of wire what happened? the lamp lit and the lamp lit because as the magnetic field decreased inside of the coil of wire the current was established in such a way to generate a magnetic field to try to restore the magnetic field inside of the loop this is not an infinite game I mean there comes a point where the energy available to the coil of wire to do that is no longer there and so it runs out of steam and it can no longer oppose the changes but this is actually a really wild thing so if you have a loop of conductor like a solenoid like this and you want to figure out well according to Faraday's law we increase the magnetic field through here which way will the current flow we can do a quick analysis to figure out which way the current will flow in order to oppose the change in magnetic field alright so let's consider a picture of a loop of wire we're going to draw the loop of wire in the plane of the white board so here is my loop of wire it's not hooked up to anything it's just sitting there and what we're going to do is we're going to to changes in magnetic field penetrating the area enclosed by the loop a so this thing has some some radius r and thus some a equals pi r squared we're going to change the magnetic field lines penetrating this area we're going to change the magnetic flux right now there is none there's no magnetic field present we're going to increase it and using Faraday's law the little opposition piece in a negative d5 bdt we're going to analyze what we would expect the loop to do the charges in the loop to do in response to the changing magnetic fields so let's imagine now that we start increasing magnetic field pointing into the board so I'm bringing maybe a north end of a bar magnet closer and closer to the board we'll start with a very weak field so we'll just put a few magnetic field lines penetrating the area so the bar magnet's still kind of far away at this point so just a few magnetic field lines are penetrating the area there were none before and now there are magnetic field lines penetrating the area so originally at say time equals zero phi b which is b a cosine beta equals zero and at some later time t equals let's call it t sub 1 now there's a little bit of magnetic field penetrating the area of the board and so now we have some new phi b at t1 which is b1 a cosine theta it's not zero anymore there's a non-zero b penetrating a non-zero a and well we can figure out what direction, how cos theta is going to look so for instance the orientation of the area here is such that the a would be parallel to b, b points into the board a can be parallel to b and so cosine of theta is just going to be 1 imagine some even later time where even more magnetic field penetrates the board we've brought the bar magnet way closer now so we have a new phi b at some time which is b2a cosine theta and theta hasn't changed we've added more b into the picture we've kept a the same b2 is greater than b1 so flux has increased we have a bigger number now multiplying a cosine theta so phi b has gone up the flux has increased what is the conductor doing this whole time in response to this well according to Faraday's law there is a little piece of opposition in there the piece of resistance if you will the increase in flux will be resisted by whatever voltage is set up inside of the wire well how is the wire going to resist this well we have magnetic field increasing into the board so what the wire will do is it will set up a current in such a way that it creates magnetic field out of the board resisting the increase into the board off some of that magnetic field increase that is now entering the board well we can use right hand rules to figure out what direction current is going to flow as a result again we're looking for the creation of a magnetic field by a current flowing in the conducting loop that sets up a magnetic magnetic field of its own that opposes that change well let's see what happens so we have magnetic field externally increasing into the board the current will be set up in such a way that it will create a magnetic field that points out to try to cancel out some or all of the change and so in order for that to happen in order for magnetic field from the wire to point out of the board it has to be the case that the current in the loop is circulating according to the direction that my fingers are pointing so current I would have to start flowing counter clockwise in this loop alright so again let's review what I did external magnetic field is increasing into the board Faraday's law which is determined by doing experiments on systems like this tells us that the voltage set up in here will create a current that creates a magnetic field that opposes that change the way to oppose that change is to add your own magnetic field pointing out of the board to cancel out the change so to do that you have to create a circulating current that makes a B pointing out and your fingers indicate the direction the current must be circulating and that's counter clockwise in the loop alright so if this loop has some resistance okay then we know from Ohm's law and let me pick a symbol for this because I've already used R for the radius of the circle you know this is going to be a little way of confusing but just be aware of the fact that when I write R in Ohm's law I'm writing resistance not radius of the circle okay we know from Ohm's law that there's going to be if there's any resistance in the conductor there's always a little resistance in conductors then the current that's set up if you take the product of those two things you'll get the voltage that's set up in the loop to create the current in the first place and we know that this is equal to the negative of the change in the flux with respect to time and we can go one step further actually we can substitute in this equation over here area is not changing with time only the strength of B is changing with time so if I now substitute in for phi with B, A then I finally can write that the voltage that's set up in here is the negative of the area times the change in magnetic field with respect to time again this is the derivative with respect to time of B times A A is not time dependent it's a constant with respect to time so the derivative does not act on it you can pull it out in front and that leaves just B we know B is changing with time at T2 the strength of B is greater than when it was at time 1 and so B is in fact dependent on time and the derivative will act on it as a result you can't escape that so this is about the simplest you can get it without knowing the time dependence the function that describes how B and T are related to one another and this will be equal to E the electromotive force that's set up in the wire in response to the induced the change in magnetic flux through the wire so it's actually a nice neat little picture you have a change in flux you can't set up a current to oppose that change while the change is happening and so you can very quickly kind of use right hand rules and math to figure out the answers to questions like for instance what direction is the magnetic field from the loop going to point therefore what direction will the current in the loop flow and you can use the right hand rule that's the direction of B is your thumb the direction your fingers curl is the direction current is circulating in the loop all nice and neat so I hope this helps to kind of get you with this picture mathematically of flux B A cosine theta what's changing is it B is it A is it theta is it some combination of those things think about that how do I then figure out the direction of the current in the loop and then finally how do I calculate the magnitude of the voltage that is set up inside the loop so if you're only interested in the magnitude of the voltage this sign here just tells you that that voltage opposes the change in magnetic flux but if you're just interested in the magnitude of the voltage you just need to take the magnitude of the change in flux with respect to time and so forth alright so let's look at this now in the case of the solenoid and actually come up with an expression for the flux magnetic flux penetrating a solenoid alright so I have here a cartoon of a solenoid again a solenoid is just repeated turns of wire that are very closely spaced I've drawn this in a very cartoon fashion to show you that there are loops and curls here but in a real solenoid as you saw earlier in the demonstrations of magnetic inductance those coils of copper wire for instance are very tightly packed together so in the cartoon here there are 1 2 3 4 5 6 7 8 9 10 11 12 13 turns so in this particular wire we have 13 turns of wire so 13 loops as it were ok and so that's the big end the number of turns in the solenoid this solenoid will have some length L over which those turns are recurring and in class when I showed you what the magnetic field for a solenoid looks like you can remember that the magnetic field inside of a solenoid is equal to mu knot and I where I is the current flowing through the solenoid divided by L so if I run a current through this solenoid I can generate a magnetic field inside the loops of wire that is equal to mu knot and I over L now the reason I want to go through this is because I want to talk about self inductance inductance is the ability for a current carrying wire to set up its own magnetic field penetrating the area of the conductor and according to Faraday's law if I drive a current through this loop and it creates a magnetic field inside of the the core of the solenoid I am now taking a B and going through an A and that is giving me a flux magnetic flux and so this is a particularly interesting phenomenon self inductance is when your own current creating your own magnetic field then creates a situation where the magnetic field can penetrate the area of the conductor like these loops creating a changing magnetic flux well if I do that if I change magnetic flux inside of that coil of wire I will induce a voltage an EMF and what does that EMF do it opposes the change in flux that is happening in the first place so what is the physics here I start pumping a current through the solenoid maybe I am using a battery or something like that so I am trying to create a magnetic field and I am pumping a current through the solenoid I start to create magnetic field inside of the solenoid as I ramp up the current so as I increase I from 0 to 5 amps I am increasing B from 0 to whatever its final value will be depending on the geometry of the solenoid well that creates a magnetic field inside of the area of the solenoid and that creates flux so what will happen is as B solenoid increases I will get a change in flux penetrating the area of the solenoid so this thing has some area A those loops of wire have some area A I will have a B dot A that is increasing with time and that will create a d phi dt and a d phi dt will create an EMF according to Faraday's law that will oppose the increase in flux so what will happen self inductance is when a turn of wire like this a solenoid for instance or even a single loop you put current into it it actually resists you putting current into it it fights back and it fights back because it is trying to oppose the change in flux through its own area this is an incredible phenomenon and it is the basis of a great deal of modern electronics if you have a device that is very sensitive to little fluctuations in electric current and you want to protect that device from those little fluctuations that inevitably are going to occur in a circuit you put a loop of wire in it and any time the current changes that will create a changing magnetic field inside the loop of wire and that change will be opposed by the loop itself because it will create an EMF that opposes the change in flux so what happens if I try to pump current into this and create magnetic field the loop will try to drive a current back against me and tamp down the increase in flux so try to drive it down so this is what is known as an inductor and a solenoid is a great example of that actually even a single loop of wire is a great example of an inductor and an inductor is capable of resisting changes in current because changes in current create changes in magnetic field changes in magnetic field create changes in flux through this loop and changes in flux create voltages that oppose the change so there is this sort of beautiful cycle of things that happens now eventually there is only so much energy for this process to occur if you keep your battery on this wire and this coil of wire eventually it will set up a steady current there will be no more change in current it will oppose and so the coil stops opposing there is no more EMF due to the self inductance anymore if you try to take current away from the loop though it will attempt to drive current back through the loop to increase it again to keep the magnetic field from changing inside of the solenoid so this is something called inductance specifically self inductance when a loop of wire is capable of regulating the current in it by using energy in its magnetic field or storing energy in its magnetic field in order to oppose or promote change in flux keep current steady always once an inductor wants to keep current steady if it starts zero it wants to keep it at zero if it starts at a non-zero level and it's constant and then it decreases or increases the inductor will resist that change and so these become like little magnetic resistors and they will set up little EMFs to try to oppose changes in circuit and again it's great for electronics because you need to protect from small changes in current that could do damage to sensitive instrumentation medical instrumentation or technical instrumentation for engineering or research purposes so what's observed when you have loops of wire like this in a circuit for instance if I were to take a battery and plug it into a loop of wire just like that solenoid up there is a symbol for an inductor in a circuit it's not an accident that it looks like a little solenoid and there's a relationship much like Ohm's law there's Ohm's law for resistors Ohm's law for resistors what do you have in Ohm's law? you have a relationship between current and voltage the amount of voltage I put on a resistor the amount of current if I double the voltage I double the current and so an Ohmic material is one where there is this linear relationship between the voltage I place on the material and the current that results in the material that constant of proportionality is what we know is the resistance R there's a similar law for inductors specifically self-inductors that is loops of wire if you try to drive current through them if you try to increase the current they will create a voltage that opposes that increase if you try to decrease the current they'll create a voltage that tries to put current back in where it's suddenly decreasing away and the law for inductors is very simple it essentially states that look there's a relationship between the flux through that loop of wire through the solenoid and the current that I drive through that solenoid if I increase the flux if I drive a less I I decrease the flux and the relationship is written as this equation phi is equal to some constant of proportionality l times the current and this quantity here is known as inductance it gives you the relationship between the current you drive through a solenoid any inductor the amount of magnetic flux that's set up inside the loops of wire in the inductor now it depends on the device that you've constructed as to what exactly l is but the good news is that for a solenoid we can actually write down an equation for this inductance fairly easily so let's go ahead and do that we're interested in calculating the flux that's established by running a current through a solenoid and to do this we need to know the following things we need to know B vector and we need to know A vector we know the magnitude of B in A the magnitude of B is just equal to mu naught and I where N is the number density of turns it's the number of turns of wire per unit length I is the current that you drive that coil of wire and of course mu naught is our constant the permeability of free space well I know the magnitude of A as well so the magnitude of A I'm just going to write that as I'm going to be cheap and just write it as A you could figure out what the radius of the turns are and plug that in but it's just some area A it could be circular or something else but it's just an area something in meters squared a total area penetrated by the magnetic field we know from looking at the solenoid that when you run a current through a solenoid it sets up a nice uniform magnetic field all the way along the axis of the solenoid penetrating each and every one of the areas involved in the many turns of wire in the solenoid so the A here is the total area penetrated by all that magnetic field and the total area is just the number of loops of wire N times the area of a single loop so you know if I have a little one centimeter radius turn of wire then I know that the area of that turn of wire is pi times one centimeter squared convert that to meters if you like and then I have many of those stacked up one after the other to make the solenoid all these many loops of wire stacked up in a cylinder to make the solenoid and just like the one again I showed you in the lecture demonstration so the total area penetrated by magnetic field is the number of loops times the area of each loop that's the total number of meters squared that that magnetic field is going to penetrate as it goes down the long axis of the solenoid alright well I'm nearly there we can rewrite this in a convenient way we can rewrite this as the number density times the length times the area of a single loop so if number density is N over L then I can write big N in terms of little N by multiplying both sides by L so I just rewrote big N as little N L and then I still have the area of each loop well this finally lets me get an equation so now I know that the flux 5V is BA what's the cosine of theta here well we have the magnetic field penetrating perpendicular to the plane of each loop so B and A the A vector is parallel to the B vector so the cosine of the angle between them is just one the angle between them is zero so this is just BA and it's the magnetic field of the solenoid mu naught N I times N L times the area of the loop alright so just to keep things straight here this L right now is the length of the solenoid alright so not to be confused with the inductance which we're going to try to figure out in a second here okay well we're nearly there so we want to find a relationship between phi B and I and whatever the thing is that multiplies I to give you phi B that's the self inductance just the inductance of this particular circuit so we can go over here and we can pick this equation up we have phi B and that's equal to mu naught which is just a constant we have two ends multiplied by one another so we have N squared we have the length of the solenoid L and we have the area of each loop of the solenoid whatever that is and then finally we have the current so this here for a solenoid this is the inductance so I took the liberty of rewriting the inductance of a solenoid the self inductance L of a solenoid just down here is mu naught N squared I've changed my notation for the length so it's a little bit less confusing so I've used a little cursive L now for the length of the solenoid and then the area of each loop now the units of this are a little funky the units of mu naught are Tesla meters per amp the units of the number density squared are just number of turns so just number one per meter squared length is meters, area is meters squared and so you wind up with the funny units of this inductance Tesla meters squared per amp well this gets its own name in honor of one of the physicists that discovered inductance in the first place and the unit of this is the Henry which is just written as an H and actually if you kind of go and work back through this what you find out is that mu naught which is 4 pi times 10 to the 7th you can actually rewrite mu naught in terms of the unit, the Henry as just Henry's per meter which is actually kind of nice that's a lot easier than Tesla meter per amp so this is a lot easier to remember you see here if you just take Henry's and divide by meters it goes away here and you just get back to the units of mu naught again so anyway that's just a little bit of an aside but inductance is kind of nice because once you define the quantity you can actually create and simplify your units for all of this stuff but it's enough to remember that when you're given inductance you'll be given the strength of an inductor regardless of whether it's a solenoid or not in Henry's so if you see, you know this is an inductor with a strength of a unit of inductance which again just to remind you is the constant of proportionality between the current you drive through an inductor and the magnetic flux that penetrates the inside of the inductor in the first place now the last concept that I want to talk about here regarding magnetic fields and this is the end of the road on magnetic fields is energy in particular energy stored in magnetic fields if I imagine that I have a circuit like this and I have an inductor hooked up to the circuit okay so I have a battery and the battery has some EMFE that it generates and I have an inductor with some inductance L 5 millihenries or something like that I can now start thinking about Kirchhoff's rules taking little loops that go through this system and then analyze what's going on in terms of energy and energy conservation and power and other concepts like that alright so this isn't as scary as it sounds what we're going to do is there's only a symbol loop in this circuit and we know that if we walk any path in the circuit that we will if we take a closed path the net change in electric potential through that closed path in any moment in time is going to be zero so let's just start our walk down here okay at point A and again for the sake of this let's imagine that there is a current I at a moment in time T alright so at some moment in time T at that moment in time the current in the circuit that's being driven by the battery now remember the next moment in time the battery might be trying to drive more current through the inductor but because of self-inductance that inductor is attempting to push back against the change in current it's trying to push back against the change in magnetic flux inside of its coils so we can analyze the circuit now just using Kirchhoff's loop lock so we'll start from point A and we'll take a clockwise walk around and we know that the net change is equal to zero in all of the the potentials that we go through well we go through the battery from the negative side to the positive side so if we take our loop we know that the sum of the V's in the loop will be equal to zero the first voltage that we encounter is the battery voltage and we'll call that just E and then we travel through this conductor no resistance, no resistance and then we hit the inductor and we know from Faraday's law that the inductor sets up an opposing voltage to the battery to attempt to resist changes in current and flux through the inductor so we know that this is going to be we know the form of this okay but we know that whatever this voltage is the sum of it plus the battery voltage is going to be equal to zero because after the inductor we come back to point A we've come full circle no other devices in the circuit and in a closed circuit this has to be true energy has to be conserved well we can look at this in a little bit more detail so for instance we can look at the actual thing that we've put in here from Faraday's law we know that this is the change in the magnetic flux with time well and from the relationship involving inductance, current and flux we know that the flux is proportional to the current that's put through the inductor and that constant of proportionality is the inductance in Henry's L right there so we can rewrite this equation one more time as L which is just a constant D I D T now don't be frightened in principle this right here is a time dependent equation involving the first derivative of the current with respect to time this is a differential equation and much like with RC circuits one would have to solve this equation not for a number but for a function that represents how the current changes with time so that depends on the details of the problem if we have a switch in the circuit we start to drive current through this at first the inductor will completely oppose changes in current and it won't let any current get set up in it but as the battery continues to push on it a little current will flow then more current will flow and eventually it will plateau out at its maximum value and it will level off though we know more changes in current and in that case there is no more change in flux so there is no more opposition from the inductor and it just sits there storing energy in its magnetic field so as we drive a current through this and it reaches its flat steady maximum value of course this thing is going to be a source of magnetic field as we continue to drive current through it this current going through a loop this is going to set up a magnetic field and that magnetic field is storing energy for us so as we move the current energy will be drained from the magnetic field to put current back opposing the change in the circuit but that won't last forever eventually it will run out of stored magnetic field and the inductor will no longer be able to keep the current going if you disconnect the battery from this for instance and then shorten the inductor across itself it will drive a current at first but that current will then decrease to zero as it runs out of magnetic field stored and then we run out of current so it's not like a capacitor but rather than using an electric field to store energy an inductor uses magnetic field to store energy so the inductor is the magnetic analog to the capacitor in electric field applications it's the magnetic field application for storing energy what we're really interested in is we're really interested in understanding exactly how it is that energy is stored in the magnetic field current given in inductance how much energy is being stored in that inductor and we can figure it out from this equation here's how you do it we know in general that power is current times voltage so if I multiply that equation by current I I can actually turn it into an equation about power power driven by the battery and power dissipated by the inductor so let me go ahead and do that if I multiply this equation through totally legit thing to do nothing has changed here except that now I have I,E on this side so this is now the power supplied by the battery and over here I have negative L,I D,I,D,T this too has units of power energy per unit time and this is the power dissipated by the inductor well great we actually have basically what we're looking for now I wanted to know energy stuff about the inductor and now I know the power that's being dissipated by the inductor I know the energy per unit time that's being put into the magnetic field taken out of the circuit put into the magnetic field and that's what I wanted so let's go ahead and write that down the power dissipated by the inductor is L,I D,I,D,T well let's think about the core definition of power power is equal to the change in potential energy change in energy of any kind with respect to time and if I think about infinitesimal changes in energy and infinitesimal changes in time this can be written as a derivative ok well I'm almost there so I know that in my inductor that the power dissipated by the inductor is equal to the change in the energy stored in the inductor with respect to time and I have the equation for that it's L,I,D,I D,T so all I have to do to get energy is to get the integral of both sides of this equation with respect to time what is the integral? remember what the integral does the integral undoes the derivative so if I calculate the indefinite integral of D,U,D,T with respect to D,T that just gives me back U it undoes the derivative the integral is the antiderivative so if I want to get U I just have to calculate the integral of this thing with respect to time alright well that's just going to be the integral of L,I,D,I D,T with respect to T the D,T is cancelled over here in this integral and I'm left with the indefinite integral of L,I,D,I well this is just the integral of X,D,X it's just a different variable so the integral of X,D,X is easy it's one half X squared so the integral of I,D,I is one half I squared and we're done one half L,I squared this is the energy stored in the inductor in its magnetic field at a given moment in time when the current has some value I so if I know the inductance L and I know the current one amp I can calculate the total amount of energy that's stored in the inductor at that time 1 amp squared is 1 so I have 2.5 joules of energy stored in the inductor at that moment in time when the current is 1 amp you can store a lot of energy in a magnetic field depending on how you design your inductors and how big you get your inductance by playing around with the parameters of a solenoid for instance you can store a tremendous amount of energy inside of a solenoidal magnetic field inside of an inductor so imagine for instance a really high magnetic field very high current superconducting magnets of an MRI machine dump all the energy that's present in their magnetic field someplace like back into the circuits that power the MRI machine in the first place it's a disaster and in fact the Large Hadron Collider the experiment that I work on experienced a tremendous accident before it switched on in 2010 finally that accident was a one year setback through the experiment what happened was the superconducting wire inside of the very high magnetic field magnets failed because the cooling failed when the cooling failed superconducting goes normal conducting and when you go normal conducting you're suddenly taking a whole bunch of I and dumping it through a big resistor and so what happened was all the energy in the magnetic field was dumped back into the circuit to oppose the change in magnetic field and it was a tremendous explosion it actually knocked out a huge section of the Large Hadron Collider completely out of commission and it took a year to bring that section down going and investigate and then replace the broken parts and then figure out if there were any more sections of the Large Hadron Collider that might fail in the same way so this might not seem like a very impressive equation but if you're putting a patient in a big magnetic field you really want to be sure that your MRI machine is functioning absolutely perfectly this is why it's essential to have good technical personnel that really understand the equipment really understand the machines be they an engineer or a physicist who can work with the physicians and make sure the equipment is functioning normally and perfectly for the safety of the patients you never want to take a big inductor like an MRI solenoid and ever have it dump all of its power from its magnetic field back into the device itself you can cause a massive explosion when that happens having discussed energy and the energy that can be stored in an inductor it's now important to go back and think about what it means to run a current through an inductor have that inductor generate its own magnetic field which then penetrates the area of the inductor causing flux and then a voltage that opposes the change in flux this isn't magic this is conservation of energy the energy has to come from somewhere to cause the magnetic field to be set up that then opposes the change in magnetic flux through the inductor in the case of a circuit the energy comes from the battery the battery provides a constant electric potential difference it's a source of energy that can drive charge in a current through the circuit and so you're essentially converting energy from the battery into energy stored in the magnetic field if you want to release the energy stored in the magnetic field all you have to do is decrease the current that's being driven through the inductor so you could do this for instance by suddenly removing the battery and just connecting the inductor across itself there's no more battery so there shouldn't be any more current drawn by the circuit but what happens is that the energy stored in that magnetic field is released this creates an electric potential difference that attempts to drive current in the direction it was originally flowing in the inductor and eventually you run out of magnetic field you run out of energy and no more current flows energy is being stored in the magnetic field in an inductor and then when the current attempts to increase or decrease energy is released from the magnetic field and put back into current or into a voltage that resists changes in current you can imagine that if you had a source of mechanical energy you could convert that mechanical energy into energy stored in the magnetic field so for example if you were to take a permanent magnet and drop it inside of the area of a conductor this would cause a changing flux that changing flux would cause an electric potential that would drive a current and that current is going to be driven through the conductor ohm's law says that that's going to cause power to be dissipated in the conductor so we can imagine if we're attempting to push or drop a magnet through a conductor as currents are set up in the conductor in response to the changing magnetic flux this will cause the dissipation of mechanical energy we can do demonstrations of this we for instance might swing a conductor through a magnetic field think of a pendulum swinging in a magnetic field if you have the pendulum made from conductor enter the magnetic field and then try to leave it that's going to cause a changing flux inside the conductor that's going to cause currents to flow and that's going to cause energy to be dissipated the energy of motion the dropping of a magnet through a conductor swinging of a conductor based pendulum through a magnetic field that's mechanical energy and it will be wasted in driving current through the conductor and that will draw mechanical energy away from the system and possibly even slow it down and stop it and we can do a couple of demonstrations that illustrate this point beautifully now for the next demonstration of magnetic induction I'm going to drop some objects I'm going to choose those objects carefully the first thing I'm going to do just to demonstrate that the objects really are the same is really drop them in front of you and you can watch them land on the carpet at the same time so the objects that I'm going to drop are a magnet and you can figure out which one of these is magnetic by trying to put it up against the blackboard which is a surface that would let a magnet stick to it if this was actually magnetic you see that this is not magnetic it just falls right off but this other thing here and I'll show these to the camera in a moment that's a magnet sticks very nicely to the blackboard it's actually quite a strong magnet too let me just hold these up to the board, to the camera for a second you can see that in all respects these things look basically identical it's hard to demonstrate this here but they have the same mass too there's really no weight difference at all in Earth's gravity between these two objects here and so what I'm going to do is I'm just going to do a drop test just to demonstrate that there's no funny business here we know that Newton's law that two objects in the same gravitational field even with different masses will maintain the same acceleration and we can demonstrate that just by doing a simple drop test you've seen these before where you drop the objects and you drop them from the same height and then they hit the ground at the same time so let's go ahead and do that okay so just to demonstrate that there's no funny business here let me go ahead and do a drop test so what I'm going to do is I'm going to release these two objects one magnetic one non-magnetic from the same height you can see that they hit the surface at the same time the clicks on the surface of the table were at the same time go ahead, rewind it, play forward again and you'll see that they hit it exactly at the same moment on the table the clicks are simultaneous between them so keep that in mind that these objects dropped in a gravitational field they respond the same way to gravity that has nothing to do with whether they fall at the same rate in a gravitational field but if we were to now put these inside of a conductor where one of these being a magnet can induce currents in the conductor and the other one being non-magnetic can't in the same way that the bar magnet was able to induce a voltage and thus a current in the conductive solenoid or portal I showed you a bit ago we're going to now examine what happens if I drop test this not in the air but through first of all an insulating material and then a conducting material so I'm going to take the magnet you can see this is the magnet because it sticks to a piece of metal and I'm going to drop it through this insulating cardboard tube you see it dropped very fast came right out the bottom let's take the non-magnet you can tell this is the non-magnet because it doesn't stick to the metal and I'm going to drop it through the insulating cardboard tube no conductors present in the cardboard tube that also fell very fast pretty much at the same rate as the magnet now what if I replace the cardboard tube with a copper tube a conductor tube now here's the non-magnet you see it's the non-magnet doesn't stick to this little piece of metal at all no funny business and I'm going to drop it in the copper tube it doesn't stick to the copper tube at all either you see no attraction and I drop it drop very fast comes right out the bottom of the tube now what if I take the magnet you see this is the magnet because it sticks to this little piece of metal here that I have boom self-sustaining you see it's not attracted to the copper at all it doesn't stick to the copper at all but what happens when I drop it in the copper tube we see that the magnet is slowed through the copper tube as it drops not because it's in contact with the copper tube but because it's creating currents in the copper tube as it goes through magnetic induction and this dissipates energy and it takes a long time for it to fall I can do this experiment again this time taking an insulator like glass and turning it into the base of a pendulum so I just clip it to the bottom of the pendulum the glass is going to swing between the north and south pole of this very strong magnet that you see here and if I just let it go you see it swings freely in between the poles of the magnet no physical contact now if I draw the glass back and I let it go we can just sort of sit here and count how many seconds it takes for this pendulum to swing down so swinging swinging swinging just a nice piece of glass swinging between north and south poles glass is not free to move is there any magnetic induction well we don't expect it and this thing does seem to be just swinging freely so I'll just stop it and I'm going to take the glass out and I'm going to replace the glass which is an insulator with a conductor I have here a little circle of aluminum you see circle a little plate of aluminum and I can go ahead and clip that right into the pendulum holder and again I want to demonstrate that this is free to swing between the north and south poles of the magnet see there's no resistance in there whatsoever and then I let it go not even one swing the mechanical energy of the falling pendulum is dissipated entirely by currents that are set up as the area moves through the magnetic field that causes a changing flux which causes a voltage which causes currents that dissipate energy now what if I could interrupt the flow of current in this disk what if instead of a solid metal disk I used one that's got all these holes cut into these little slots see how this is slotted the aluminum's been interrupted by gaps of air so now there isn't a clear path for current to circulate inside of the area what will happen well if I swing the pendulum back if I pull it back and then let it go look at that it's almost swinging freely see it wears down pretty quickly compared to the glass but it gets at least several swings in before it stops and this is the principle by which you can detect for instance cracks in the skin of an airline fuselage by moving a magnetic field over the skin you can detect voltages and currents and you can look to see if there are interruptions in the current flow in the surface that would then be indicative of a crack in the surface of the metal skin of the plane so this is a very common technique for instance for looking for flaws in the skin of metal aircraft a really neat demonstration of the beautiful things that you can do with magnetic induction is illustrated by this little simulator in the world around us we take for granted the fact that we use we use the movement of electrons and electric current to light our lights power our laptops charge our phones heck even power our cars in some cases these days but where does that electricity come from well it comes from magnetic induction as Faraday recognized if I can change the density of magnetic field lines penetrating an area enclosed by a conducting loop of some sort any change in that quantity any change in that flux will result in electricity flowing in the conductor and if I can find a way to sustain that change of magnetic flux well then I should be able to drive an electric current through a conductor and then hook things up to that and do work and power devices and so forth the very thing we do in the modern world this little demonstrator illustrates the principle I have here a mechanical source of energy this is a spigot when I flip the switch on a flow of water will come out of the end of the spigot I have a paddle wheel now you've probably seen these decorating yards or parks or maybe even doing serious work near hydroelectric power plant of some sort usually they're much fancier and they're called turbines in that case but the principle is the same hook by the flow of water the wheel will spin now a fix to the paddle wheel is a magnet and so you can now imagine what's going to happen if I turn on the water okay so I could imagine harnessing a river to do this turn on the flow of water hook a turbine or a paddle wheel up to that flow attach some magnets to the turbine and then have those magnets move in the presence of a loop of wire I should be able to according to the principle of magnetic induction I should be able to induce a voltage and thus a current in this conductor and light a light bulb and in fact what we're going to see is this is the exact basis of something called alternating current watch very carefully the ions inside the wire as they move they'll move back and forth but every time they move they do work on the light bulb and the light bulb lights and if you control this process carefully enough yes the light bulb is flickering but you can make it flicker so fast that human eyes can't see it turn on the flow of water here look at that so we have our paddle wheel now spinning at 100 revolutions per minute fixed to that as a bar magnet spinning at 100 revolutions per minute we can imagine in our minds actually we don't have to we can show the magnetic field as it's being changed in the presence of this conducting loop of wire and there we have it we have a changing magnetic field through a fixed area B vector dot B vector is changing with time and whenever that happens you set up a voltage inside of a resistive device you can establish a current that can do work and in this case light a light bulb so this is the principle of electric power generation this is in fact a generator you can take one source of energy in this case mechanical energy from the pressing of water on paddle wheels you could burn fuel and make a piston move using expanding gases and do the same thing but the idea is always the same you take one form of energy you turn it into motion of a magnet the motion of the magnet induces a current and that current can then be used to power devices like lamps computers, refrigerators and so forth these things are great in an emergency for powering home appliances that you really need things you need to boil water for instance they're also simply useful for generating all of the power that's needed by our electrical devices throughout the entire electrical grid you