 Hi, I'm Zor. Welcome to a new Zor education. I would like to continue talking about alternating current in different circuits. The previous lecture was about circuits which contain resistor that's one part and then resistor and capacitor that was the second part of the previous lecture. Now in this lecture instead of capacitor I will consider inductor. So let's consider we have a circuit which contains the generator of alternating current, resistor and inductor. That's what will be considered in this lecture and I will derive similarly to previous lecture. I will try to derive the value of the current in this circuit based on the generated EMF. Okay, now this lecture is part of the course called Physics for Teens. It's presented on Unizor.com. Every lecture on this website contains detail notes so I recommend you to watch this lecture from the website rather than if you found it on YouTube or wherever so you go to the website you choose the Physics for Teens course and then this is the part which is called electromagnetism and the chapter is the Ohm's law for alternating current. Okay, now there is a prerequisite course, Physics for Teens. It's called Math for Teens on the same website. I do recommend you to either study this course or be comfortable with whatever concepts are in this course. Basically it includes a little bit more of calculus than usually in high school. It's probably up to the first maybe year in the college but anyway this is necessary because for instance for this particular lecture and the previous one I'm using differential equations. So Math is a must for studying Physics. Alright, so let's go directly to this particular problem which we have at hand. So let's consider that we have a generator of alternating current which generates the oscillation, oscillating voltage according to sinusoidal oscillations. This is my generated electric motor, electric motor force EMF or voltage on this particular generator. Now this voltage goes from one end to another and then since alternating it goes back so this is what is described by this particular equation where t is obviously the time and omega is an angular speed of rotation of the rotor inside the generator and that's what actually makes this alternating current to have this sinusoidal oscillations. Now as the electric current goes around this circuit in this direction or in that direction doesn't really matter. At any moment the EMF which is basically a difference in potential between these two ends, since this is a connection in series, it's actually drops. So this difference between potentials between this and this is drops here and drops here. So these are voltage drops here, difference between potential between this and this and this is difference in potential between this and this. So obviously e of t is equal at any moment of time. So the difference between this and this is equal to difference between this and this plus difference between this and this. It's plain arithmetic actually. It's like 5 is equal to 2 plus 3. Now what do we know about these two voltage drops? Well this is the voltage drop on a resistor and at any given time it actually obeys the regular Ohm's law. This is the current and this is the resistance of this particular resistor. This is the regular Ohm's law. Voltage equals to current times resistance. So that's easy. Okay. Now I didn't mention it but it's kind of obvious that since this is a closed circuit the current which goes through this is exactly the same as current which goes through this one. So that's why I'm using I t not without any kind of a indices here because I of t here and I of t here is exactly the same. Now let's talk about the voltage drop on inductor. Okay. First of all why the voltage drops on the inductor? If it's a straight wire for instance without all these loops the voltage wouldn't drop. It would just you know there is no resistance actually but there is some resistance if this is a coil. Now if you remember there is a self induction effect. Whenever you have a coil you have certain electromagnetic flux which is the result of the fact that this is loop after loop after loop because every loop has certain electromagnetic field inside this loop and when the flux is changing it generates its own electromotive force that's called self induction which is always goes against the change of the voltage which comes on the ends of this. So that's why it's always written as a rate of change of the magnetic flux is basically generating this particular voltage and it's always written with a minus sign because it resists the original voltage. Now in our case we don't really need the minus sign because all we need is actually the drop of the voltage. We need this negative voltage because that's exactly what we are adding drop here and drop here to get the voltage over there. So in our case we don't need this minus. Now both are functions of time and now let's talk about electromagnetic flux. Well electromagnetic flux obviously depends on the current which goes through this particular inductor and it also depends on certain properties of the inductor which are called inductance. So inductance is that coefficient which characterizes the properties of the inductor and it obviously depends on for instance the radius how many loops we have what kind of a material it's made of etc. So this thing is equal to L times I of t where L is inductance which is supposed to be given in as much as the resistance is given if we want to know what exactly is happening with the circuit. So this is basically a definition of the inductance. Inductance is that coefficient which contains the current with the change of the flux the rate of change of the flux. Okay I made a mistake I need not just current I need change of the current obviously. Current by itself doesn't really cause anything change of the current rate of change of the current is causing change of change change of the value of the flux and that's what generates the self induction and the voltage drops. So again change of the rate of the current is causing change of the rate of the flux rate of change of the flux and that's what causing the self induction which is basically a functional equivalent of the resistance because self induction is directed against the change of the voltage. So we have actually expressed these two in terms of the values which are kind of given R and L and the one which needs to be defined this is the current. So this is given because this is the formula and these are all contains only one unknown function i of t the current. So it's a differential equation right so let's just write it down. vR t equals vR which is R times i of t plus vL which is L times i of t. So I'm using just the little slash to signify the derivative just easier. Well obviously what we have to do to solve this particular equation is bring it into some relatively familiar kind of format. Okay I'll divide everything by L so and I will use y of t equals to i of t. I'll use the different letter and I'll explain you why in a second. So in this particular case if I will divide everything by L my equation would be y plus R divided by L so I'll use a equals to R divided by L so it will be a y of t equals and I will use b is equal to e0 divided by L b sin omega t. Now if you look at this equation and if you which my previous lecture where instead of inductor we had the capacitor the equation was exactly the same I was just using different values for a and b different constants doesn't really matter but the resulting equation is exactly the same. Now I was discussing how to solve this particular equation in the previous lecture and the notes for the previous lecture have even more detail explanation of how to solve this equation. So right now I'm just referring you to that lecture and notes for that previous lecture to familiarize yourself how to solve this equation. Now I will just write the solution which I have borrowed from the previous lecture and the solution is y of t is equal to b a times sin omega t minus omega cosine omega t divided by a square plus omega square and plus constant some kind of constant. Okay now first what I will do I will change it slightly what I will do I will use the following is equal to R divided by L right so what I will do is the following I don't use this picture anymore. I will use a is equal to sin a divided by a divided by square root of a square plus omega square is equal to sin of some angle psi and omega divided by square root of a square plus omega square is equal to cosine of psi right these are two various sum of squares of them is equal to one so I can always find the angle psi which is equal to arc tangent of a divided by omega this is a simple trigonometry this is less than one by absolute value this is less than one by absolute value and sum of square is equal to one so that's why I can always find such an angle psi equals to arc our tangents of this and obviously sine is equal to this and cosine is equal to this so what does it bring me to well here if this is sine and this is cosine of some angle psi then this is sine times sine minus cosine times cosine which is what which is a cosine of omega t plus psi and that I have to multiply by b divided by now this is square of this times square of this right one square is used to assign these values so another square root of this remains uncovered and this is my y of t well y of t is actually i of t it's a current I just used the y to have the same equation as in the previous lecture so this is done and plus constant so that's what I have very much similar to the previous lecture actually this okay now if we will put our values back into this oh I think I forgot the minus sign yes we had sine times sine minus cosine times cosine this is minus cosine there some I'm sorry about that yeah now it's good I would like to change slightly minus cosine of omega t plus sine I would like to change it to sine why why why do I want to change it to sine because my EMF is expressed as an oscillation of sine and I would like actually my current also to be in terms of sine with some phase shift but I would like to change it a little bit so I will have similarly sine here and sine for IFT so for this I will do a little trigonometric trick so minus cosine okay now sine and cosine are related in this way sine of alpha is equal to cosine of pi over 2 minus alpha right now using this I will put minus sine of pi over 2 minus omega t minus sine which is equal to sine now sine is odd function so if I change sine here I have to change sine here so it would be omega t minus pi over 2 minus psi my rate omega t would be with a plus pi pi over 2 would be with a minus and this would be with a plus correct now what is my function what is my angle psi I know the tangent of psi is equal to a over omega right now if I have pi minus 2 pi over 2 minus psi well tangent of this is equal to cotangent of that right remember this is my right triangle if this is a this is omega this is psi tangent is equal a over omega this is pi over 2 minus psi so cotangent basically of this angle is equal to a over omega or tangent is equal to omega so tangent of pi over 2 minus psi is equal to omega over a and what I will do I will just use the more familiar letter phi for this and now I can see that the whole thing is equal to sine of omega t minus phi where tangent pi is equal to omega over a okay so that's it now I will use this to substitute into this formula so what do I have I have minus b b is e0 now that was minus so minus actual goes here I don't have minus anymore I have this so I have b times sine of this so b is e0 over l times sine of omega t minus phi divided by square root of a square which is r square divided by l square plus omega square equals e0 sine omega t plus sorry minus minus phi divided by if I will put l inside the square root it will cancel this one so I will have r square plus and it will be multiplied by this one square now another little kind of recollection l times omega again if you remember it's one of the previous lecture we were using l times omega as something which is called inductive reactance of the inductor which is equivalent to its resistances if you will take a look at the units of measurement it will be ohms and again it's similar in some way to x with a index c which is reactance of the capacitor capacitive reactance so capacitor has certain characteristic which is equivalent to a resistance and it's called capacitive reactance and inductors have characteristic similar to resistance and it's called inductive reactance and that's what it is so as a result we have the formula and plus constant by the way which I still have it right okay so now this is basically kind of a representation of the ohms law for AC for alternating current which includes resistor and inductor this is actually very much like for the capacitor at the for the capacitors I have E0 divided by r square plus xc square both for inductors they have xl square here so this represents this square root of r square plus xl square it represents something which is equivalent to the resistance for the whole circuit if the whole circuit contains resistor and inductor so their entire resistance in a way is the square root of this and it looks like the ohms law for direct current you have some kind of a voltage which is E0 times sin omega t and you have some kind of equivalent to a resistance there is this angle phi this is a phase shift so the oscillation of the current are shifted from the oscillation of the electromotive force now if it's minus phi it means it shifted to the right now if you remember the shift was different for a capacitor in the previous lecture I had plus here it was sin of omega t plus phi so if it's only a capacitor with with a resistor then the oscillation are shifted to the left if there is no resistor if you remember it was minus pi over 2 now in this case absolutely similarly if there is no resistor it will go to minus pi over 2 and again we were addressing this in one of the lectures where we didn't have resistor at all so just as a check so when we didn't have a resistor we had minus pi over 2 now if we do have a resistor it's a bit more complicated but if I will use this now omega over a is omega over a is rl which is xl divided by r so that's my tangent so if my r is equal to 0 if there is no resistance it's infinity tangent of the angle is infinity and that's what pi over 2 actually is the tangent of pi over 2 is infinity so that would be exactly minus pi over 2 if there is no r here now obviously you understand it when I'm saying something like tangent of pi over 2 is equal to infinity it's not exactly correct mathematical statement there is no such thing tangent is not defined at pi over 2 however if my angle is approaching pi over 2 it's tangent approaching infinity okay that's more kind of mathematical expression but for physics actually they do the simplified versions without any problems so that's basically it this is what what the current in in in the circuit which contains resistor and inductor with reactance xl that's how it looks and the phi is defined as arc tangent of xl over r over r okay so that's it so now we have basically two cases about three cases plane r plane resistor then in the previous lecture I was talking about resistor and capacitor so it's RC circuit now this lecture is rl circuit and that's the formula difference between capacitor and inductor in this circuit is this minus sign minus here for inductor and plus for capacitor and obviously for capacitor I have xc and xc if you remember is 1 over c times omega xl is omega l so these are equivalent to resistance for resistor and they're both called reactants this is capacitive reactants and this is inductive reactants they're characteristic of both the device itself whether it's capacitor or inductor that's c and l and also it depends on the frequency and on the speed of rotation so the higher speed of rotation the higher reactants or resistance if you wish of the inductor and as far as the capacitor is concerned it's just the other way around the higher the frequency of rotation angular speed the smaller resistance of this particular capacitor and if my alternating current is oscillating faster and faster the capacitor presents less and less resistance inductor is the other way around the more frequently my oscillation oscillations are going the more resistance the current will feel from the from from this inductor okay basically that's it for today I do suggest you to read the notes for this lecture and the next one would be probably kind of a combination of whatever we know it will be resistor and inductor and capacitor in one circuit I specifically decided to go through all these three different variations with gradual increasing of the complexity because there is some mathematics here and some people might feel a little bit uncomfortable so I decided to approach the same thing from few different sides so you will basically feel how alternating current is going through all the different kinds of circuits that's it thank you very much and good luck