 Hi, I'm Zor. Welcome to Inizor Education. Today we will continue talking about magnetism and electric current and their mutual dependency on each other. Sometimes I prefer to talk about certain purely theoretical issue in terms of a problem. So today I call this the problems, but basically they are kind of theoretical lectures. Because during this discussion about this particular problem, during solving these problems, this new theoretical material is presented. Now this lecture is part of the course called Physics for Teens, presented on Unizor.com, where there is another course called Math for Teens, which is prerequisite for this one. You really have to know math to learn physics. Like in this particular case we need vector algebra and trigonometry. I mean, there are different aspects of mathematics which are used. Well, all aspects of mathematics are used in physics, okay? To make it simple. All right. And the site, by the way, is completely free. I do suggest you to use this site to access the whole course rather than any individual lecture which you might find on YouTube or somewhere else. Okay, so we go to magnetism and the electric current problems. Number two, we have two problems here. Now before going into the problem I would like to remind you something which we have learned before. Now if you have a straight line wire and there is some kind of a amperage electric current going through this, direct electric current, there is a magnetic field which is formed around this particular wire in the circular form. Let's say the direction is this one, so this is plus, this is minus. And then using the corkscrew rule, my direction of the magnetic lines would be like this. Like this. So this part is on this side of the board and everything else is behind the board. So these are magnetic field lines which basically represent the magnetic force field. At any point the direction of the force is tangential to a magnetic line. Now the magnitude of this force is equal to mu zero times i divided by 2 pi r, where r is the distance. So we did talk about why it's 2 pi r, obviously it's the length of the circumference. Now it's supposed to be proportional to amount of electricity which is going through the wire per unit of time. So the unit of electric current is ampere. Ampere is one coulomb per second. So it's amount of electricity per unit of time. And mu zero is permeability constant of the vacuum, considering everything is happening in the vacuum of course. It's just a constant. Now B is measured in Tesla's and we did define what is exactly Tesla in a previous lecture. I don't want to talk about this, but this is basically a strength or intensity of the magnetic field. Well it's a vector. It's a vector. This is the magnitude and the direction is tangential to a circle. Circle lies in the plane which is perpendicular to the current. So this vector is always, since the vector is in the plane which is perpendicular to the line, obviously it's perpendicular to this line of the electric current as well. And if you connect the center of this with this, obviously it will be perpendicular to the radius because tangential line to a circle is perpendicular to the radius. Okay, so this is something which we have already discussed before. I just want you to know this because this will be used in these problems. So what's my problem number one is the following. Now you have two very long, mathematician would say infinitely long and very thin. Again, mathematician would say infinitesimally thin wires. And there is an electric current I1 and I2 going in opposite direction. So my problem right now is to evaluate the force which is in between these two lines. And the force, by the way, is repelling as it was explained in the previous lecture. And I will talk about this again. Now, well, actually if we are talking about the whole force, now these are infinitely long lines. So the force is obviously infinite but we are not really talking about the whole force which is acting from one wire onto another. We are talking about a force per unit of length. So we will use all the laws which are applicable to the infinitely long lines but we will apply these laws only to a unit of lengths like meter, for instance, in the C system. Okay? Alright, so that's what we have to determine. Alright, so first of all we have to talk about magnetic field. Now around this one, magnetic field would be like this. So I put it this way. Now in this case it would be like this. Now, what we consider first is how this particular line, electric current, affects the unit of lengths of this one. Okay, so first of all we do know the Lorentz field, the Lorentz force. Remember this? The Lorentz force is basically in the definition of the Tesla as a unit was the force is equal to, now this is the force of the magnetic field, some, any magnetic field, but in this particular case it's a uniform magnetic field. So a uniform magnetic field and there is electric current in it, which is perpendicular to the field. So if you have two magnets north and south, okay? So the uniform magnetic field is, it means that all the magnetic lines are from north to south parallel and magnitude of the force is the same. Now if you have some kind of a line here and there is an eye here, then the force will be proportional to the eye, proportional to lengths of this piece of wire and proportional to, well actually it's vector product, but in case of perpendicularity I can put a regular one. Because if it's not perpendicular I have to multiply by sine of angle for obvious reasons and in this case if it's perpendicular sine of 90 degrees one, so it's just plain multiplication. This is not a vector so to speak equation, this is a scalar. So we're talking about magnitude of this vector, magnitude of electric current and magnitude of the force of the magnetic field. Now what situation do we have here? So let's just think about it. Our field goes this way. Now if we will consider a piece of one meter here, what is the magnetic lines which are crossing this particular direction of the current? Well if magnetic field does this way, so it goes back to the board and then comes out here. So this is the radius. So let's say the distance between these two wires is d. So we have a magnetic field which has certain value, whatever the value is b, and it's perpendicular. Now it goes from behind the board towards us, right, because the lines go this way. So when they cross this particular wire, they are all perpendicular to the wire, are going from behind the board towards us. So it's perpendicularity and now we can calculate the direction of the force. The force which is acting on this unit length of the current can be found using the right hand rule. The direction of the force I mean. So what is the direction? If it comes to here, my right hand should have the magnetic field coming into it, which is like this. Then my thumb should go to the direction of the current and then this is the direction of the force. This is the direction of the force. This is the force 1, 2, from 1 to 2, from I1 to I2. And what's the value of this? Well the value is the current which is coming through this, which is I2. Well then you can say L, whatever the L is, maybe it's 1 meter, but in any case for instance it's any kind of a piece of the wire. And then the magnetic field intensity. Now what is magnetic field intensity? Well as we were saying before, if you have a wire and you have a magnetic field, its intensity is mu 0i divided by 2 pi r, right? So r is basically g, mu 0 and i is basically I1, right? In this particular case. So it's 2 pi, sorry, 2 pi goes there, 2 pi d and on the top we have mu 0 times I1. So this is the force which is acting on the piece of wire of the length L. But if we are talking about 1 meter, then we have to per unit, well forget about the word meter, unit of length, whatever the unit is. So if this is the unit L and per unit of length it should be, I'll put this line on the top. That means per unit of length. Obviously we have to divide by L, right? So it would be mu 0 pi 1 i2 divided by 2 pi d. Okay? So this is my force which is acting from this wire per unit of length on this wire. Great. Now let's talk about another force. Another force is from this wire onto this one. Let's do exactly the same thing, because it's really the same thing. It's completely symmetrical, right? So first of all, what's the direction of the magnetic line? So it's this way. So at this location lines are coming from behind the board towards us. And then using the rule of the right hand, I have to do this. And my thumb goes to direction. So magnetic lines go into my hand. My thumb goes down. So the force goes this way. So obviously it's a repelling force. And we know that if direction is opposite, then there will be repelling force. That was a previous lecture. So this is the force 2 1 from the second to the first. So what is it equal to? Well, it's equal to I 1. I 1 times L, if this is the piece of L lengths, times B. And what's the B? B is in this case magnitude of the field of this wire at this location. And it's the same thing as before. So it's 0. In this case it's I 2 divided by 2 pi distance D. The same thing. Now, and if we're talking about per unit of lengths, we have to divide it by L. And we will have 0 by 1 by 2 divided by 2 pi D. So as we see, it's exactly the same thing by magnitude, but directed opposite. Well, which is not a surprise because, you know, the third Newton's law, the way, if there is a force of action, there is a counteraction. So let me just get another marker. So this is basically the result. So these are the forces which are acting in this particular case. So if you have two parallel wires, the end direction is opposite. So repelling force is the multiplication of their currents divided by 2 pi times the distance between them. And there is a constant of proportion analogy, which is basically a property of the environment where they are. It's a vacuum. It's some kind of one constant. If it's something else, it's something else. Okay, that's my first problem. And basically this is the result, which is kind of theoretical result, which I wanted to present as a problem. Now, the second problem is kind of practical. I briefly mentioned, basically, the approach, which I'm going to explain right now before. Just imagine yourself as a, whatever, 17th, 18th century physicist, which understands that there is something which is called electricity. There is something which is going through this wire. And you would like to measure it in some way. And you don't have really, you don't know how to measure it. So you have certain experiments and you found out that there is a repelling force, like I was just talking about between two wires. There is a repelling force, which depends on the electric current, amperage I1 and I2. So what can be an experiment, which would allow you to determine what's the amperage by measuring the force? Well, the force to measure the force is also not very easy, right? In any case, something like whatever I'm going to explain right now might be an experiment, which would lead to measuring the strength of electric current. So I imagine myself as a 18th century physicist and I'm arranging the following experiment. If you consider two wires and one wire, now both wires are perpendicular to this board, they are parallel to each other and they are hanging on some kind of threads. Now, what happens if I will put electric current through these two wires in opposite direction? Let's say plus on this side of the board and minus on that side of the board for this wire and the opposite minus here and plus behind the board for this wire. Well, we know that opposite direction of the wire would reduce the force of repelling, right? So the wire will swing this way. If we assume that the wires are identical, they have identical mass per unit of length. They are relatively thin and relatively long. We can use all these formulas which we were using before and now measuring this angle, let's say the angle is 5 and it's the same because we are putting the same source of electricity from some kind of a battery. We connect both ends to the same ends of the battery plus this and plus of this guy behind on this side of the board and plus of this guy behind the board. We connect to one terminal of the battery, the positive one and another two ends to another. So the battery produces exactly the same amount of voltage to both of them. So we have exactly the same current. So let's go back to the formula which we have before. F equals to, what was it? Mu is 0, I1, I2 divided by 2d, right? That was my formula from the previous problem. 2 pi d, sorry. So that's what we derived in a previous problem. So now I would like to measure I, knowing F. And how do I know F? Well, if I know the angle phi and know the mass of the wire per unit of length I'm talking about, then we can determine very easily F. So now in our case what happens here is the following. Now if my initial distance is d, my final distance between these two, when they are in equilibrium basically, it's capital D, which is equal to d plus these two pieces. This one and this one. So I need the length obviously. Let's say it's L. So this piece is L times sine of phi. So we have 1 and 2. We have 2 these pieces, sine of phi. So that's my new distance. So I have the distance and I have the equilibrium, right? Now what does it mean? Well, equilibrium means that the tension here, so this is my weight mg. I'm talking about per unit of length, so it's not infinite. Now tension of the lengths, again, you can consider that every meter has its own thread and they're all going in the same in the same line. So the whole wire is swinging on all these threads at each meter interval, okay? So it's per meter. And here we have this force. So these three vectors, tension, magnetic force of repelling and weight, these are supposed to be in equilibrium, right? Okay, so let's just do it in writing all these equations and we will come up with a solution, right? So first of all, if this is a tension and it's supposed to be directed this way and this way, right? We will represent the tension as vertical component and horizontal component. Now horizontal component is supposed to be balanced with F and vertical component is supposed to be balanced with my weight P, right? Don't pay attention to this, this is a short vector and this is a long one. Obviously it's just my drawing, but they are supposed to be of the same length to be in equilibrium, right? So my T times, okay, what is this vertical component? This is phi and this is phi as well. So this is T times cosine phi. That's equal to mg. Now T sine phi is equal to F. And what is F? F is mu zero times, we are putting exactly the same current in different directions, but the magnitude of the amperage is the same. So it's I square, right? I one and I two are equal to each other and that's I divided by two pi and the distance, the distance is the big one, B, which is also known. I mean initial distance we know, length we know, phi we know. So this is basically sufficient to determine I because what we have here is two equations with two unknowns. T is unknown and I is also unknown. Well, the best way is just to divide the second one by the first one and we will see, okay, let me get rid of this, so I have more space. We will have, if we will divide on the left, we will have tangent phi, sine divide by cosine and here we will have mu zero I square divided by two pi d and G, from which I square is equal to two pi d and G tangent phi divided by mu zero and finally I is equal to square root of this, which is equal to square root of two pi and G. D is D plus two L, what was it, sine phi and tangent phi divided by mu zero. So that's my final expression for I. So that's how you measure the strengths of the current in the units which we know, we know mass, we know acceleration of the free-falling. D is the distance, initial distance between the wires, L is the length of the wires, phi is the angle from the vertical when both wires are repelled each other and mu zero is a constant. So that was the problem, but in theory again I wanted you to immerse yourself in the situation of experimentator, experimentator, whatever. The physicists of a long time ago they were actually talking about how to measure this electricity sink which they have discovered and apparently they had to bring the new entity, the electricity back to the old ones, mechanical ones, like the force which they have already learned from the Newton's time, which is like 17th century. All right, that's it. I do suggest you to go to the website Unizor.com, go to this lecture, it has notes as every other lecture has notes, very detailed ones and this problem is obviously the solution to this problem is fully presented in the textual part of this lesson and good luck.