Force between two parallel current wires

say we have two long infinitely long imagined current carrying wires separated by some distance d then we know that these wires are going to generate their own magnetic fields the magnetic fields will of course be everywhere but here's the thing current carrying wires inside a magnetic field experience a force so this wire second wire will experience the force due to the magnetic field generated by the first wire and similarly this wire will experience a force due to the magnetic field generated by this wire which means these two wires are going to sort of pull or push on each other and in this video we're going to figure out exactly how much that force is in magnitude and direction yay fun stuff so let's start by concentrating on one of the wires let's figure out the force on the second wire let me get rid of the magnetic fields all right how do i calculate the force on the second wire well since we are calculating force on a current carrying wire due to the magnetic field this is the lorentz force and we've talked about this before so this force will be in terms of current it'll be i times l cross b and if you don't know where this is coming from well we've talked a lot about this in our previous videos on force on a current carrying wire inside magnetic field so feel free to go back and check that out so this is basically coming from the lawrence force q times v cross b all right now before we continue there are two currents there are two wires and there are two magnetic fields so can you pause and think about which this which is this i1 or i2 l1 or l2 and b1 or b2 can you pause and try that could be confusing so pause and think clearly about this all right remember we're calculating force on the second wire which means the current should be of the second wire what about the length since you're talking about the second wire this should be the length of the second wire but you might say hey mahesh you just said it's an infinitely long wire so if this is infinity force also becomes infinity what's going on well of course if you have an infinitely long wire then if you add up all the forces you will get infinite force right each piece is experiencing some force you add them all up you get infinity so what we can do is instead of substituting infinity let's keep it as l2 itself meaning let's calculate the force on some length of this wire not the entire wire let's say we're only calculating this much this is our l2 okay so we're only calculating force on this much length of the wire let's see what to do about that little bit later okay what about the magnetic field is this b1 or b2 well might seem like b2 because everyone everything is two but remember this is experiencing a force due to the magnetic field generated by the first wire let me bring back the magnetic fields remember see this is experiencing a force due to the magnetic field generated by this wire so this is b1 so let me write that so this should be b1 okay now let's figure out the direction of the force what about the direction well it's in the direction of l2 cross b1 and we know how to calculate direction of the cross product we can use our right hand for that so l2 direction is upwards and b1 direction is what's the direction of b1 well b1 is the magnetic field created by i1 this current and how do i get that direction oh i again use my right hand rule oh this is this is so much fun okay so first we can use a right hand rule to figure out what the magnetic field direction is at this point due to this current then we can again use our right hand to figure out what the cross product direction comes out to be so again can you pause the video and see what you end up with okay so first let's figure out the direction of the magnetic field b1 i use my right hand thumb rule thumb points in the direction of the current and the encircling fingers gives me the direction of the magnetic field and so notice the magnetic field is sort of like coming out of the screen and then goes into the screen so everywhere to the right it'll come out of the screen and go into the screen and the same thing will happen over here so magnetic field is into the screen this is our b1 okay now let's do the cross product l2 is upwards b1 is inwards if you didn't try this before maybe now you can try okay let's do this so let me redraw that so in three dimensions l2 is upwards b1 is into the board i can again use my right hand and i arrange my right hand such that my palm starts with l2 and then i can cross towards b1 and this is what it looks like here goes start with l2 and move towards b1 see and now thumb gives me the direction of the force so my force will be to the left that means this wire is being pulled to the left this is the direction of the force let's get rid of the hands now now let's figure out its magnitude so if i look at just the magnitude f2 that's going to be i2 times what's the magnitude of l cross b it's going to be l b l b sine theta where theta is the angle between these two so what is what is the angle well this is upwards l2 is upwards b is inward so the angle is 90 degrees again here's my awesome drawing in three dimensions as you can see that's 90 degrees right so sine 90 is 1. yay no science cyanide is one okay so i know what i2 is i know l2 is i've taken some random link but it's okay but what is magnetic field b1 do i know that can i calculate that well yeah because magnetic field b1 is generated by i1 and this is a long current and hey in previous videos we figured out the formula for magnetic field due to long straight wire carrying current so we can plug that in and we'll get our answer so again feel free to pause at any moment if you feel that you want to try this on your own so let me go ahead and substitute that so f2 is going to be i2 times l2 times what was that b1 we got that from ampere's circular law if you remember so that would be mu 0 times i by 2 pi r that's what we got if you remember now what is i over here well this is the current that's generating the magnetic field that's i1 and what is r r is the distance is that given to us yeah that's d so this is d so if you put it all together the force f2 is going to be i'm going to put mu naught first that's our constant so i1 times i2 times l2 divided by 2 pi d now if i want to calculate that force i know the value of mu naught i 1 and i 2 if i know the values i can plug in d is something that is known to me i can plug that in but l2 is i don't know what that what do i do with l2 well think about it if l2 is larger the force will be more of course right if you take a longer section of the wire when you add up all those forces you'll end up with more total force so how do i represent this i don't want l2 so you know what we like to do we divide by l2 on both sides so let's get rid of that l2 from this side and the l2 would come over here and we can just represent it this way and we could say hey this is the expression for not force but force per unit length think of it as an si unit every meter of wire if i put l2 is equal to 1 then every meter of wire will experience this much force so this is the expression we like to represent it in terms of force per unit length now here's my question what would be the direction and the magnitude of the force acting on the first wire well we can redo the whole thing and i encourage you to try redoing it yourself but there's a shortcut we can use newton's third law if this wire is pulling this wire this wire must also pull this wire so immediately we can say that the two forces must be equal and opposite so the force acting on this should be in this direction and the force acting is equal and opposite so if i take one meter of this wire and it experiences a force of i don't know 100 newtons if this is 100 for one meter then even this one meter should experience exactly the same 100 newtons of force right and therefore immediately just by using newton's second laws newton's third law i can say this should be the same for f1 over l1 so in general we can say if you have two long wires carrying current you know at some distance then the force per length on any of those wires would be so much newtons per meters right this is the unit will be now newtons per meters now here's a question for you in this case we found when both currents are going upwards they attract each other right with this force per unit length what if we change the direction of the current so what if they're both going downwards or what if one was going up and one was going down interesting cases so i have three other cases what if both are going down what if one goes down one goes up what is another one this one goes up and this one goes down in each case can you pause and think about what will be the direction of the forces all right there are multiple ways to answer this question what i like to do is just think about what happens to the signs of these if both the currents are flipped the magnetic field flips so there'll be a negative sign over here think of it that way but l2 will also flip because current is also going down this current i2 is also doing downward so this will also flip so minus into minus nothing happens so they compensate so force direction remains the same in this case they will keep pull so they'll pull each other same what about this case in this case magnetic field becomes negative in the sense i'm opposite of this direction but l2 stays the same so i'll get a net negative meaning now the force will be in the opposite direction so this will experience the force outwards and if this is pushing this this will also push this newton's third law and so the same thing will happen over here they'll push each other so what's interesting is that when the current is in the same direction the wires attract each other with this much force per unit length and when they are the currents are in the opposite direction they repel each other with this much force per unit length finally before winding up ampere did a lot of work he was one of the pioneers of figuring out this force law and what was so important was that using this we were able to define what one ampere was this si unit of current was defined using this and if you're wondering what does that even mean we'll talk about that in future videos