uh hi everybody physics ninja today what i want to do is i want to look at how you calculate uh the magnetic force when you have two currents next to each other there are two uh situations to consider uh the first that i'm going to look at is when currents are parallel to one another how do you apply my force equation to calculate the direction and the magnitude of the force of one wire acting on the other wire in the second case i'm going to flip one of those currents so the currents are anti-parallel all right in the second video i'm going to make which i'll link in the description i'm going to apply what i've uh taught you in this video in order to consider a more complicated arrangement of wires i'm going to consider three wires on the vertices of an equilateral triangle and how would you calculate the magnetic force of two wires acting on the third all right like with all my videos if you like it give it a thumbs up consider subscribing to my channel okay let's get started all right in order to solve this problem i'm first just going to review two right hand rules that i use to solve this now our force equation is written as uh this equation right here it's the current multiplied by the length okay and it has a vector here this simply means that that vector is going in the direction of the current there is a cross product now with the magnetic field and it's the magnetic field where the current is located so i have to know how to evaluate the direction of this cross product and for this i use right hand rule number one it's depicted here in that little cartoon you take your index finger you point it in the direction of the first vector that is the direction of the current now the second vector the index vector sorry the index finger points in the direction of the magnetic field okay and my thumb gives me the direction of this cross product this will give me the direction of the magnetic force acting on that current okay so this is how you have to find the direction of the force okay this is one of many methods this is the method i prefer to use all right let's try a simple example to make sure we understand this property here all right let's practice with this simple example here i have a current a positive current moving to the right and it's placed in a magnetic field that is into the page right hand rule number one says you take your index finger you place it in the direction of the current my middle finger should point in the direction of the magnetic field that's into the page and now look at the direction of my thumb my thumb is pointing up when i do this that means that this current here would experience a magnetic force f that is acting upward all right let's look at right hand rule number two now all right our magnetic force equation has a very important term right there has to be a magnetic field in space if there's going to be a force acting on a current so we have to be able to understand how to find the direction of the magnetic field all right so magnetic fields are themselves produced by currents so how would we find the direction for example of the field produced by a current that is flowing in some arbitrary direction let's start with the upward direction right hand rule number two is used to find the direction of the magnetic field so all you have to do for right hand rule number two again it's not to find the force that's right hand rule number one right hand rule number two says you put your thumb in the direction of the current the positive current and just curl your fingers around that current what you see is you have that the field is in the direction of your fingers so if i have a long wire carrying a current upward the magnetic field produced by this current is simply circumferential around that wire okay it would simply form circles around that wire that would be the direction of the magnetic field so it's always changing as i go around a circle around that wire okay now let's go back to our original problem and apply everything we've learned all right we start with our first case where we have parallel currents our goal is to calculate the magnitude in the direction of the magnetic force but on which object we have two objects here i am going to start with finding the force on the current i2 so if i look at my force equation i'm just going to write a 2 here it's the force on object 2. that means it has a current i2 which in this case i'm going to assume is 0.5 amps it's going to have a length okay that is the length of object 2. now what about this last term and this is the important part this current here is placed in space it's placed next to another current it's placed in the magnetic field produced by this guy okay so it's really the field produced by object one at this location so we really have to use two right hand rules for this problem we first have to use a right hand rule to find the magnetic field produced by current i1 and then well we go back and we use my right hand rule in order to find the direction of the force we have to evaluate this cross product between the two vectors so let's first look at the current i1 and find the direction of the field everywhere in space so for this we start by just looking at the current i1 i am going to apply my right hand rule again this case was right hand rule number two is used to find the direction of the magnetic field so you place your thumb in the direction of the current that's going up and my fingers now curl around right so the field is changing direction and it is circumferential like this however after i'm going to be placing a current over here in this region over here somewhere to the right of the first wire but if you look at the field everywhere on the right hand side of this current we're going to notice that the magnetic field here is always into the page how do you represent a vector into the page again you simply write it as something like this everywhere in space now anywhere to the right of uh the current i1 the vectors into the page look at what these vectors are going on this side right these vectors here are coming out of the page and i'm looking at a two-dimensional view here all right so we know the direction of the magnetic field how now do we find the magnitude of the field okay i've done videos on this before you can do you can look this up in your book the magnitude of the field is pretty straightforward it's mu zero that's the permeability of free space multiplied by the current in this case it's the current i1 in this we're looking at the field produced by that current divided by two pi multiplied by the distance from the center of that wire okay so we're going to have current i2 that is a certain distance away from the first wire okay so that would be the distance i would substitute in here all right let's put it all together now and calculate the force all right so now we put everything together so i have the direction of the magnetic field now produced by the current i1 going into the page on the right side coming out of the page on the left side right and it does these it's circumferential around that wire the field is expression is given by this magnitude okay so it gets smaller the further i am away from the source the source is i1 so we're now interested in finding the direction of our magnetic force so for that we apply the right hand rule number one so what do you do you place your index in the direction of the current my middle finger now should be into the page and if you do that correctly you should find that their magnetic force has to act in this direction this is the force f2 okay let's go ahead now and find the magnitude this is a cross product remember and if i have any two vectors if i want to find what the magnitude is imagine you have a vector a and i have a vector b over here there is some angle here between both of those vectors if i'm worried about how big the vector c is the equation to find the magnitude you simply multiply the magnitude of each one of those vectors and it gets multiplied by sine of the angle theta between both of those vectors so we're interested in the magnitude of the force f2 how big is that force so i applied our equation here so it's the magnitude of the first vector that's simply i2 multiplied by the length of this second current and multiplied by the magnitude of the field b1 that is our expression up above here the permeability of free space the current i1 divided by 2 pi r and r in this case is the distance from center to center of these two wires i'll just write it as d right here and multiplied by sine of the angle okay so what is the angle now between both of those vectors one vector is pointing up it's the direction of the current the second vector is into the page actually so this angle theta here is actually 90 degrees for our case so sine of 90 is equal to 1. so that gets simplified all right so our magnitude of the force f2 is i'm going to just rearrange this so i'll bring mu 0 here 2 pi these are constant values you can see it depends on the length right how long is that current multiplied by both currents right this um how much current is in the magnetic field and also the current i1 which is the source of the magnetic field it also it depends on the distance between those two wires right if the distance is far i'd expect the force to be small and again i know longer need to worry about that sine of theta term because that's equal to one so a lot of times for these problems we're interested in the force per unit length so what you could do is simply divide through each side by the length l2 and that'll give you an expression for the force per unit length right you can just eliminate this term and then you're left with that okay so let's go ahead now and just substitute our values just to get a numeric value for the force per unit length acting on this current i2 all right so substituting our numbers now um you just simply have to look some of these up right if you don't know them mu zero is four pi multiplied by ten to the minus seven so that makes it pretty small divided by 2 pi and now i substitute the values of the currents here 0.5 for the current i2 1.2 for the current i1 and they're separated by 3 meters and remember this is the force per unit length that i'm calculating here so i put that in the calculator and i get 4 times 10 to the minus 8 this is in newtons per meter right that's the force per unit length acting on the current i2 right so if you know the length you can simply bring that to the other side and find the total force in newtons all right so let's have a look at this problem so we just did this calculation we found that there was a force a magnetic force that was acting in this direction that was the force of object one on object two all right but what if the problem was calculate the force now on object one well how would you do it okay so let's consider what we have so i'm going to use my equation to find the force on object one this is going to be due to object two however the current that i'm considering here is i1 it's going to have a length l1 and it's placed now in a magnetic field that is produced by object two what is the direction of the field produced by object two well again i use right hand rule number two and i should find right the field is circumferential so it goes into the page on one side comes out of the page on the other side so in this position here the field produced by current i2 should be out of the page everywhere where current i1 is located the field produced by the current i2 is out of the page the magnitude of the field produced by current i2 is mu 0 i2 divided by 2 pi r remember these wires are a distance d apart so let's put everything back into the equation all right let's first find the direction actually we can find the direction using right hand rule number one all right we're going to use right hand rule number one so we take our index finger you place it in the direction of this positive current the field is now coming out of the page right you can see by these dots here these symbolize a vector that's pointing out of the page if you do this correctly you should see that the magnetic force my direction of my thumb should point in this direction so this is the direction of the force f1 so we know the direction of f1 if you're interested now and how big is that force now we substitute in all our magnitudes so we have i1 l1 uh the magnitude of b2 is mu zero i 2 divided by 2 pi over d okay again this sine of the angle theta but it's sine of 90 degrees in this case so you see that if i'm looking at the force per unit length for this for this wire i'm going to get the exact same expression as what i previously had mu naught over 2 pi i have both currents now i 1 i2 divided by d and sine of 90 is 1 so we get to the exact same expression that we previously had the other important thing is that look it if we have currents in the same direction right so parallel currents that means that we are going to have an attractive force between both of those wires the forces are the same magnitude they're in opposite direction and they're on different objects okay let's go look now at the anti-parallel case all right we now consider the opposite current case now it's going to go a lot faster so again the field produced by the current i1 goes like this that means everywhere on the right hand side we have a feel that is going into the page okay so the field at this position is going to have the same magnitude because the field is given by mu zero it's produced by the current i1 and we're measuring it always the same distance from center to center all right so now we apply right hand rule number one because we want to find the direction of the magnetic force you take your index finger you point it down now remember now i have to rotate my hand a little bit because my middle finger has to point into the page if you do that carefully and you apply right hand rule correctly you should find a magnetic force that is acting like this to the right make sure you'll actually get that okay if you consider the opposite case now you have the current i2 flowing down this current produces a magnetic field that goes around that wire okay now you have to apply right hand rule number two right to find this direction of the magnetic field produced by a current if i do that my thumb's in the direction of the current going down here i should get my fingers that are going into the page so a vector that goes into the page is like this and on this side it's coming out of the page right so everywhere to the left of the current i2 the field should be going into the page okay so again this is the field produced by current two how would i find now the direction of the magnetic force acting on the current i1 again we apply right hand rule number one to find the direction of this magnetic force so let's go ahead and apply it we have a current going up you place your index finger going up you place your middle finger in the direction of this magnetic field this is going into the page and your thumb if you did this carefully your thumb should now be pointing to the left this is the direction of the magnetic force produced by the current i2 on i1 so what do we have we have opposite currents are going to repel each other okay this is a very very important topic in magnetism okay um currents that flow in opposite directions repel each other currents that flow along the same direction are going to attract each other and that is a consequence of this magnetic force equation and the direction of the field produced by each one of those currents all right and if i'm interested now in the magnitude of these forces i'm going to find the same magnitude they're just in different directions and i'm going to get to the exact same expression that i had for the parallel case okay so i'll just write the general equation like this again depends on both currents depends on the distance between them okay and this gives me the force per unit length acting on each one of those wires alright thanks for watching folks hopefully you've learned something in this video