Right hand Rule
PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
The right hand rule is a way to predict the direction of a force in a magnetic field. To predict the behavior of positive charges, use your right hand. To predict the behavior of negative charges, use your left hand. If your thumb points in the direction of the velocity and your fingers point in the direction of the magnetic field, your palm points in the direction of the force.
So let's talk about the Right Hand Rule. This is one of the most major things that comes up when you're studying magnetic fields for the first time and really it first comes up when you do cross products maybe in pre-calculus but people kind of forget or maybe haven't taken pre-calc so let's talk about it because it's not difficult but its easy to kind of mess up if you're not used to how it works and I'll show you 3 different Right Hand Rules actually kind of 4 but really 3 all the same and then one is a little different.
Let's just go through it and just see how it works. Alright so we start off with the Lorentz force law f equals qv cross b. Alright cross products work like this, you take your right hand, you put your thumb in the direction of the first vector your fingers in the direction of the second vector and your palm points in the direction of the cross product so when we're doing this with the Lorentz force law, first vector velocity so that means my thumb always has to play the role of the velocity. Second vector magnetic field, so that means my fingers have to play the role of the magnetic field the cross product gives the force so that means my palm is always in the direction of the force.
Alright, so let's do a little bit of work with this but first of all I have to show you open major convention that you may or may not be aware of. Magnetic fields have to be in three dimensions but look I'm drawing everything on the board, that board only represents a two dimensional space so I can indicate over I can indicate up but how do I indicate out or in. The way that we do that is we have this convention we say look whenever you see a cross that means that you're talking about a vector that is pointing into the board okay? Basically you can think about it like you know when I put a vector like that it's a arrow what would it look like if the arrow was pointing into the board? Well you'd see the feathers and so that's what the cross is, the feathers. What if it's pointing out of the board? Well now I'm going to see the arrow tip so I just make a little dot now sometimes I'll circle that to indicate that its not just an errant dot that I just put on there but sometimes I'm not really that worried about it for example if I've got lot's of them, it's obvious that this represents the magnetic field so in this case I've got a positive charge moving downward in a magnetic field that's directed into the board. Alright here we go thumb is the velocity, fingers are the magnetic field and notice that my palm is now pointing to the right so that's the direction of the force on this charge to the right.
Alright, let's do this one. What about if the magnetic field is down but the positive charge is going into the board? Alright, thumb, fingers and now I've got a force that's directed to the left alright. What about here? This is weird because now I don't have the velocity what I've got instead is the force and the magnetic field but that's still fun I can still do exactly the same thing. I don't have a velocity so I don't know what I'm doing with my thumb yet, but I do have a magnetic field so that's comming out right? I've got a force so that means my palm has to point down and look at that! My thumb is now pointing in that direction so that must be the direction of a positive charge yet it feels a force down alright? One more a little twist, what if it's a negative charge? Now there's a really easy answer to that, you just pretend it's a positive charge and then just do whatever is the opposite of that but there's another way which is actually more useful in practice because the electrons have negative charge so a lot of times on these exams you'll be asked about electrons a lot and you don't want to have to always do it as if it was positive and then just not listen to it, so what you do instead is you use your left hand alright? So negative charges you use your left hand positive charges you use your right hand and as soon as I recognize that I'm going to use my left hand, everything goes exactly the same way and now the force into and that's the way that it goes.
Now you might wonder what happens to the charge after it goes into the magnetic field, well it turns out that because the force is always perpendicular to the velocity, charges that are moving in magnetic fields always move in circles that's called l'armoire precession so we can actually see that in each of the examples so it's a really easy idea if I've got a charge that's coming down and a force that's going to the right boom that's the l'armoire circle alright? What about here? Well I got a charge that's going in force to the left so here it is l'armoire circle alright? What about here? Now I'm going this way force is down l'armoire circle and how about this guy? Force is into so it's going be l'armoire circle I can't write that one right? But you see that it will always circle around the magnetic field lines alright that's the first and probably most useful form of the right hand rule but let's look at a couple of the other ones over here.
Alright, the first one that I want to mention and this one's really the exactly the same really is what happens when I've got a current in a magnetic field. Well currents are moving charges so that means that just got a lot of charges moving in this magnetic field. Current is going to be in the direction of the velocity so I just say, okay instead of velocity my thumb is the current boom boom left done, very very very simple and basically the same it's just instead of velocity my thumb now represents current. Most of the time we take the convention that the arrow here associated with the current is the direction of positive charge so it's right hand all the time unless they tell you explicitly that negative charges are moving in this direction and then of course just left hand.
Alright now, there's two other right hand rules and these are associated with magnetic fields that come from currents so this is associated with something called the biosovart law or something called amperes law so the idea is that whenever you've got a current like this, there will be magnetic field associated with it so if I got a current that goes like that, there's going to be a magnetic field that circulates around this current alright so this is it's a different physical situation we can't expect the right hand rule to be exactly the same but hopefully in this case it's almost the same. Thumb current, fingers again are the magnetic field but rather than keeping them out like that here's what we're going to do we're going to act like we're grabbing the wire alright? So we're going to grab the wire and our fingers are the magnetic field so that means that in this case the magnetic field will circulate around just like that in exactly the way that my fingers are circulating around it if I grab it so that means that above the current the magnetic field is coming out of the board and below it's going into so there it is, I've got magnetic field circulating around my wire in exactly that way.
Alright, here's the last one and this one is kind of the, the most different alright, but it's also very useful. What if I have a current loop? Alright, well I could play this game just like we just did and I could say "alright well let me grab the wire"5 okay? Well if I grab the wire like that with my thumb in the direction of the current then the magnetic field inside will be coming out of the board and outside will be going into the board so this is exactly the same as we just had no difference so why I'm I saying it's different? Well because will apply the right hand rule in a slightly different way here okay. You don't have to do this, you can always do it this way but sometimes it's more useful to instead put your fingers in the direction of the current and then your thumb will point in the direction of the magnetic field at the center of the current loop out, of course it gives us the same answer that we got the other way but this is associated with something called a magnetic moment and so you might be asked to think about magnetic moments and these current loops and it's easier when you're focusing on that to use the right hand rule when now your current is your fingers and the thumb is the magnetic field.
Alright that's the right hand rule.
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