**Concept:** Force on Moving Charges & Right Hand Rule

Hey guys. In this video we're going to talk about forces, magnetic forces on moving charges and I'm going to introduce the right-hand rule, let's check out. Alright, so if you have a charge that is moving through an existing magnetic field and it's going to look like, this you have a magnetic field. Remember, fields are usually represented with multiple lines magnetic fields is B and you have a charge, let's say q, that's moving, let's say this way with a speed V, that charge will experience a magnetic force due to the fact that it's moving in a magnetic field, okay? The magnitude of that force will be given by this equation Fb, equals q, V, B, sine of theta, where q is just a charge, V is the speed and it's a vector, or I guess the magnitude of the velocity vector, this is the magnitude of the magnetic field and times sine of theta, where theta is the angle between the two vectors, q is not a vector V and B are vectors and directions, so the angle will be the angle between the two things with directions between V and B, for example, here, V is going this way and B is going this way, so the angle here is this, which would have been or which is 90 degrees, okay? By the way, this is called Lorentz force named after mr. Lorenz, and you should know that the units for a magnetic field is Tesla, named after mr. Tesla, cool? And then one Tesla is one Newton divided by one ampere times meter, if your professor likes you to memorize these things, let's do a quick example and calculate some magnitudes here. So, here we're saying a 2 Coulomb charge, let's write that down, the charge is 2 coulombs, moves perpendicular to a magnetic field, moves perpendicular to magnetic field, perpendicular means 90 degrees, it doesn't tell us, the problem doesn't tell us who's moving where or which thing points in what direction. So, we can just kind of make it up as long as they're 90 degrees apart. So, we can say that the velocity is going to go this way and the magnetic field is this way because we're told that the angle between the two of them has to be 90 degrees, cool? Let's say here, that it's moving with 3 meters per second. So, that's our V, 3 meters per second, and it feels a force, that's our magnetic force, of 4 Newtons, these are unrealistic numbers but I just wanted to keep it really simple, what must be the magnitude of the magnetic field? Magnetic field is big B, and I want to know what is big B, so the question is, is there an equation to tie these four variables together, and obviously there is, that's the one we just looked at FB equals q, V, B, sine of theta. Now, we have everything but we're looking for B. So, all we have to do is solve for B. So, I move it over here, B is FB divided by q, V, sine of theta, sine of theta is easy, the angle between q, the end angle between V and B is 90, and the sine of 90, you should know, is 1. So, this whole thing just becomes a 1. So, we don't have to worry about it, the force is 4, the charge is 2, the speed is 3, this is 4 over 6 or 2 over 3 or 0.67, we are talking about magnetic field strengths, so this is 0.67 tesla, cool? That's it, that's all I got to do with that equation. Now, one thing we haven't talked about yet. So, we talked about the magnitude but we haven't talked about the direction and direction in magnetism problems are always going to come from the right hand rule and there's going to be a few variations of the right hand rule, we'll tweak the rule to make it work for different problems, right hand rule abbreviated RHR, in case you see around, you know you're getting comfortable with things when you start using abbreviations. So, before we get into the right hand rule which is massively important, a lot of people get confused here. So, we got to go slowly through this, I want to warn you that there's a bunch of different rules, in fact, most books and most professors will use some version like this, which you might have seen in class, okay? Where it's like you're shooting someone but then like your middle finger sort of like sticks to the side, it's hard to do this on camera, but this is the most popular one and there's an engineering reason or sort of a more advanced physics reason why this is kind of a clever thing to do, I don't like that version, I use a different version, a long time ago I sort of thought about all the different ones and I've settled on the one I'm going to explain to you guys, it's got a bunch of advantages, you can't fully appreciate the advantages unless I explain the whole thing to you and that's too much. So, you just have to kind of trust me or you can use whatever your professor uses or whatever else other valid method you find that you may like. So, whatever you do, you have to pick one and stick with it but if what you're doing is different from your professor, different for me you just have to make sure that they match, what I mean by that is, if I'm solving a problem and I got a direction to the left and you're using a different method, your method should still give the same answer same thing with your professor. So, if you use my method and your professors using a different method, you have to make sure that you're actually getting the answers that match his answers otherwise something's wrong, okay? So, please be careful, pick one and then go with it. So, when we saw, the reason the right hand rule is because things are now going to be in three dimensions and if you have two dimensions you can be going up or down, that's one dimension the second dimension is left or, right? But now, we're going to have a third dimension. So, one dimension is going to be, let's say the x axis and you could be going right or left, that looks ugly but whatever. and then here, you can be going up or down but now we're going to have sort of the z axis here, that is going to be going either away from you or towards you, okay? Now, doing this on camera is weird and I want to explain this so that you don't get confused, when I. Away from myself I'm coming towards you, okay? So, I don't want you to look at me I want you to imitate me, okay? So. Away from yourself and towards yourself, those are the two directions, hopefully there's no one around, this is going to get weird, right? So, the direction away from you, again, do this with me, right? Away from you, is the same thing as into the page, into the page. So, get your clutch hand out, right? Look at the page, away from you is into the page or into the computer screen, okay? That's going into the page or into the plane and towards you, towards you. So, go ahead and point yourself, right? Like an idiot, towards you, just like I'm doing, we're all being idiots, is going to be out of the page, out of the page. So, here's the page, right? And, if I point It myself you can think of it coming out of the page towards me. So, go ahead and do that, again, this is probably super simple but I want to make sure that we go slowly here, there's symbols for these things, just like how up looks like this and down looks like this and right looks like this and left looks like this, totally obvious, the symbols for these things for into and out of are not as easy, so the symbol for away from you is an x, and the symbol for towards you is a dot, and the classic way of remembering this is, if you are looking, this is my drawing of an eye, if you are looking, I can't draw, but if you're looking at an arrow, okay? So, if you're looking at an arrow and the arrow is going away from you, what you see is sort of a little x in the back of the arrow. So, as you see an arrow going away from you, you see the x right here, okay? If the arrow is going towards you, you don't see the x instead you see this front of the arrow here, which looks like a big dot, okay? Because you see this here. So, that's how you're supposed to remember, if you have a better way, cool? Just remember, that away from you is into the page and that is an x because it's the back of an arrow, cool? Alright, so those are the things you need to know.

Now, there are three things that have direction and I already talked about two of them here, the velocity has a direction, the magnetic field has a direction but the force also has a direction, okay? So, we're going to do is going to use the right hand rule to figure out these directions. Remember, I mentioned the magnetic field is typically drawn with lots of arrows, therefore, we're going to use our four fingers, hopefully got all four still, we're going to use four fingers to indicate the direction of the magnetic field, okay? So, this is going to be the direction of B, by the way, you're supposed to keep them together, don't do weird stuff like this. So, four fingers like this and that's going to be B, speed is usually represented with a single arrow. So, that's going to be our single thumb, right? Hopefully you don't have two of these you guys. So, single thumb which is going to go this way, and what's left is the palm of your hand, which is going to be the direction of the force. So, fingers will be B, the thumb will be V, again, fingers are B because there's multiple lines. So, there's multiple fingers, thumbs single line and then what's left is the palm, which is going to be the magnetic force, and one of the ways to remember this, is that you may want to hit someone, right? Hit someone with the palm of your hand, right? Hit something, right? With the palm of your hand. So, B, V, right here, and then the force, right? Slap something, so the palm of your hand. Now, this works with your, right? Hand, I have two versions of the right hand here, one, this is with the palm looking towards you, okay? So, I want you to do this, right? I want you to look at your palm, I'm going to go real slow here, because I wanna make sure you nail this, you're looking at your palm. Notice that your thumb is to the right, right? Again, I have to sort of do this with me, look at your palm pull it out and then notice that your thumb is to the right just like in this picture, so the direction of the force will be towards you in this case, okay? Because your palm is pointing towards you. So, would this be an x or a dot, and I hope you're thinking that this would be a dot, okay? And in here, this is the back of your hand, this is the back of your hand. which again, here is the back of my hand, I'm looking at it, follow me, don't look at me, imitate me, right? So, do the same and then you're going to see the back of my hand, which means that the palm of your hand is actually going out that way, it's going away from you and away from you is x, okay? So, this is out of the plane, of the page or plane, and this is into the page, awesome, that's the right hand rule.

The last thing I want to say is that all of this crap works for positive charges. So, if you have negative charges everything still behaves in the same exact way, except instead of using the right hand you're going to the left hand, okay? But everything stays the same, people have other different ways of doing this, I like just switching hands, positive hand negative hand okay, cool? Same rule, let's do an example and what I want to do is I'm going to two of these and I want you to do two of these and let's make sure that we can nail it, okay? So, the first one, we want to just find the direction of the magnetic force, which remember, is the direction of your palm, on a moving charge in each of the following situations, remember, charges have to be moving for you to have a force, force equals q, V, B, sine of theta, if you don't have a V you don't have an F, okay? So, in all these cases are moving. So, here we have a proton and then here an electron, what's the significance of this? Well, proton is positive, which means, we're going to use the right hand, okay? And an electron is obviously negative, which means, we are going to do the same thing but with the left hand, okay? So, the proton is moving left. So, let's just draw moving left means the direction of the velocity. So, moving left this way, and the B field is pointing up, I'm going to draw a few lines here, B field is pointing up, what is the direction of the force? So, you do, you now have to get your hand in the same, with the same setup that is described here, and to make this a little bit easier, I'm going to switch screens here and I'm going to do this over here, on paper, okay? Because you need to be able to do this on paper for your test. So, you're going to draw that you have a V this way and you have an FB this way and you want to find, I'm sorry, just a magnetic field B, and you want to find the direction of the force, which way is the force, okay? So, remember, multiple arrows means multiple fingers B, right? So, that's the direction of B right there, and V is to the left so this is really easy, when you do this you now have to look at the direction of the palm of your hand, the palm of my hand is going into the page, into the page and away from me, so the direction of FB is into the page, which into the page is, remember, the arrow going away. So, it's an x. So, FB is going in the x direction. So, if you want to, what you can do is you can put a little x here and say, that's the direction of FB, okay? And then this is our V right here, cool? So, this is A, this is part A, I'm going to do Part B and I want you to try C and D on your own. So, let's do Part B, Part B, we have an electron which means, we're going to use the left hand rule, which is just the right hand rule with your left hand. So, left hand ready, pull yours out, the electron is moving down. So, moving down means that, here is the electron, it's moving down, okay? And in a B field that points out of the page, out of the page, out of the page. So, if you look at the page and then you point out of the page you're pointing towards yourself, right? Again, I want you to do this, make sure you point out of the page that means that you're pointing towards yourself, okay? Out of the page, out of the page, this means that the arrows is coming at you, which means symbol is a dot, so the way you would represent is, you put lots of little dots everywhere, okay? Little dots everywhere and you'll see this is direction of my magnetic field because the field exists sort of everywhere, right? In lots of places. So, how do we do this? Well, let's get our left hand and position the left hand according to this. So, V is a single arrow. So, it's going to be much thumb so this is my V right there and if I do this I end up with something like this, okay? But my B is supposed to be into the page, right? Now, my B is pointing to the right That's not right. So, I have to make sure, I'm sorry, out of the page. So, I have to do something so that my fingers are going to point towards me while keeping this guy pointing down, I'm being really slow here, because I want you to totally get this, okay? If you can do this, I hope you see what happens, right? I want you to be imitating me at this point, the B's are coming towards my face because it's out of the page and the V is going down, when you do this your palm is now pointing to the right, okay? Your palm is pointing to the right, the only way to get rid of this is to do it a bunch of times. So, this means that the magnetic force will be to the right, okay? That's it. So, I want you to pause the video if you have to and do C and D, I'm going to keep rolling here, but I hope you pause the video, even if you already know this stuff, just do it real fast, make sure you got it, I'm going to use left hand rule here, because this is an electron and the electron is moving down, the electron is moving down, V, and the B field points left so this is my B lines, left hand, I have to make my left hand point left and notice how this gets kind of weird. Now, I got to do this and you can't see me but I'm sort of contorting here, if you try to do something like this you'd sort of struggle as well, right? So, something like this but notice that now my finger is up here, that's a problem, my V is down. So, what I got to do is sort of do this, weird contortion, and if you're doing this the way I'm doing you're all kinds of weird now, right? So, this is my B, this is my V but what you see happens is my palm is up my palm is up, it's coming out of the page, therefore, it's pointing towards me. So, I can say that the magnetic force is out of the page, which is given by a dot. So, if you want you could draw right here that the FB is out of the page, hopefully you got that right, let's do one more.

And then. Now, I have a proton. So, we're back at positive, right hand rule, positive, you try it yourself, okay? Hopefully, this is getting easy now, proton is moving into the page, so the velocity is into the page, if something is going into the page you see the back of the arrow right here, into the page, right? So, this is the direction of V and the magnetic field is pointing right. So, it looks like this, V, B and we want to know what is the direction of the force, which is the direction of your palm. So, let's do this. V should be into the field so it should point my thumb into the page but then I need, I need the, I need a magnetic field to be to the right, so this is really hard for you to see and I'm trying to contort here, but if you do this, okay, there you go, if you do this, hopefully you're seeing that the, my palm is pointing down on my page, okay? My palm is going this way, my palm is going this way, so this is the magnetic force, is going to down, okay? So, again, I have up, down, right, left, duh, but you also have into and out of the page, okay? So, that's it for this one, hopefully you got it, the visuals are a little bit tricky here, but you know, hopefully that was enough practice, let's keep going.

**Example:** Force on Charge Moving at an Angle

Hey guys. So, in this example, we are looking for the force on a charge, that's moving through a magnetic field in three different scenarios, let's check it out. So, we want the magnitude and direction of the magnetic force, so we want the magnetic force on a 3 Coulomb charge, so q equals plus 3. So, it's positive, so we're going to use the right hand rule for direction, we would use the left hand if it was negative and it's moving with this velocity here, v equals 4 and it has a 5 Tesla magnetic field, that's the strength 5 Tesla and it is that field is directed in the positive x axis, okay? So, that's the field right there, and we want to know what is this force, if the charge is initially moving in these three directions here. So, in all three cases B is going to the right, but the direction of the velocity is different, here the velocity is going up, here the velocity, because it says positive Y axis, here the velocity is going to the left because it's negative x axis and here, it makes 30 degrees with the y axis. Now, the positive y axis over here, this is a little bit ambiguous because you could make 30 degrees with the positive y over here, right? This guy is 30 degrees away from the positive y but this guy is also 30 degrees away. So, we'll talk about that when we get there. So the equation we're going to use is the only equation that makes sense, is the equation for force on a moving charge, right? Which is q, v, B sine of theta, I know q, v and B, weÕrehich is going to plug those, so the challenge here is just making sure we find the right angle, the correct angle. So, q is 3, v is 4, B is 5, those are given, theyÕre up here and the angle we should use is the angle between the two vectors, between v and B, is the angle we should use, v is up, B is to the right, v is directly up, B is directly to the right, so they're exactly perpendicular to each other, they make an angle of 90 degrees, so sine of 90 degrees, sine of 90 by the way is 1, so the answer is just 60 Newtons, okay? And, what about the directions? Well, we're going to use the right-hand rule. So, remember, my fingers represent multiple lines, so itÕs my B field, it's going to point up like this and, it's actually like this, right,? aAnd my velocity should go up. So, it's already up, so this is the direction I should be looking, for, most ofnotice that my palm is out, my palm is away from me and you got to do this yourself, looking at your page, right? If you put your hand in front of you and you see that your palm is away from you, it's going into the plane or into the page, okay?

So, the direction is into the page, okay? So, we wanted the magnitude, we got it, and we wanted the direction and that's the direction. What about here? FB is going to be the same thing, 3, 4, 5 times sine of theta, but here the angle between v and B is 180, right? They're anti parallel to each other and the sine of 180 is 0, that means that there is no force at all, okay? And, if there's no force then there's no direction for you to worry about, right? Now, how can you remember this? One way to remember this, is if you look at your right hand rule, this should serve as a reminder, that B and v are supposed to be at 90 degrees and what I mean by are Bby supposed to be, is this is the scenario in which you get maximum force, okay? If your vB moves a little bit, now you have less force, you've lessft the maximum but youwe still got some force and if you go all the way, right? As you do this you're decreasing the magnetic force all the way to here, and when you get here, which is parallel, to 0 degrees, right? Parallel, now you have 0 force. Same thing, if you go all the way over here and you are at 180, I can't really do that, that hurts, then that's going to be 0 force as well, cool? Maximum force, little less force, 0 force, cool. So, let's jump into this one here. Here, I would talk about how there's two directions, because it's not clear, it's ambiguous, but actually doesn't matter because the magnitude would be the same, okay? The magnitude would be the same. So, if you want, you could have calculated the two different angles, right? This, the distance, the angular distance between this red arrow and this blue arrow here is 60. Remember, you don't necessarily use the angle thatÕs given to you, you use the angle depending on the definition, definition of the angle should be the angle between v and B. So, got to be very careful whenever you see an angle, okay? So, that's one angle you could have used, letÕs call that theta 1 or you could have used the angle all the way to this blue arrow here and that theta 2 it's 90 degrees right here, plus the 30. So, 90 plus 30 is 120. You could have used either one of these guys and you would have gotten the same answer because sine of, IÕm going to use 60, right? Sine of 60 and you can plug this in to double check equals sine of 120, right? So, it's the same thing, you get the same answer no matter what, and that answer is 52 Newtons. Remember, I told you that if you're slightly at an angle, you're going to get less than maximum, this is the maximum force you can get, for this arrangement, this is a little less than maximum, right? And this here is the minimum which is just 0, okay? Now, what about the direction? Well, B is this way and you can have your charge even move, either moving that way or this way, right? Either way, for both situations my palm is away from me. So, if you look at your page and your palm is away from you, which means you're looking at the back of your hand, that means that the force is going into the page as well, okay? So, the direction here is into the page for both of those situations, cool? Let me get out of the way, it's into the page, cool? ThatÕs it for this one guys, let«s keep going.

**Problem:** An electron is moving in a straight line (red line below) when it enters the horizontal 0.2 T magnetic field (blue lines). The angle shown below is 37°. If the electron experiences a 10^{-12} N force upon entering the field, how fast must it be moving?

At a given instant in time, an electron and a proton are moving directly toward each other at the same speed in a uniform magnetic field that is oriented 90° from the velocities of the particles. They experience magnetic forces which are:

A. equal in magnitude but perpendicular to each other.

B. identical

C. equal in magnitude but opposite in direction.

D. in opposite directions and differing in magnitude by a factor of 1840.

E. in the same direction and differing in magnitude by a factor of 1840.

Watch Solution

A +2 μC charge is at rest in a magnetic field of 2 T pointing along the +x-axis. What is the force acting on this charge in the magnetic field?

A) -4 μN

B) +4 μN

C) +2 μN

D) 0

Watch Solution

A particle having a mass of 0.195 g carries a charge of -2.50x10 ^{-8} C. The particle is given an initial horizontal northward velocity of 4.00x10 ^{4} m/s. What are the magnitude and direction of the minimum magnetic field that will balance the Earth's gravitational pull on the particle? __Hint:__ The velocity and the gravitational force are perpendicular.

Watch Solution

A 2.5 *μC* charge with a mass of 250 *μg* enters a uniform, 1.5 T magnetic field. If the charge moves in the +x-direction at 150 m/s and the magnetic field points upwards, answer the following questions:

a. What is the magnitude of the force on the charge?

b. What is the acceleration of the charge?

c. After 10 ms, what is the speed of the charge?

Watch Solution

A -5 *nC* charges moves with a velocity *v = * (10 m/s) î - (20 m/s) ĵ in the presence of a uniform magnetic field *B *= (-1.5 *mT*) î + (0.5 *mT*) ĵ. What is the magnetic force acting on the charge, both magnitude and direction?

Watch Solution

A positive charge is moving to the right and experiences a vertical (upward) magnetic force, in which direction is the magnetic field?

A) to the right

B) upward

C) to the left

D) into the page

E) out of the page

Watch Solution

A particle with negative charge is traveling to the left with speed v. The particle passes through a region between two horizontal plates that carry equal and opposite charges, as shown in the sketch. Gravity can be neglected. The direction of the minimum magnetic field in the region between the plates that allows the particle to pass through the region undeflected is

A) into the page

B) out of the page

C) to the left

D) to the right

E) toward the top of the page

F) toward the bottom of the page

Watch Solution

A particle with negative charge *q* is moving to the right and enters a region where the magnetic field is uniform and directed into the page. If the particle moves through the region with constant velocity, the electric field in the region has direction

(a) into the page

(b) out of the page

(c) to the left

(d) to the right

(e) toward the top of the page

(f) toward the bottom of the page

Watch Solution

A uniform magnetic field of magnitude 0.80 T in the negative z-direction is present in a region of space. A uniform electric field is also present. In Figure 1, the electric field is set at 11,200 V/m in the positive y-direction. An electron is projected with an initial velocity v_{o} = 1.4 x 10^{4} m/s in the positive x-direction. The y-component of the initial force on the electron is closest to:

A) +2 x 10^{-15} N

B) -2 x 10^{-15} N

C) +4 x 10^{-15} N

D) zero

E) -4 x 10^{-15} N

Watch Solution