 In this video I want to compare the magnetic force acting on a charge with the electric force that would act on a charge. And as we do we will see there is so much information packed in these tiny equations and there are such huge differences between the two forces. So let's begin. So to begin with if we just look at the equations we find the main difference is over here. The magnetic force, the Lorentz force has V cross in it. And boy that itself causes a huge difference in way the two forces behave. So we'll first compare their strengths and then we'll compare their directions. Okay so let's start with the strength of the magnetic force. And for that let's assume that let's take an example. Okay let's say that we have a uniform magnetic field and in that magnetic field let's say we have four protons. One two three four and let's say this first proton is moving this way with some speed. Second proton is moving this way with the same speed. Third one is moving this way with the same speed. And the fourth one is not moving at all. And what I want you to think about just by looking at this equation is what will be the strength of the magnetic force on each one of them? Will it be the same? Will it be different? Why can you pause and think about it? Alright to figure out how strong the force is we have to look at the magnitude of this equation. So if I just look at the magnitude of this force it's going to be magnitude of this thing. Will that be? The magnitude of that is just going to be Q into magnitude of V cross B and the magnitude of V cross B will be magnitude of the velocity times the magnetic magnitude of the magnetic field times sine of the angle between the two. So that matters. So the angle between the velocity and the magnetic field matters. So let's look at each one of them. If I look over here notice the angle between the two is 90 degrees. So over here sine 90 is going to be one and that's maximum which means over here this proton is going to actually experience the maximum magnetic force. What about this one? Well it is the same charge, same speed, same magnetic field but look at the angle. The angle is smaller over here and therefore sine theta is going to be smaller than one and therefore this force will be less than before. So it will be less. What about this one? Well over here they're in the same direction which means theta is zero. If sine theta is zero the magnetic force goes to zero which means this charge is not going to experience any force. No force, zero. What about this charge? Well it's not moving at all which means there is no velocity therefore this also does not experience any force. What we see right in front of our eyes is that the magnetic force highly depends on the direction of the velocity. It's important that it should be moving but more importantly it should be moving making some angle with the magnetic field. So let's write that down. This is, this looks important. So what we find is that the magnetic force is maximum when, when the two are perpendicular, when velocity is perpendicular to the magnetic field. And we find that the force is a minimum which is zero when either the velocity is zero or when they are in the same direction. So when velocity is parallel to the magnetic field or when velocity is zero. Okay now let's look at the electric field. Something we've seen before. The good old familiar electric field. What if I ask the same question? Same four protons moving with same, same four conditions. On moving this way, on moving this way with some speed, same speed. While moving this way with the same speed this not moving at all. What do you think will be the electric force on each one of them? Will it be same or different? Okay can you pause and think about it? For that we look at the magnitude of the electric force and the magnitude of electric force will be magnitude of q which is just q into magnitude of e which is just going to be e. And so what we find is that there is no velocity in this equation and therefore electric field says I don't care about the velocity. I don't care about the direction of the motion. It's going to put the same force. Which means they're all going to experience the same force. So right in front of your eyes the big difference. The first big difference is this depends on the velocity whereas the electric force does not, does not depend on the velocity. Does not. Alright now let's look at their directions. And this time I start with the electric field. So if I were to ask you what is the direction of the electric force here, here and here and here. What direction is it? Well for that we're just going to look at the vectors. It says the direction of the electric force will be in the same direction as the electric field. If the charge is positive it'll be in the opposite direction if the charge is negative. So over here all of them because they're protons will experience the force in this direction. In this direction. And so what we find is that the direction of the electric force will always be parallel. So it's in the same direction or in the opposite direction but always parallel to the electric field and that's nice. If I know the direction of the electric field that's all I need and I will know the direction of the electric force. Positive charge same direction. Negative charge in the opposite direction. Simple. What about the direction of the magnetic force? This is where the cross product comes in. The direction of the magnetic force is not so straightforward. The direction is going to be in the direction of the cross product of the velocity and the magnetic field. How do we do that? Well we've talked about it but let's quickly recap how do you do a cross product. So if I take say this example I'm going to redraw it over here somewhere. So here is my magnetic field and here is the proton moving upwards with some velocity let's say v. Now to figure out the direction of the magnetic force I have to cross from v to b. So for that I'm going to use my right hand and the way I'll position my right hand is going to be somewhat like this so that my palm is facing the velocity vector it's ready to cross towards b. Notice as it crosses the thumb points in the direction of the cross product so the thumb points in the direction of the magnetic force so in this case the magnetic force will be inwards. It will be into the screen this is the direction of the magnetic force and more importantly what this means is that look the direction of the magnetic force is not parallel to the field it's perpendicular to the field it's also perpendicular to the velocity. So over here what we find is that the magnetic force magnetic force will always be perpendicular is perpendicular to velocity and is perpendicular to the magnetic field. So again huge difference here very straightforward in the same direction over here you need to know both the direction of the magnetic field and the direction of the velocity to figure out what the direction of the force is going to be. But most importantly this part the fact that the magnetic force is perpendicular to the velocity has a huge consequence and to understand that again let me take an example let's say I had a very complicated non-uniform magnetic field I don't even know how it is so imagine very complex magnetic field okay and in that I throw a proton it could be any charged particle but let's say I throw a proton and as a result because it is so complex let's say it takes some very weird crooked path through it and comes out like this okay my question to you is do you think the speed of this particle do you think it can change as it moves through the magnetic field very specific question I want you to think about it pause and think about it. All right well my first thoughts would be of course why would it not change if we have like a very complex magnetic field and you know it's taking a very complex path its direction is changing there are forces acting on it so sure I mean clearly surely the speed might change right but guess what the speed of this particle will never change your magnetic field will never be able to change the speed of a particle regardless of how complex the field is and the question is why you may be wondering because the magnetic force is always perpendicular to velocity how does that make any sense well let me give you an example not an example let's think a little bit about it so if I were to ask you let's consider imagine you have some object that's moving this way okay my question is if you want to increase or decrease or change the speed of this particle what should you do well you should push it you should put a force on it what direction should you put a force on it so let's say you want to increase then you should push it in the same direction right you should push it in the same direction is there any other direction you can push it increase the speed yes you can push it this way and you can increase the speed because when you put a force this way there is a component of that force the force has an effect in that direction and that effect is going to accelerate the particle and increase the speed. So as long as your force has a component in the direction of the velocity, it can change the speed, it can increase it or decrease it. But what if your force is perpendicular to the velocity? Now it has no component in the direction of the velocity. As a result, this force cannot change the velocity. All it can do is change the direction, but it cannot change the speed. And since our magnetic force will always be perpendicular to the velocity according to our equation, that means our magnetic force can never change the speed of our particle. Another way to look at it is in terms of work and energy. If you think about the work done by a force, when the force is perpendicular to the displacement vector, that's zero. You've probably seen that in mechanics. So another way to think about it is that our magnetic force does no work and therefore it cannot change the kinetic energy. So a big consequence of this is no change in speed or kinetic energy. And this means regardless of how complicated, this will always hold true. This will always hold true regardless of whether you're dealing with the uniform or non-uniform magnetic fields. And therefore it doesn't matter how complex it is, the speed will never ever change. What a beautiful result, right? Okay, what about over here? Can this electric force, can the electric force change the speed of a particle? Again, I want you to pause and think about it. The answer is yes, because the electric force does not depend on the velocity direction. So say in this, at this moment in time, yes, it may not change the speed, but over here, definitely it can change the speed. Here it'll definitely change the speed. So electric force can change the speed of the particle. It can do work. It can do work. In fact, that's one of the reasons why we think about potential, electric potential, because it can do work and talk about that. But magnetic forces cannot do work. Huge difference. And so I hope you can appreciate how much information is packed in these little equations. And so you don't have to remember any of these. If you can just deeply understand the equations, everything, all these differences, all the nuances, follow immediately.