a. Starting from rest, how long does it take an electron to move ...

Question

a. Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

Accepted Answer

Welcome back. Everyone. In this problem, we want to calculate the time taken by a hydrogen ion that is initially at rest in a uniform electric field of magnitude E equals 1.25 kg newtons per kilo to cover millimeters. For our answer choices. A is 1.44 microseconds. B is 8.35 microseconds, C is 2.9 microseconds and D is 0.914 microseconds. Now let's draw a sketch of what's going on here to better understand what we're seeing. So we have a hydrogen ion that's in an electric field. OK. And let's say that we'll treat our hydrogen ion, which is a proton as an electric charge. OK. So this is our electric field, right? That's acting on the ion. And if it's in the electric field, that means it will experience a constant electric force and accelerate uniformly because we want to look at the ion from its starting point and in red to when it moves by millimeters. So we can imagine that our ion is going to end up here. OK. Now, what else do we know? Well, let's assume the hydrogen ion moves in the positive X direction. OK. Let's assume it's moving this way where we'll call that force the force in the x direction of the ion. And we will also assume that the electric field points in the positive X direction. So it has a force of the electric field in the X direction. OK. Now, what do we know we know that the magnitude of our electric field? So let's make a note of that. We know that the magnitude of our electric field equals 1.25 kg newtons per kilo. So in other words, 1250 newtons per kilo, we also know that the distance it's supposed to cover the in the X direction is 50 millimeters. So in si units, that would be 50 times 10 to the negative 3 m, we also know that it's initially at rest. So its initial velocity is 0 m per second. And we know that we are trying to find the time, yeah, the time taken by our hydrogen ion to cover 50 millimeters. So the time is what we don't know, is there anything else that we're missing here? Well, we know that we're talking about, we're assuming that the, we, we're treating the hydrogen ion as a point charge. So we also know that its charge Q equals 1.6 times 10 to the negative 19 Coomes as a constant. And we also know that since it's a hydrogen ion that we're treating as a proton, the mass of a proton equals 1.67 times 10 to the negative 27 kg. Now, since we're talking about the movement of a charge, then we can think about our kinematic equation. We need a kinematic equation that relates all of these variables. Remember we're looking for at the time and we already have as it relates to the kinematic movement, we already have the distance that it covers and the initial velocity velocity. So let's think what kinematic equation relates to those three variables. Well, remember that we can use the equation that says a change in distance. OK. The change in distance equals the initial velocity multiplied by the time plus half A T squared. And that acceleration would be in the X direction. The good thing is we already know that it's moving from rest. So we know our initial velocity is zero and we know the distance that it travels. So when, when our velocity is zero, when our initial velocity equals zero, then our change in the X direction equals zero multiplied by T which is zero, half A T squared, that's acceleration in the X direction. If we want to solve for our time, then this tells us that if we multiply both sides by two, then our time squared will be equal to two multiplied by the change in X divided by the acceleration in the X direction. So the time would be equal to the square root of two multiplied by the change in X divided by the acceleration. But what's the problem here? We don't know the acceleration in the X direction. So we need to figure that out. So let's go back to what we know. OK. We know from our list of variables, we also know the mass of the hydrogen ion. OK. And when we think about it, what relates mass and acceleration? Well, again, recall that force equals mass multiplied by acceleration. So the force in the X direction will be equal to the mass multiplied by the acceleration in the X direction. And that means the acceleration will be equal to the force divided by the mass. Again, we have the mass, but we don't have the force. So we'll have to dig deeper once more. Let's dig again. What do we know about the force that's being exerted on our, our, our, our, our force that the hydrogen ion is exerting rather? Well, we have the charge. And we also have, let me make a note of that. We have the charge and we have the magnitude of the electric field. So we can dig a little deeper. Again, to recall that the magnitude of an electric field equals the force divided by the charge. So that tells us then that the force in the x direction would be equal to the charge multiplied by the magnitude of the electric field. Now, that's pretty important because we can use this idea for the force to find the acceleration in the X direction. So therefore, acceleration would be equal to the force which is the charge multiplied by the magnitude of the electric field divided by the mass. And we know that we have the charge, we have the electric field and we have the mass. So that means we can substitute that into our equation for T. So T is now going to be the square root of two multiplied by the change in X divided by the acceleration in X which is Q multiplied by the magnitude E divided by M. No. Do we have all of these variables? Actually, we do. So we can substitute therefore to find the value for the time. So what does that mean? Well, I can rewrite my equation as two multiplied by the change in X multiplied by the mass. And all of that will be divided by Q multiplied by the magnitude of the electric field. Now, by substituting the values the change in X, remember we had said that it is, let me write it over here. We had said that the change in X is 50 multiplied by 10 to the negative 3 m. OK. We know that the mass of our ion of our hydrogen ion is 1.67 multiplied by 10 to the negative 27 kg. And we are dividing that by the charge multiplied by the electric field. The charge we are, we know that is 1.6 times 10 to the negative 19 Coomes and the electric field, the magnitude of the electric field is 1250 newtons per Coolum. So that is what is going to be equal to the time it takes. And when we work that out, if you were to substitute these values into your calculator, you will find that the time would be equal to 9.14 times 10 to the negative seven seconds. Now, since that's the, the, the value for our time, we can rent it in si units for our problem. So this would be equal to 0. multiplied by 10 to the negative six seconds, which gives us 0. microseconds. Therefore, that's the time it would take for the charge to move the distance of millimeters. When we look back on our answer choices, that would have been answer choice. D Thanks a lot for watching everyone. I hope this video helped.

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a. Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

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