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The figure shows a graph of electric potential versus position along the $x$ -axis. An electron is originally at point $A$, moving in the positive $x$ -direction. H...

Question

The figure shows a graph of electric potential versus position along the $x$ -axis. An electron is originally at point $A$, moving in the positive $x$ -direction. How much kinetic energy does it need to have at point $A$ in order to be able to reach point $E$ (with no forces acting on the electron other than those due to the indicated potential)? Points $B, C$, and $D$ have to be passed on the way.

The figure shows a graph of electric potential versus position along the $x$ -axis. An electron is originally at point $A$, moving in the positive $x$ -direction. How much kinetic energy does it need to have at point $A$ in order to be able to reach point $E$ (with no forces acting on the electron other than those due to the indicated potential)? Points $B, C$, and $D$ have to be passed on the way.



Answers

The figure shows a graph of electric potential versus position along the $x$ -axis. An electron is originally at point $A$, moving in the positive $x$ -direction. How much kinetic energy does it need to have at point $A$ in order to be able to reach point $E$ (with no forces acting on the electron other than those due to the indicated potential)? Points $B, C$, and $D$ have to be passed on the way.

Hi there. So for this problem we need to calculate the magnetic and direction of the magnetic field that this election produces at some given points. Each of them are are the distance are of 2 μ And the electron has the velocity of 0.1 times the speed of light. So the first question is to calculate the magnetic field at a point A. And be first we can see from the figure the point B is a sum given angle from the electron. Um To obtain them and get a feel for all of these points. We are going to use the magnetic field produced um by a moving charge. And that formula or equation is following where the angle tita is that the angle between the the back tour of the velocity with the vector of the distance to the point where we want to measure for the electric field. So the magnitude can be obtained from this expression and the direction and can be obtained from in the right hand rule. So first we're going to attain the magnitude in this case for a we're gonna call this for a yes music. And the uh permeability of frizz Hayes Over four. The charge in this case is the same for everyone and we are trading with electron and we can see that's the back tour are from okay the electorate the point. Okay. Form an angle of 30 degrees. Because If this angle it sets the degrees and we know that um the angle that formed the to access these 90° which is um The threat from 90°, sets the degrees and uh and that give us The 3rd degrees that we want. Um And everything else is just what we are given for these problems. So we are going to substitute all of those values. We know that they um permeability of free space. It's four pi times 10 to the minus second. Newton's need transparent 2nd Square or Column Square. All of these over for pie. And The charge is it corresponds to the charge of an electron which is 1.6 times 10 to the -19. And the velocity of this tron is given a 0.1. The speed of light. We also know that the speed of light is three times 10 to the eight m/s. So we put that in here. zero Stand to the eight m/s. And the angle as we said is 30°. And the distance to that point. And To me, Crow m, which is two times 10 to the -6 m. With all these, we obtain a value off. I'll remember that distance is to the square and this give us Six times 10 to the -8 Tesla. That's the units of the electric field. So that's for this point. And the direction could be attained by their right hand side. And that is given by the product between the back tour the velocity back tour and the our back door. So the right hand sign is that you put your run high in sand and right hand hand in the direction of the be back tour and you go from there too. She is them are better and you will see that your tamp is pointing inward so it is inward. The we put like another assets. It is going to point in this direction in work and if we said that as we said this as positive and this is negative, this positive in this negative then it will be negative and direction in work or negative if you said it in that way now for part being well for the same part but to obtain the magnitude of the electric field in for the point being we will have that the angle between the the better be and yeah, angle between the distance. The bet were The position that we're off. The point being it's we need to some all of this into some 90° every another colour. You need to some these 9° and also test 60°. So we will have 90, 90 plus assisted degrees. That's 150°. So with that in mind we just simply need to put all of those values in here. So we have the permeability of risk pace over for pine. Um, everything else is just the same as before. The velocity, the speed of the lecture and the same. But what changes is the angle between these two is 150°. and 50 degrees. And the distance is also the same because it is a state in the problem that all these points are at a distance of two times 10 to the -6 m square of course. And if you introduce all this information into the calculator, we obtain a value off Again six times 10 to the -8 Hessler. Now the direction again, we put our right hands our right hand in the direction of being and we go from there to the better art and we will see that again that direction is inward. So we will have something like a back tour in work. So as I said before, if we said that as the negative, you will have a negative inward. Yeah, or in one of the page, as you wanted to put it now, part B of this problem is the bet the a magnetic field in the point is C. So this is more straightforward. The point C is in the X axis set. So it will form with Better be 90°. So we have that and we're gonna put everything just as before. Um Yes, so we will have the same. The only thing that changes is the uncle between them that we said it is 90° and we know that the sign of 90°. It's just one. So from here we obtain a value off 1.2 times 10 to the -7 Tesla and the direction. Again we put our hand and the velocity backdoor and go from there to the our bed board. And again it will give us a tour in work, no direction in worth. If we set as the negative we attain a negative bar. And finally parsi of this problem is the fact tour the magnitude of the magnetic field in a point D. So that we can see from the figure that we need to go to obtain the angle we need to go from the all the way down to this point. So we know that this is 90° and then again this is 90°. So the distance to that point is um it will be the same as before this saying All of this but only changes design and it will be the sign of 180 and we know that this it's zero. So by multiplying zero byte everything else is just zero. So this is the magnetic field at the point. The self. Because the magnitude of the magnetic field at this point is zero. We don't need to calculate the direction because The battery is just zero. So this is it for this problem. Thank you

So it's starting from rest, which means initial velocity zero. We want to accelerate electro. So a speed off zero point 98 seat by a difference in difference of putting show. So we want to increase the energy off electric in this case, the connecting energy because he has the speed by a difference in potential. So this is just the charge of the electron times the difference in potential. So this is the creation that we have to use. So first of all, in the first item, we want to calculate, I turn a we want to calculate what should be the difference in pretension. I can be. We want to calculate what should be the absolute value off the connecting energy off the electron. So first of all, we do not have any ways to calculate the difference in potential in first. So let's calculate the connecting energy because it's easier. We know by definition that the connective relativistic energy is just gamma M C square minus M c square, so we can see that the connecting Aaron and you relativistic energy is just m C square. There multiplies gamma minus one. So we just have to calculate gamma and then calculate M C Square and we'll be fine. So this is just let's see and see square. We know that it's this one one minus 0.98 See, divided by sea square in the square root so we can cross, see and hear minus one. So this is the connecting energy one miners, you don't buy 98 here we know the mass of the electron We know the speed of light. So this is just the connecting energy is just four point 0 25 m c. Square. So this is just equal two three point 95 2 95 actually 2 95 times 10 to the power of minors, 13 jowls or we can see in the electoral votes. This is two points, you know, six lev because to convert jowls to a lateral votes, we just have to multiply by six points two for two time stand to the power off 18. So this is value off the connecting energy off the electron. So now that we have the connecting energy of the electron, we can finally calculate the difference in potential that is necessary. So the difference of potential. It should be out of e e close the connecting energy divided by the charge of the electron. So this is simply 2.6 times 10 to the power off six votes. Just remember that the connecting energy that we years in is that three point 2 95 time Stan, our miners 13. So as we can see the value off, the difference in potential is just the energy in electoral votes is closed to damage of electoral votes because here we have Mager, he would have Maga. And here we have 10 to the power of six. So it's pretty goes, but it's not equal. So this is the final answer to the problem. The first answer, the answer to the value of the connecting energy is 3.2 95 Time stand to the power of minors 13 and the value of the difference in potential is this one. And here this is a different potential. And that's all. Thanks for watching

In this problem. We have a electron moving at a constant velocity and were asked to find the magnetic field at four different points a through D on to find the Cartesian coordinate system with ex horizontal. Why barnacle and see out of the page were given at a velocity of 10% of the speed of light to three significant figures as well as each position would like to find us two microns away from the electron. So to find this, we use the abuse of our law for a constant a particle with constant velocity which is be of our it's equal to charge of the moving particle times the permitted ity constant, uh, multiplied by the magnitude of the velocity over four pi times, gift magnitude of the distance squared and this is all multiplied by the cross product between the direction of the velocity in the direction where we're trying to find the magnetic field. Now we know for every point in this problem, this whole quantity is a constant. We can call this be not for the velocity. It is for each point going to be pointing in the white hat direction and the distance to the point we're trying to find the magnetic field. We can write this as co sign data are ex hat direction plus sign. What's data are in the Why had direction? Now we know that. Why Hat Cross? Why Hat is going to be zero So we can on Lee the only concern ourselves with the X component of this position vector to find the magnetic field. And if we plug in the values for our at point A we get be of our a is equal to do you not times co sign 60 degrees times Why hat cross except and this evaluates to negative Z hat and co sign of 60 degrees is 1/2 so we can write this as negative be happy be knocked over to in the Z at direction And since our B has the same X component as our A, we know that this is also the same as the magnetic field at RB. Now for our C, The position on Lee has an ex Capone component so we can write this as negative. Be not see him. And for our d this on. Lee has a Y hat component and so we can also right that be r D is zero. So now we have to plug in all of the given values for the distance, the velocity, the permeability constant, a swell as the charge of the electron. With this, we get that the constant be not is equal to negative 1.20 times 10 to the minus seven Tesla on This is negative because the charge of the electron is negative and notice when we plug this into all of these values for B r, A, b, R B and B R C, it cancels with this negative sign and we get that all of these fields point in the positive Z at direction out of the screen on. We wrote this as with three significant figures because that is all the significant figures were given for the velocity in the distance.

In this problem, potential versus distance in X and Y direction is soon by the graph. And we have to find for Shannon electron kept in xy plane under this potential field. Okay, so before starting to solve this problem, let us first recall the concept that is a related to the potential and electric field, How potential is related to electric field. So we know that electric field E is equals two minus D P by D R. Okay, so basically rate of change of Putin's here is indicated by the electric field. And once we know the value of electric field, then we can find the force since force will be caused to charge into electric rate. So first we will use this relation minus Davey, where D. R. To get the value of electric field from the graphs. And then we will find what is the net electric field since their electric field exists in X and Y dimensions. And from the help of, with the help of net electric field we can calculate the net force acting on the electron. Okay, now, what does this physically signifies this derivative? So this derivative basically signifies the slope of the potential versus distance curve. Okay, so how we can write this term? So electric field in X direction will be equals two minus db by dx. Similarly electric field in Y direction will be closed to minus D V by divine. Okay, now, how to get the value of divided by So it is the slope D B, y d excess slope. So slope of V versus excursion. So here we can see that the slope is indicated by anti data. So let us say this angle is theta. Then how to get the value of 10 theatre. So we will use this right angled triangle in this right angled jungle the basis of 0.4 unit. Okay, now what is the value of this height? So this height is minus or two times V. S. And in the problem V. S. Is given as 500 world therefore. And the slope will be minus. What is the slope? So minus Two years. So instead of years we can write 500 world divided by 0.4 m. Okay, so it comes out to be minus and the minus will be plus. So 1000 divided by 0.4 which will be 2500 world per meter. Okay, so this is the value of electric field in X direction. Similarly, we can do the similar process for the white direction. So in white direction, this will be the key to So we can use the right angled jungle. Here, we can use this right angled jungle. So as we can see here that each unit is 0.1 m. So this is 0.10 point 20.30 point four. And this one will be 0.5. And this 0.5 will the potential corresponding to this distance is V. S. Weep and V. S. S 500 gold. So what we can right here, That electric field in white direction will be close to minus 500 Divided by distance is 0.5 m. So it will be involved per meter unit. So electric field in white direction will be equals 2 -1000 world per meter. Okay, now we know the value of electric field in X direction and we know the value of electric field in my direction as both these fields are perpendicular to each other. Therefore, net electric field will be caused to Squire route over electric field in X direction plus a Squire of electric fielding. Why direction? So it'll because two square root over 2500 fully square Plus -1000 Holy Square. So it comes out to be 20 692.58 World Per meter. Okay, world permit er can also be written as Newton per column. So it will be 2692.58. Newton per column. Since both of these units are equal world permit er is equals to Newton per column. Now we know the value of net electric field. Then we can find the value of net force. So net force F will be equals two Q into E. Now, what is the charge or as the electron escaped. Therefore, charge on on an electronic 1.6 into 10 days. To the power -19. Coolum And it will be multiplied with electric field which is 2692.58 Newton per column. So columns get cancelled here. So the electric force left comes out to be 4.3 into 10 days, to the power -16 Newton. So this is the force that electron will experience under the given field. So this is the answer of the problem.


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