1. A point charge q with mass m is released from rest at the origin in...

 

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1. A point charge q with mass m is released from rest at the origin in an external magnetic field and an external electric field. Both of the external fields are uniform but the electric field points in the z-direction and the magnetic field points in the r-direction: = Eok B = Bo where Eo and Bo are constants. The electric field accelerates the point charge along the z-axis, but once the point charge has acquired a non-zero velocity along the 2-direction it will also experience a magnetic force. a) Using Newton's second law and the Lorentz force, show that the acceleration of the point charge in each direction must satisfy the following: dr dt where w is the cyclotron frequency d'y dt dz dt dz dt Eo dy dt qBo TIL and r(t), y(t), and z(t) give the position of the point charge along each axis as functions of time. b) The above equations are called coupled differential equations because they mix derivatives of y with derivatives of z (and vice versa). Show that these equations can be de-coupled by taking another time derivative. By de-coupled, I mean you should have an equation containing only derivatives of y with respect to t and an equation containing only derivatives of z with respect to t. c) The de-coupled equations are easier to solve, but let's not go through the actual steps. Instead I will just give you the general solution to the equations given in part a): x = Ct+C y=C3 cos wt + Ca sin wt + + C5 Ent Bo z = C4 cos wt - C3 sin wt + C6 where C, C2, C3, C4, C5, and Co are arbitrary constants. Show that these functions solve the equations from part a). You do NOT need to derive them, just plug them directly into the equations from part a). d) Recall that at t=0 the point charge is at rest and located at (x, y, z) = (0,0,0). Use these initial conditions to determine the values of the arbitrary constants. e) (Optional) Now try to sketch the motion of the charged particle. This can be tricky. It is helpful to take your expressions for y(t) and z(t) and notice that you can manipulate them into the equation for a circle, but one in which the center of the circle moves along the y-axis with constant speed. This is called cycloid motion and is identical to a wheel rolling at constant velocity. 1. A point charge q with mass m is released from rest at the origin in an external magnetic field and an external electric field. Both of the external fields are uniform but the electric field points in the z-direction and the magnetic field points in the r-direction: = Eok B = Bo where Eo and Bo are constants. The electric field accelerates the point charge along the z-axis, but once the point charge has acquired a non-zero velocity along the 2-direction it will also experience a magnetic force. a) Using Newton's second law and the Lorentz force, show that the acceleration of the point charge in each direction must satisfy the following: dr dt where w is the cyclotron frequency d'y dt dz dt dz dt Eo dy dt qBo TIL and r(t), y(t), and z(t) give the position of the point charge along each axis as functions of time. b) The above equations are called coupled differential equations because they mix derivatives of y with derivatives of z (and vice versa). Show that these equations can be de-coupled by taking another time derivative. By de-coupled, I mean you should have an equation containing only derivatives of y with respect to t and an equation containing only derivatives of z with respect to t. c) The de-coupled equations are easier to solve, but let's not go through the actual steps. Instead I will just give you the general solution to the equations given in part a): x = Ct+C y=C3 cos wt + Ca sin wt + + C5 Ent Bo z = C4 cos wt - C3 sin wt + C6 where C, C2, C3, C4, C5, and Co are arbitrary constants. Show that these functions solve the equations from part a). You do NOT need to derive them, just plug them directly into the equations from part a). d) Recall that at t=0 the point charge is at rest and located at (x, y, z) = (0,0,0). Use these initial conditions to determine the values of the arbitrary constants. e) (Optional) Now try to sketch the motion of the charged particle. This can be tricky. It is helpful to take your expressions for y(t) and z(t) and notice that you can manipulate them into the equation for a circle, but one in which the center of the circle moves along the y-axis with constant speed. This is called cycloid motion and is identical to a wheel rolling at constant velocity.

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a Using Newtons second law and the Lorentz force we can write the equations of motion for the point charge in each direction In the xdirection m dxdt ... View the full answer