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lovelymusiclady
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Hi all! I'm super lost on this homework question. I tried asking the professor but was kind of brushed to the side. My vector calculus knowledge is pretty limited (I had an unfortunately experience in that class). Anybody have any ideas on how to go about solving for this?
It's a problem out of Classical Mechanics-by Taylor, Chapter 2 Section 52.55 ***
A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields,
E pointing in the y direction and B in the z direction (an arrangement called "crossed E and B
fields"). Suppose the particle is initially at the origin and is given a kick at time t= 0 along the x axis
with Vx = Vxo (positive or negative).
(a) Write down the equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane z =0.
(b) Prove that there is a unique value of Vxo, called the drift speed Vdr, for which the particle moves undeflected through the fields. (This is the basis of velocity selectors, which select particles traveling at one chosen speed from a beam with many different speeds.)
(c) Solve the equations of motion to give the particle's velocity as a function of t, for arbitrary values of
Vx0. [Hint: The equations for (Vx, Vy) should look very like Equations (2.68) except for an offset of Vx
by a constant. If you make a change of variables of the form Ux = Vx —Vdr
and Uy=Vy, the equations for (Ux, Uy) will have exactly the form (2.68), whose general solution you know.]
(attached is equation 2.68)
(d) Integrate the velocity to find the position as a function of t and sketch the trajectory for various values of Vxo
It's a problem out of Classical Mechanics-by Taylor, Chapter 2 Section 52.55 ***
A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields,
E pointing in the y direction and B in the z direction (an arrangement called "crossed E and B
fields"). Suppose the particle is initially at the origin and is given a kick at time t= 0 along the x axis
with Vx = Vxo (positive or negative).
(a) Write down the equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane z =0.
(b) Prove that there is a unique value of Vxo, called the drift speed Vdr, for which the particle moves undeflected through the fields. (This is the basis of velocity selectors, which select particles traveling at one chosen speed from a beam with many different speeds.)
(c) Solve the equations of motion to give the particle's velocity as a function of t, for arbitrary values of
Vx0. [Hint: The equations for (Vx, Vy) should look very like Equations (2.68) except for an offset of Vx
by a constant. If you make a change of variables of the form Ux = Vx —Vdr
and Uy=Vy, the equations for (Ux, Uy) will have exactly the form (2.68), whose general solution you know.]
(attached is equation 2.68)
(d) Integrate the velocity to find the position as a function of t and sketch the trajectory for various values of Vxo
