Charged Particle in Crossed E and B Fields 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 vo (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 (b) Prove that there is a unique value of vgo called the drift speed 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 vo Hint: The equations for (va, Uy) should look very like what we derived in class except for an offset of v by a constant. If you make a change of variables of the form Un 0. 2 vdr, for which the Vdr and uy = vy, the equations for (ug, uy) will have exactly the Vx form of what we derived in class, whose general solution you know. (d) Integrate the velocity to find the position as a function oft and sketch the trajectory.
Charged Particle in Crossed E and B Fields 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 vo (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 (b) Prove that there is a unique value of vgo called the drift speed 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 vo Hint: The equations for (va, Uy) should look very like what we derived in class except for an offset of v by a constant. If you make a change of variables of the form Un 0. 2 vdr, for which the Vdr and uy = vy, the equations for (ug, uy) will have exactly the Vx form of what we derived in class, whose general solution you know. (d) Integrate the velocity to find the position as a function oft and sketch the trajectory.
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Transcribed Image Text:Charged Particle in Crossed E and B Fields 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 vo (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
(b) Prove that there is a unique value of vgo called the drift speed
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 vo Hint: The equations for (va, Uy) should look very like what we
derived in class except for an offset of v by a constant. If you make a change of variables
of the form Un
0.
2
vdr, for which the
Vdr and uy = vy, the equations for (ug, uy) will have exactly the
Vx
form of what we derived in class, whose general solution you know.
(d) Integrate the velocity to find the position as a function oft and sketch the trajectory.
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