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The Wien (E x B) Filter
Developed by E. Behringer
This set of exercises guides the students to compute and analyze
the behavior of a charged particle in a spatial region with mutually perpendicular electric and
magnetic fields. It requires the student to determine the Cartesian components of hte forces
acting on the particle and to obtain the corresponding equations of motion. The solutions to
these equations are obtained through numerical integation, and the capstone exercise is the
⃗⃗ × 𝐵
⃗⃗ (Wien) filter.
simulation of the 𝐸
Exercises
Exercise 1: Forces acting on a charged particle and the equations of motion
Imagine that we have a particle of mass 𝑚, charge 𝑞, and velocity 𝑣
⃗ = (𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 ) with 𝑣𝑧 >>
^.
⃗ = 𝐵𝑥 ^
⃗ = 𝐸𝑦 𝑦
𝑣𝑥 , 𝑣𝑦 entering a region of uniform magnetic field ⃗𝐵
𝑥 and uniform electric field ⃗𝐸
As shown below, 𝐵𝑥 > 0 and 𝐸𝑦 < 0. The length of the field region along the 𝑧-axis is 𝐿.
Alt Figure
Neglecting any other forces (e.g., gravitational forces), show that the Cartesian components of
the combined electric and magnetic forces are
𝐹𝑥 = 0
𝐹𝑦 = 𝑞𝐸𝑦 + 𝑞𝑣𝑧 𝐵𝑥
𝐹𝑧 = −𝑞𝑣𝑦 𝐵𝑥
resulting in the equations of motion
𝑥̈ = 0
$$ \ddot{y} = {{q}\over{m}}\Bigl(E_y + v_zB_x\Bigr) $$
$$ \ddot{z} = - {{q}\over{m}}v_yB_x $$
where the dot accents indicate differentiation with respect to time. Note that the particle will
not experience any transverse acceleration if 𝑣𝑧 = 𝑣𝑝𝑎𝑠𝑠 = −𝐸𝑦 /𝐵𝑥 .
(a) Assume that 𝐸𝑦 = −105 V/m and 𝐵𝑥 = 2.00 × 10−3 T, and that all other field components
are zero. Calculate, by hand, the Cartesian components of the acceleration at the instant
when a Li + ion of mass 7 amu and kinetic energy 100 eV enters the field region
^ = (𝑥
^+𝑦
^ + 100 ^
traveling along the direction 𝑢
𝑧 )/√10002.
How will these acceleration components compare to those for a doubly ionized nitrogen ion
(N + )?
(b) Write a code to perform the calculation in part (a). Note that, as soon as the particle
enters the field region, the velocity components will change, and therefore so will the
forces. To calculate an accurate trajectory, it is necessary to repeatedly calculate the
forces, a task for which the computer is very well suited.
(c) What do you expect the trajectory of this ion to look like as it traverses the field region?
Explain your answer.
Exercise 2: Computing the Trajectory
Solve the equations of motion to obtain the trajectory of the Li
traverses the field region from 𝑧 = 0 to 𝑧 = 𝐿 = 0.25 m.
+
ion from Exercise 1 while it
(a) On separate graphs, plot 𝑥, 𝑦, and 𝑧 versus time.
(b) Plot the trajectory in space. What does the trajectory of the ion look like? What did you
expect (Exercise 1)? What happens if you reduce the initial kinetic energy of the ion by a
factor of 100? A factor of 10,000?
(c) What is the kinetic energy of the ion at the end of its trajectory? How does it compare to
its initial energy?
⃗ × ⃗𝐵
⃗ (Wien) Filter, Part 1
Exercise 3: The ⃗𝐸
Regions of mutually perpendicular electric and magnetic fields can be used to filter a collection
of moving charged particles according to their velocity. If we assume that a particle of velocity
𝑣𝑝𝑎𝑠𝑠 = −𝐸𝑦 /𝐵𝑥 enters the field region traveling exactly along the 𝑧-axis, the particle will
experience zero net force and therefore zero acceleration and zero deflection from the 𝑧-axis.
If a small, circular aperture of radius 𝑅 is placed on the 𝑧-axis at 𝑧 = 𝐿, then this particle will be
transmitted through the aperture.
(a) Use your program from Exercise 2 to determine the maximum value of 𝑣𝑧,𝑚𝑎𝑥 = 𝑣𝑝𝑎𝑠𝑠 +
𝛥𝑣 for which an aperture of radius 𝑅 = 1.0 mm will transmit the Li + ion (now assuming
that 𝑣𝑥 = 𝑣𝑦 = 0). What is the value of 𝛥𝑣/𝑣𝑝𝑎𝑠𝑠 ?
(b) Repeat (a) for an aperture of radius 𝑅 = 2.0 mm. What is the value of 𝛥𝑣/𝑣𝑝𝑎𝑠𝑠 ?
⃗ × ⃗𝐵
⃗ (Wien) Filter, Part 2
Exercise 4: The ⃗𝐸
As an extension of Exercise 3, now assume that the particles entering the field region at the
origin have a normal distribution of velocities directed purely along the 𝑧-axis. The center of
the distribution is 𝑣𝑧,𝑝𝑎𝑠𝑠 and its width is 0.1𝑣𝑧,𝑝𝑎𝑠𝑠 .
(a) Allow 40,000 particles from this distribution to enter the field region at the origin. What
is the resulting histogram of the scaled velocities 𝑣𝑧 /𝑣𝑧,𝑝𝑎𝑠𝑠 of the particles transmitted
through a circular aperture of radius 𝑅 = 1.0 mm centered on the 𝑧-axis? How does it
compare to the histogram of the initial velocities?
(b) Repeat part (a) for an aperture of radius 𝑅 = 0.5 mm.
It is worth noting that an actual source of ions will not only be characterized by a distribution
of velocities, but also distribution of directions (no ion beam is strictly mono-directional, just
like a laser beam is not strictly mono-directional). This is an additional fact that would have to
be considered to accurately simulate the performance of a real Wien filter.