PHYS 102 – Lab 4 – Motion of charged particles in E and B Fields
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Motion of charged particles in E and B Fields
Measurement of e/m / Mass spectrometer
1. Objective
The objective of this laboratory is to explore design approaches and measure a fundamental physical
quantity - the charge-to-mass ratio of an electron.
2. Physical considerations of experimental design
Single mass analysis: Magnetic force on an electron and measurement of e/m
•
An electron beam is created in vacuum through thermionic emission in which electrons escape
from a heated filament (e.g., tungsten at ~2200ºC) that ejects thermally-excited electrons from the
metal.
Figure 1 - The electron gun consisting of a
heated cathode, protective metal grid, anode
and power source to the accelerating
voltage.
•
The filament is referred to as the cathode and an accelerating plate
with a slit to allow the electron beam to pass through is called the
anode which is at higher potential than the cathode. This is called
an electron gun. Emitted electrons are accelerated by an electric
field created by applying a potential difference, ΔV, between the
cathode and the anode. Other electrostatic elements may be
employed to focus the electron beam. The formula
!
𝐾 = " 𝑚𝑣 " = 𝑒(Δ𝑉)
•
Equation 1
relates the kinetic energy, K, and the electron velocity, v, to the potential difference, ΔV, where m
is the mass of the electron and e is the magnitude of the charge on the electron.
Thus
" $ %&
𝑣=*
•
'
+
!(
"
Equation 2
The electron beam enters a region with a nearly uniform magnetic field generated by Helmholtz
coils. The Helmholtz coil configuration consists of two circular coaxial coils each with N turns
separated by a distance equal to the radius of the coils. The two coils have the same number of
turns, carry the same current in the same direction. This arrangement produces an axial field that is
highly uniform in the region between the coils. The magnetic field can be expressed as:
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PHYS 102 – Lab 4 – Motion of charged particles in E and B Fields
𝐵) (𝑥) =
*! + , - "
"
/
!
"
./)0-("1 0- " 2
#$
"
+
!
"
#$
"
./)3-("1 0- " 2
1
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Equation 3
67
where 𝜇4 = 4 𝜋 × 1035 8 is the permeability of free space, N is the number of turns in each of
the pair of coils (R = 0.15 m; N = 130 in this apparatus), I is the current through the coils, and R is
the radius of each coil.
Figure 2 - Helmholtz coil geometry
At the center of the Helmholtz coils, x = 0 and Equation 3 reduces to
𝐵) (𝑥) =
*! + ,
-
9
*:+
;(
"
,
= (1.168 × 1039 T m/A) -
Equation 4
with I in A, R in m, and 𝐵) (𝑥) in T.
•
A⃗ according to:
A charge q moving with velocity 𝑣⃗ experiences a force 𝐹⃗ in a magnetic field 𝐵
A⃗
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
Equation 5
A⃗ is measured in units of Tesla (T).
where the magnetic field 𝐵
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PHYS 102 – Lab 4 – Motion of charged particles in E and B Fields
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The direction of 𝐹⃗ can be determined by first finding the
A⃗ using the right‐hand rule for vector products.
direction of 𝑣⃗ × 𝐵
A⃗ for a
The direction of 𝐹⃗ is the same as the direction of 𝑣⃗ × 𝐵
A⃗
positive charge and opposite to 𝑣⃗ × 𝐵 for a negative charge.
Note that the force in Equation 5 is perpendicular to both the
instantaneous velocity and the magnetic field. This means that
the magnetic field does no work on the charge; thus the
magnetic field changes the direction of the charge but not the
magnitude of its velocity. With the velocity of the charge
perpendicular to the magnetic field, the magnetic force acts
radially inward and the charged particle will move in a circular
orbit in a plane.
Figure 3 - Force on and path of
positive charge moving in a
uniform transverse magnetic
field
Applying Newton’s second law:
A⃗C = 𝑞 𝑣 𝐵 = ' <
C𝑞 𝑣⃗ × 𝐵
=
where
𝑟=
<"
=
"
Equation 6
is the centripetal acceleration. Then
'<
Equation 7
>?
From Eqns. [3] and [5], the ratio e/m is expressed in terms of measurable quantities.
$
'
$
'
•
Equation 8
" %&
= (= ?)"
Equation 9
Equation 8 is interpreted as follows:
•
•
•
<
= =?
A⃗, increasing the velocity of the charged particle (greater ΔV)
For a given magnetic field 𝐵
increases the radius r of its circular orbit.
For a given velocity, as the magnetic field is increased, r, the radius of the circular orbit is
reduced.
$
is the ratio of two fundamental constants: the electron charge and the electron mass. In the
experiment measuring the charge-to-mass ratio, e/m is determined by directly observing the orbital
radius, r, of a beam of electrons of speed v moving in a magnetic field of magnitude B.
'
$ (= ?)"
From Equation 9, we see, 𝛥𝑉 = '
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"
so, if we plot ΔV vs
(= ?)"
"
$
, the slope will be '.
PHYS 102 – Lab 4 – Motion of charged particles in E and B Fields
Page 4 of 4
The e/m Apparatus
Figure 4 - The e/m apparatus consisting of a pair of
Helmholtz coils, vacuum tube and control box
containing power supplies for the tube heater, coils
and accelerating voltage; circular electron beam
observed in the experiment.
The e/m apparatus, depicted in Figure 3, consists of a partially
evacuated glass bulb placed between the Helmholtz coils.
Inside the glass bulb, electrons are emitted from a heated
cathode connected to the negative terminal of a high-voltage
power supply. The cathode is partially shielded by a
surrounding grid that allows electrons to pass through. The
electrons are then attracted to an anode mounted below the
grid and connected to the positive terminal of the high-voltage power
supply. The electrons escaping through the grid are accelerated
downwards towards the anode by an adjustable potential, V. The
accelerated electrons emerge from the electron gun with a velocity v and
enter a region of uniform magnetic field B. Because the velocity of the
electron beam is perpendicular to the uniform magnetic field direction, the
A⃗ resulting in a near
beam is deflected by the magnetic force 𝐹⃗ = 𝑒 𝑣⃗ × 𝐵
vertical circular orbit. The electron beam is visible since the energetic
electron beam excites low pressure He gas in the tube which produces
visible light; thus, the orbital radius of the electron beam can be measured
with a ruled scale placed inside the tube.
Considerations
A couple of challenges in the measurements arise because of non-idealities in the electron gun energy loss
of the electrons as they collide with and excite the He atoms in the tube.
• Electrons are accelerated by the potential difference, ΔV, between the cathode and anode. The
aperture in the anode that allows electrons to pass through causes some non-uniformity of the
accelerating electric field resulting in an overestimation of the accelerating potential.
• Electrons lose energy when they collide with He atoms resulting in a reduction in the radius of the
electron path. Measuring radii of the outermost edges of the circular electron beam (electrons that
have had fewer collisions) will reduce the error in this component of the e/m analysis.
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