Download Manipulation of charged particles in discharge tubes

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9.4
FROM IDEAS TO
IMPLEMENTATION
#1.3
CATHODE RAYS:
MANIPULATION OF
CHARGED PARTICLES
FIELDS
In order to understand the behaviour of cathode rays is essential that you know about
the general behaviour of charged particles in electric and magnetic fields.
But is meant by the term field?
A field is a region of influence. For example, an object because of its mass experiences
a force in a gravitational field. A charged object experiences a force in an electric field.
In a scalar field, each point in space can be assigned a
number to represent the strength of the field. For
example, a map of Australia shows the variation of
temperature measured in oC.
In a vector field, each point in space can be assigned a
number to represent its strength and arrows or lines
are used to indicate the direction of influence of the
field. For example, a map of Australia shows the
variation in wind velocity (magnitude in knots/s and
direction). Electric charges have an electric field
surrounding them and charges in motion have a
magnetic field surrounding them as well.
A vector field can also be shown as a pattern of lines. The lines indicate the field
pattern and the density of the lines indicates the strength of the field. The closer the
field lines are together, then the stronger the field and hence the stronger the force.
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Electric field
The electric field can be represented by a pattern of lines. An electric field line
originates on positive charges and terminates on negative charges. The direction of the
electric field is in the direction of the force that would act on a positive charge at that
location. The greater the density of the electric field lines than the greater the electric
field intensity.
A particle with charge q experiences a force F in an electric field E . The direction of
the force on the charge q is tangential to the electric field line.
F
E
F
(1)
F = E q or F  qE
A radial electric field surrounds a point charge and a uniform electric field is found
between two parallel but oppositely charged conductive plates.
large electric field intensity
small electric field intensity
+Q
+ + + +
+
-
d
-Q
-
POSITIVE charge
+
NEGATIVE charge
-
-
-
V
-
uniform electric field
If the distance between the two oppositely charged plates is d and the potential
difference is V, then the electric field intensity E between the charged plates is
(2)
E
V
d
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Magnetic Field
A magnetic field surrounds a permanent magnet and a wire carrying an electric current.
A magnet has a north pole and south pole. The magnetic field of a
magnet can be shown by a set of continuous loops that exit from the
north pole of the magnet and enter at the south pole. The B-field lines
indicate how a small magnet (or compass) will align itself in the field. The
magnetic field is strongest where the field lines are closest together at the two
poles.
strong field:
high density of B-lines
B
S
weak field:
low density of B-lines
N
S
N
compass needles / magnets
align along B-field
A current (moving charges) through a wire alters the properties of space near it such
that a piece of iron will experience a force. Hence, surrounding a wire carrying a
current is a magnetic field. It is the interaction of the magnetic field and the iron that
leads to the force, rather than the current and iron acting upon each other.
The magnetic field surrounding a straight conductor carrying a current I can be
visualized as a series of circles. The closer the lines are together, the stronger the
magnetic field. A compass placed near the wire will align itself with the field lines.
The direction of the magnetic field is determined by the right hand screw rule. Using
the right hand: the direction of the thumb represents the current (direction in which
positive charges would move) and the curl of the fingers represents the direction of
the magnetic field. The magnetic field strength is given by the vector quantity B where
B stands for the B-field or magnetic induction or magnetic flux density. The S.I. unit
for the B-field is the tesla [T].
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I
B
The right hand screw rule is used to determine the direction of the
magnetic field produced by moving charges.
The B-field surrounding a conductor carrying a current I at a distance R from the
conductor is given by equation (1)
(1)
where
B=
0 I
2 R
0 = 4 × 10-7 T.m.A-1
0 is a constant called the permeability of free space.
direction of B-field given by right hand rule
B-field increases with increasing I
B
B
R
I
B-field decreases with increasing R
B
current I into page
R
B-field surrounding straight conductor carrying a current.
current I
Magnet field due to a
current loop. The
direction of the
B-field is given
by the right hand
screw rule.
B-field
B-field
stro
ngB-field
strong
Bfield
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A solenoid is a conductor wound into a long set of coils
Solenoids are often used as electromagnets where a
ferromagnetic substance placed inside the coils greatly increases the strength of the
magnetic field. The magnetic field patters for an air filled solenoid and when a rod of
ferromagnetic material is placed inside the coils.
Magnetic field patterns for a solenoid (air and ferromagnetic cores).
The magnetic field of a solenoid is very similar to that of a bar magnet.
Magnetic field patterns for a bar magnet and solenoid.
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MOTION OF CHARGED PARTICLES IN ELECTRIC AND
MAGNETIC FIELDS
A particle of mass m and charge q in an electric field of intensity E experience a force F
(1)
F=Eq
and hence acquires an acceleration a
(4)
a
qE
m
A uniform electric field is found between two parallel but oppositely charged
conductive plates. If the distance between the plates is d and the potential difference
is V, then the electric field intensity E is
(2)
E
V
 constant
d
When an electron (-) enters a region of uniform electric field E between two parallel
but oppositely charged plates, it will be drawn towards the positive plate and repelled
from the negative plate. Since the electric field is uniform, the force acting on the
electron is constant and hence its acceleration is also constant. The motion of the
electron is similar to that of a projectile where the constant force of gravity acts upon
the object. We will consider two cases for the motion of an electron in a uniform
electric field:
(i)
If an electron is initially at rest and moves in the direction of the electric
field lines from the negative plate to the positive plate through the distance
d then work W is done on the electron to increase in its kinetic energy
Work = change in kinetic energy
V 
W  F d  qE d  q   d  qV
d
2
1
EK  2 mv
1
2
mv 2  qV
If there is an aperture in the
positive plate to which the electrons
are attracted, then, some of the
electrons will pass through the
aperture and proceed as a beam of
particles moving with a constant
velocity v. This is the principle of the
electron gun in a cathode ray tube.
V
+Q
-Q
E
F
v
d
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(ii)
When an electron enters the region where the uniform electric field acts
perpendicular to its motion, the electron is deflected from its initial
direction and moves in a parabolic path. Charged metal plates within a
cathode ray tube can be used to deflect the electron beam. The greater the
strength of the electric field intensity, then, the greater the deflection of
the electron beam.
v
+Q
F
E
-Q
straight
line path
parabolic path
straight
line path
Experiments show that moving electric charges are deflected in magnetic
fields. Hence, moving electric charges experience forces in magnetic fields and
they are accelerated. The magnetic force on a stationary charge is zero. A
force experienced by a charge in motion in a magnetic field does not change
its speed or kinetic energy, the acceleration produces only a change in
direction of motion since the direction of the force is perpendicular to the
velocity of the charge.
The direction of the force on a moving charge can be found by using the right
hand palm rule. Using only the right hand: the fingers point along the
direction of the B-field; the thumb points in the direction in which positive
charges would move (if a negative is moving to the left, then the thumb must
point to the right); the face of the palm gives the direction of the force on the
charged particle. Applications of the right hand palm rule are illustrated in the
following figures.
+q
v
+I
F
out of page
B
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F
palm face
B fingers
motion of a
positive charge in a
magnetic field

v
v  q 
thumb
v   q  thumb
v (+q)
B fingers

v  q 
motion of a
negative charge in
a magnetic field
F palm face
The direction of the force on a charged particle is determined by the right hand
palm rule.
Magnetic forces on charged particles have important implications from the functioning
of electronic devices to phenomena in astrophysics and plasma physics. A bar magnet
can be used to deflect an electron beam in a cathode ray tube.
e
S
N
B
right hand palm rule
thumb
palm
facing down
v  q 
v  q 
fingers
B
F
The direction of the force on an electron beam in a cathode ray tube is given by the
right hand palm rule.
In most cathode ray tubes found in television sets and cathode ray oscilloscopes, a
magnetic field produced by currents in coils are used for the deflection of the electron
beam rather than the deflection that would be produced by the electric field of two
parallel oppositely charged plates.
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From such deflections the magnitude of the force FB is found to depend upon the
strength of the B-field B, the magnitude of the charge q and the velocity of the charge
v (direction w.r.t. the B-field  and magnitude v). The magnitude of the force FB is
given by equation (3)
FB  qv B  q v sin  B
(3)
FB  qv B sin
The angle  is measured between the directions of the velocity and the magnetic field.
The component of the velocity perpendicular to the B-field is
v  v sin
When using the right hand palm rule, the thumb must point in the direction along v
for a positive charge. If the motion of the charge particle is along the direction of the
B-field ( B v ) then the magnetic force on the charge is zero since  = 0 and sin = 0.
The S.I. unit for the B-field is called the tesla [T], in honour of Nikola Tesla who made
important contributions to electrical energy generation.
Example
An electron beam travels through a cathode ray tube. A south pole of a bar magnet is
placed above the beam causing the beam to be deflected. The magnitude of the B-field
at the location of the electron beam is 0.0875 T. The beam of electrons has been
accelerated by a voltage of 6.65 kV. What is the magnitude of the force acting on the
electrons and which way is the beam deflected?
Solution
How to approach the problem: ISEE
Category: work done on charge by a potential difference
qV  12 mv 2
force on a charged particle in a B-field
FB  qv B sin 
Diagram:
direction of force – right hand palm rule
Sketch the physical situation
Choose a [3D] set of axes for the direction of B, v and F
Summary of given information
Summary of unknown information
Solve the problem
Evaluate your answers
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N
+y
thumb
v  q 
fingers B
S
B
e
FB
-
+x
palm face
+z
v  q 
beam will be deflected
to the right
right hand palm rule
F
B
magnitude: electron charge q = 1.602x10-19 C
electron mass: me = 9.101x10-31 kg
accelerating voltage: V = 6.65 kV = 6.65x103 V
B- field: B = 0.875 T
FB = ? N
v = ? m.s-1
From the diagram, using the right hand palm rule the electron beam is deflected
towards the right.
The work done by the accelerating voltage increases the kinetic energy of the electrons,
therefore, we can calculate the speed of the electrons in the beam.
qV  12 me v 2



(2) 1.602  1019 6.65  103
2 qV
v

m.s-1  2.965  107 m.s-1
31
me
9.109  10
A moving charge in a magnetic field experience a force



FB  qv B  1.602  1019 2.965  107  0.0875 T  1.56  1013 T
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B
B
I
A conductor carrying a current or a charge moving through a magnetic field will
experience a force whose direction is determined by the right hand palm rule.
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