Download The Cyclotron…

The Cyclotron…I
1. Motion of the charges occurs in two
semicircular containers, D1 and
D2…referred to as the Dees
2. The Dees are evacuated in order to
minimize energy loss from collisions
3. A high frrequency alternating voltage
is applied to the Dees along with a
uniform magnetic field directed
perpendicular to the plane of the
Dees.
4. Positive ions released at the center of
the magnet move in a semeicircular
path and arrive back at the gap in a
time T/2, where T is the period of
revolution
The Cyclotron…II
1. The frequency of the applied voltage,
V, is adjusted such that the polarity of
the Dees is reverseed in the same
time it takes the ions to complete one
half of a revolution
2. If the phase of the applied voltage is
adjusted such that D2 is at a lower
potential than D1 by an amount V, the
ion will accelerate across the gap to
D2 and its kinetic energy will increase
by an amount qV
3. The ion continiues its semicircular
path in D2 but at a larger radius
because of its increased velocity
4. After a time T/2, it is once more at the
gap, the potential across the gap has
been reversed (D1 is now at a lower
potential), and the ion is given
another “kick” across the gap
The Cyclotron…III
1. The process repeats itself and at
each half revolution, the ion gains an
additional qV of kinetic energy
2. Important to note that the operation
of this device is based on the fact
that the time for one revolution is
independent of the speed or radius of
the path of the ion
3. The maximum kinetic energy of the
ion when it exits the cyclotron is in
terms of the radius R, of the Dees.
4. Recall that the radius of circular orbit
is given by
from which we get…
First successful cyclotron built by E. O. Lawrence and his graduate
student M. Stanley Livingston, accelerated a few hydrogen molecule
ions to an energy of 80,000 electron volts. Since each ion received an
accelerating kick twice in a circuit as it entered and left the single flat
semicircular electrode or 'dee', those that managed to reach full energy
and fall into the collecting cup 4.50 cm from the center of the instrument
had made no fewer than forty turns.
The Hall Effect:
Reprise
A conducting slab of
width w carrying a
current I is placed in a
magnetic field B oriented
perpendicular to the
direction of the current.
r
E
The magnetic force qvdB on
the charge carriers pushes
them to one edge of the slab.
This separation of charge produces an electric field, which opposes
any further separation of charge.
Eventually these two forces come into equilibrium:
qE = qv d B
The electric field produces a potential difference V = Ew :
V
q = qv d B ⇒ VH = wv d B the "Hall voltage" measures v d
w
The Hall Effect II
The drift velocity may be obtained by measuring the Hall voltage. Recall the
expression for the drift velocity…
From a knowledge of the drift velocity, which came from an experimental
measurement of the Hall voltage, we can obtain n… the number of charge
carriers per unit volume (charge density)
Now, A is the cross sectional area of the conductor. Substitution of the
expression for the drift velocity into our expression for the Hall voltage gives
The Hall Effect III
Because A is equal to the width, w, times the thickness, t…NOT to be
confused with time in this case!!! We have A=wt. This gives us…
DO NOT confuse t with time here!!! The quantity 1/nq is what is known as
the Hall Coefficient, RH
Because all quantities other than n and q in the expression above can be
measured the Hall coefficient is readily obtained.
1. The coefficient gives the sign of the charge carriers and their density
2. In most metals the values obtained from Hall effect measurements
are in good agreement when n is approximately equal to the number
of valence electrons per unit volume
3. This is a CLASSICAL model and for metals like Fe, Bi, and Cd or for
semiconductors like silicon and germanium a model based on the
quantum nature of solids is needed.
Example 1…Cyclotron
Calculate the maximum kinetic energy of
protons in a cyclotron of radius 0.50 m in a
magnetic field of 0.35 T
Example 2
A copper wire of cross sectional area 3X10-6 m2
carries a current of 10 A. Find the drift velocity
of the electrons in the wire. (The molar mass of copper
is 63.5 g/mole and the density of copper is 8.95 g/cm3)
Example 3
A rectangular copper strip 1.5 cm wide and 0.1
cm thick carries a current of 5 amperes. A 1.2 T
magnetic field is applied perpendicular to the
strip. Find the resulting Hall voltage. (The molar mass
of copper is 63.5 g/mole and the density of copper is 8.95 g/cm3)
Example 4
The magnetic field of the earth at a certain location is
directed vertically downward and has a magnitude of
0.5X10-4 T. A proton is moving horizontally towards the
west in this field with a velocity of 6.2 X 106 m/s.
1. What are the direction and magnitude of the
magnetic force the field exerts on this charge?
2. What is the radius of the circular arc followed by this
proton?
Example 5
A singly charged positive ion has a mass of 3.2 X 10-26 kg.
After being accelerated through a potential difference of
833 V, the ion enters a magnetic field of 0.92 T along a
direction perpendicular to the direction of the field.
1. Calculate the radius of the path of the ion in the field.
Example 7
Calculate the cyclotron frequency of a proton in a
magnetic field of 5.2T.
Particle tracks from a Bubble Chamber
Example 8
A singly charged positive ion moving with a speed of 4.6 X
105 m/s leaves a spiral track of radius 7.94 mm in a
photograph along a direction perpendicular to the
magnetic field of a bubble chamber. The magnetic field
applied for the photograph has a magnitude of 1.8T.
Compute the mass (in atomic mass units) of this
particle and from that, identify the particle
Example 9
The picture tube in a television uses magnetic deflection
coils rather than electric deflection plates. Suppose an
electron beam is accelerated though a 50-kV potential
difference and then passes through a uniform magnetic
field produced by these coils for 1 cm. The screen is
located 10 cm from the center of the and is 50 cm wide.
When the field is turned off, the electron beam hits the
center of the screen. What field strength is necessary to
deflect the beam to the side of the screen?
Example 10
The Hall effect can be used to measure the number of
conduction electrons per unit volume, n, for an
unknown sample. The sample is 15 mm thick, and when
placed in a 1.8 T magnetic field produces a Hall voltage
of 0.122 µV while carrying a 12 A current. What is the
value of n?
Example 11
In an experiment designed to measure the earth’s
magnetic field using the Hall effect, a copper bar 0.5 cm
in thickness is positioned along an east-west direction.
If a current of 8 A in the conductor results in a
measured Hall voltage of 5.1 X 10-12 V, what is the
calculated value of the earth’s magnetic field? (Assume
that n=8.48 X 1028 electrons/m3 and that the plane of the
bar is rotated to be perpendicular to the direction of B)
Example 12
A cyclotron designed to accelerate protons is a provided
with a magnetic field of 0.45 T and has a radius of 1.2 m.
1. What is the cyclotron frequency?
2. What is the maximum speed acquired by the
protons?
Example 13
A certain cyclotron designed to accelerate alpha particles
has a diameter of 1.52 m and operates at a frequency of
9.37 MHz.
1. What is the maximum kinetic energy of the alpha
particles?
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