1. Motion of the charges occurs in two semicircular containers, D 1 and D 2 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 I
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 D 2 is at a lower potential than D 1 by an amount V, the ion will accelerate across the gap to D 2 and its kinetic energy will increase by an amount qv 3. The ion continiues its semicircular path in D 2 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 (D 1 is now at a lower potential), and the ion is given another kick across the gap The Cyclotron II
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. The magnetic force qv d B 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 : r E q V w = qv d B V H = wv d B the "Hall voltage" measures v d
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, R H 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 m 2 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/cm 3 )
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 10 6 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 10 5 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 10 28 electrons/m 3 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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