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Applications involving charged particles moving in a magnetic field

A charge moving with a velocity v in the presence of both an electric field E and a magnetic field B experiences both an electric force qE and a magnetic force qv x B. The total force (called the Lorentz force) acting on the charge is Lorentz force acting on the charge

Velocity selector
In many experiments involving moving charged particles, it is important that the particles all move with essentially the same velocity. This can be achieved by applying a combination of an electric field and a magnetic field oriented as shown in Figure 29.22. A uniform electric field is directed vertically downward (in the plane of the page in Fig. 29.22a), and a uniform magnetic field is applied in the direction perpendicular to the electric field (into the page in Fig. 29.22a). For q positive, the magnetic force is qv x B upward and the electric force qE is downward. When the magnitudes of the two fields are chosen so that qE = qvB, the particle moves in a straight horizontal line through the region of the fields. From the expression qE = qvB, we find that velocity selector only those particles having speed v pass undeflected through the mutually perpendicular electric and magnetic fields. The magnetic force exerted on particles moving at speeds greater than this is stronger than the electric force, and the particles are deflected upward. Those moving at speeds less than this are deflected downward. velocity selector diagram

The Mass spectrometer
A mass spectrometer separates ions according to their mass-to-charge ratio. In one version of this device, known as the Bainbridge mass spectrometer, a beam of ions first passes through a velocity selector and then enters a second uniform magnetic field B0 that has the same direction as the magnetic field in the selector (Fig. 29.23). a mass spectrometer Upon entering the second magnetic field, the ions move in a semicircle of radius r before striking a photographic plate at P. If the ions are positively charged, the beam deflects upward, as Figure 29.23 shows. If the ions are negatively charged, the beam would deflect downward. From Equation 29.13, we can express the ratio m/q as mass to charge ratio Using Equation 29.17, we find that determine mass to charge ratio Therefore, we can determine m/q by measuring the radius of curvature and knowing the field magnitudes B, B0, and E. In practice, one usually measures the masses of various isotopes of a given ion, with the ions all carrying the same charge q. In this way, the mass ratios can be determined even if q is unknown.

A variation of this technique was used by J. J. Thomson (1856–1940) in 1897 to measure the ratio e/me for electrons. Figure 29.24a shows the basic apparatus he used. Electrons are accelerated from the cathode and pass through two slits. They then drift into a region of perpendicular electric and magnetic fields. The magnitudes of the two fields are first adjusted to produce an undeflected beam. When the magnetic field is turned off, the electric field produces a measurable beam deflection that is recorded on the fluorescent screen. From the size of the deflection and the measured values of E and B, the charge-to-mass ratio can be determined. The results of this crucial experiment represent the discovery of the electron as a fundamental particle of nature. thomson apparatus for measuring charge to mass ratio When a photographic plate from a mass spectrometer like the one shown in Figure 29.23 is developed, the three patterns shown in Figure 29.25 are observed. Rank the particles that caused the patterns by speed and m/q ratio. rank particles by speed and mass to charge ratio

The Cyclotron
A cyclotron can accelerate charged particles to very high speeds. Both electric and magnetic forces have a key role. The energetic particles produced are used to bombard atomic nuclei and thereby produce nuclear reactions of interest to researchers. A number of hospitals use cyclotron facilities to produce radioactive substances for diagnosis and treatment. schematic diagram of a cyclotron A schematic drawing of a cyclotron is shown in Figure 29.26. The charges move inside two semicircular containers D1 and D2, referred to as dees. A high frequency alternating potential difference is applied to the dees, and a uniform magnetic field is directed perpendicular to them. A positive ion released at P near the center of the magnet in one dee moves in a semicircular path (indicated by the dashed red line in the drawing) and arrives back at the gap in a time T/2, where T is the time needed to make one complete trip around the two dees, given by Equation 29.15. The frequency of the applied potential difference is adjusted so that the polarity of the dees is reversed in the same time it takes the ion to travel around one dee. If the applied potential difference is adjusted such that D2 is at a lower electric potential than D1 by an amount ΔV, the ion accelerates across the gap to D2 and its kinetic energy increases by an amount qΔV. It then moves around D2 in a semicircular path of greater radius (because its speed has increased). After a time T/2, it again arrives at the gap between the dees. By this time, the polarity across the dees is again reversed, and the ion is given another “kick” across the gap. The motion continues so that for each half-circle trip around one dee, the ion gains additional kinetic energy equal to q ΔV. When the radius of its path is nearly that of the dees, the energetic ion leaves the system through the exit slit. It is important to note that the operation of the cyclotron is based on the fact that T is independent of the speed of the ion and of the radius of the circular path.

We can obtain an expression for the kinetic energy of the ion when it exits the cyclotron in terms of the radius R of the dees. From Equation 29.13 we know that ν = qBR/m. Hence, the kinetic energy is kinetic energy of ion in cyclotron When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play. We observe that T increases and that the moving ions do not remain in phase with the applied potential difference. Some accelerators overcome this problem by modifying the period of the applied potential difference so that it remains in phase with the moving ions.

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