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Charged Particles In Circular Orbits

Learning Objectives. Book Reference : Pages 113-115. Charged Particles In Circular Orbits. To understand that the path of a charged particle in a magnetic field is circular To equate the force due to the magnetic field to the centripetal force

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Charged Particles In Circular Orbits

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  1. Learning Objectives Book Reference : Pages 113-115 Charged Particles In Circular Orbits To understand that the path of a charged particle in a magnetic field is circular To equate the force due to the magnetic field to the centripetal force To examine practical applications of circular displacement of particles

  2. In the previous lesson we have seen that a moving charged particle is deflected in a magnetic field in accordance with Fleming’s left hand rule Path of a Charged Particle in a Field 1 Magnetic field coming out of the page Initial Path of the Electron Conventional Current Conventional Current Conventional Current Force Force Force Electron Gun Electron Gun Electron Gun • The force acts perpendicular to the velocity causing the path to change.... The force acts perpendicular to the velocity causing the path to change.... The force acts perpendicular to the velocity causing the path to change.... Circular motion is achieved

  3. The force on the moving charged particle is always at right angles to the current velocity... Circular motion! • As always with circular motion problems we are looking for a force to “equate” to the centripetal force • From the last lesson we saw that the force on a charged particle is F= BQv • BQv = mv2 / r • r = mv/BQ • The path becomes more curved (r reduced) if the flux density increases, the velocity is decreased or if particles with a larger specific charge (Q/m) are used Path of a Charged Particle in a Field 2 BQv Velocity

  4. A beam of electrons with a velocity of 3.2x107 ms-1 is fired into a uniform magnetic field which has a flux density of 8.5mT. The initial velocity is perpendicular to the field. Explain why the electrons move in a circular orbit Calculate the radius of the orbit What must the flux density be adjusted to if the radius of the orbit is desired to be 65mm [21mm, 2.8mT] Problems 1

  5. The magic A2 crib sheet quotes the following : Charge on an electron (e) = -1.6 x 10-19 C Electron rest mass (me)= 9.11 x 10-31 kg However it also quotes... Electron Charge/Mass ratio(e/me) = 1.76x1011 Ckg-1 Two things.... Do not be phased by this since they could quote this for another particle... Simply inverting it (m/Q) for this sort of calculation saves one step in the calculator! Notes

  6. Last lesson we saw the CRT (Cathode Ray Tube) Applications 1 : CRT revisited Such devices can also be called Electron guns Thermionic devices The cathode is a heated filament with a negative potential which emits electrons, a nearby positive anode attracts these electrons which pass through a hole in the anode to form a beam. This is called Thermionic emission. The potential difference between the anode and cathode controls the speed of the electrons.

  7. These are machines which can be used to analyse the types of atoms, (and isotopes) present in a sample Applications : Mass Spectrometers 1 r = mv/BQ The key to how it works is the effect that the mass has on the radius of the circular motion while keeping the velocity and flux density constant

  8. First we need to ionise the atoms in the sample so that they become charged... Electrons are removed yielding a positive ion Applications : Mass Spectrometers 2

  9. A component known as a “velocity selector” is key to obtaining a constant velocity Applications : Mass Spectrometers 3 Collimator with slit The +ve ions are acted upon by both an electric field & a magnetic field. Only when they are equal & opposite do the ions pass through the slit Positive Ion The electric field is given by F = QV/d & the magnetic field given by F = BQv Only ions with a particular velocity will allow QV/d = BQv & hence only ions with that particular velocity make it through the slit. Note that it is also independent of charge since the Qs cancel

  10. Cyclotrons are a method of producing high energy beams used for nuclear physics & radiation therapy Applications : Cyclotrons 1 An alternating electric field is used to accelerate the particles while a magnetic field causes the particles to move in a circle, (actually a spiral since the velocity is increasing) Compared to a linear accelerator, this arrangement allows a greater amount of acceleration in a more compact space

  11. Two hollow D shaped electrodes exist in a vacuum. A uniform magnetic field is applied perpendicular to the plane of the “Dees” Applications : Cyclotrons 2 Charged particles are injected into a D, the magnetic field sets the particle on a circular path causing it to emerge from the other side of this D & to enter the next. As the particle crosses the gap between the Ds the supplied current changes direction (high frequency AC) & the particle is accelerated, (causing a larger radius)

  12. Assuming Newtonian rather than relativistic velocities apply... • The particle leaves the cyclotron when the velocity causes the path radius to equal the radius R of the D • v = BQR/m • The period for one cycle of the AC must approximate to the time for one complete circle (2R) using s=d/t for t • T = 2Rm/BQR The frequency of the AC will be f=1/T • f=BQ/2m Applications : Cyclotrons 3

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