Transcript Section 1
2.2 MOTION OF CHARGED PARTICLES IN MAGNETIC FIELDS Magnetic Force on a Particle The force on a charged particle moving through a magnetic field is given by: πΉ = ππ£π΅π πππ πΉ = force (newtons, π) π = charge of particle (coulombs, πΆ) π£ = velocity of particle (metre per second, π. π β1 ) π΅ = magnetic field (tesla, π) π = angle between the field and velocity (degrees, °) Magnetic Force on a Particle Hence, in a uniform magnetic field, there is no magnetic force on a particle if: The particle has no charge The particle is stationary (π£ = 0) The particle moves parallel to the field (π = 0° or 180°) Right Hand Rule (for Charged Particles) B magnetic field F force (negative charge ) Out of back of hand F force (positive charge) Out of palm v velocity Example 2 State the direction of the force on the particle in each of the following situations. up out of the page out of the page no force Example 1 Calculate the force on an electron fired at 4.0 × 106 π. π β1 at 60° to a magnetic field with strength 1.5T. v - B 60° Motion of a Charged Particle at ππ° to the Field A charged particle will experience a force of constant magnitude at right angles to its velocity. This force provides a centripetal acceleration, making the particle move in uniform circular motion. Motion of a Charged Particle at ππ° to the Field Motion in Magnetic Field Simulation Motion of a Charged Particle at ππ° to the Field F F F Loop generating the magnetic field Derivation: Radius of the Circular Path πΉπππππ’πππ = πΉππππππ‘ππ ππ = ππ£π΅π πππ ππ£ 2 = ππ£π΅π πππ π ππ£ = ππ΅ π€βππ π = 90°, π πππ = 1 π ππ£ = ππ΅π ππ£ π= ππ΅ Note: Increasing π or π£ will increase the radius. Increasing π or π΅ will decrease the radius. Example 3 An electron enters a 0.4 T uniform magnetic field moving at 2.0 × 108 m.s-1 as shown. a) Find the force it experiences as it enters the field. F b) Draw the path taken by the electron on the diagram. c) Find the radius of the path it takes. Example 4 A positive B negative D C negative no charge Particles moving at other angles to the field Particles moving at other angles to the field The particle only experiences a force perpendicular to the field. Therefore, its motion parallel to the field remains constant. Application: The Cyclotron A cyclotron is a device used to accelerate ions (charged particles) to high speeds. Application: The Cyclotron These ions can then be used to create radioactive isotopes. The isotopes are used for medical diagnosis and treatment (commonly PET scans). SAHMRI in Adelaide A vial of radioactive glucose produced by the $4 million cyclotron. evacuated chamber ion source magnet magnetic field D-shaped electrodes (dees) magnet target electric field (between the dees) deflecting magnet Ion Production A heated filament produces electrons by thermionic emission 1 electron less positive ion Ion Production Modern cyclotrons use an electric arc (a plasma) to produce the ions. How a Cyclotron Works Ion source The magnetic field bends the path of the ion into a semi-circle. The electric field accelerates the ion each time it crosses the gap. The dees. One is positively charged and the other is negatively charged. How a Cyclotron Works Every time the ion crosses the electric field (the gap between the dees), it is accelerated and gains kinetic energy. βπΎ = π = πβπ Example 5 dees Find the direction of the magnetic field if the ion source is (a) positive out of the page (b) negative into the page F v Practice Find the direction of the magnetic field if the ion source is positive. magnetic field is into the page F v Derivation: Period of the Ionβs Circular Motion π= ππ£ ππ΅ but 2ππ π π β΄π= ππ΅ π2ππ β΄π= ππ΅π 2πππ β΄ ππ = ππ΅ 2ππ β΄π= ππ΅ π£= 2ππ π This shows that the time it takes the ions to circle the cyclotron once is independent of their speed. Hence, the potential difference alternates at a constant frequency π= 1 π = ππ΅ 2ππ Derivation: Kinetic Energy of Emerging Ions ππ£ ππ΅π π= βΉπ£= ππ΅ π 1 πΎ = ππ£ 2 2 1 ππ΅π β΄πΎ= π 2 π 2 2 2 2 π π π΅ π β΄πΎ= × 2 π2 π2 π΅2 π 2 β΄πΎ= 2π This shows that the final kinetic energy of the ions does not depend on the electric field but rather the magnetic field and the radius of the cyclotron.