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Charged Particles in Electric and Magnetic Fields

Charged Particles in Electric and Magnetic Fields. Motion of charged particles Lorentz Force Examples: cyclotron, mass spectrometer. Recall:. in an electric field. in a magnetic field. In general:. “Lorentz Force”. +. Magnetic Fields: F = q v  B. q. v.

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Charged Particles in Electric and Magnetic Fields

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  1. Charged Particles in Electric and Magnetic Fields • Motion of charged particles • Lorentz Force • Examples: cyclotron, mass spectrometer

  2. Recall: in an electric field in a magnetic field In general: “Lorentz Force”

  3. + Magnetic Fields: F = qvB q v • The force is perpendicular to the velocity (and path), so: • no work is done by B • kinetic energy is constant • speed is constant x x x x x x x x F B Only the direction of the motion changes due to the magnetic force.

  4. + 1) UniformB, vperpendicular to B q Magnetic force (v ) is Fm = qvB(constant) x x x x x x x x r But Fc = mv2/r,so the radius of circle is : Motion is a circle.

  5. Example: A proton is moving in a circular orbit of radius 14cm in a uniform magnetic field of 0.35T directed perpendicular to the velocity of the proton. Find the orbital speed of the proton.

  6. 2) Uniform B, v not perpendicular to B x z + + y • path is a circular helix along a field line • the component v (parallel to B) is constant On a large scale, the particles follow the field lines.

  7. Frequency [rad/s] Angular frequency is independent of speed and radius. = 1/T (period) The kinetic energy of a particle at radius r is

  8. + The Cyclotron : accelerates particles to very high velocities x x x x x x x x x x B - “dees” + AC Voltage Supply The electric field across the gap is reversed each time the proton arrives, so that its speed in the gap continually increases. Because the time for each half-circle is the same for any proton speed, the voltage supply can just be set to a constant frequency.

  9. Quiz A cyclotron can accelerate protons to a maximum kinetic energy of 1 MeV. You want to design a new, improved model with a higher maximum proton energy. Which of the following changes would help? • Double the magnetic field • Double the diameter of the dees • Double the proton current • Double the voltage of the voltage supply

  10. Example: A cyclotron needs to accelerate electrons to a speed of 0.95c for experiments. Given that it can provide a magnetic field of 0.5T: a) what is the cyclotron frequency?b) what is the radius of the cyclotron?

  11. + The Velocity Selector: allows to select particles that move at the same velocity x x x x x x x x x x E Source Detector v B Find: Conditions so that the path is straight.

  12. x + Free-body diagram: Need for straight path B v E for straight path

  13. x + Question: How are the charges deflected that are either traveling too slow or too fast to travel in a straight line? B v E

  14. Mass Spectrometer : separates ions according to their mass:charge ratio x x x Uniform B x x x x x x x x x radius depends on mass Ion Source Exercise: Assume 12C and 14C have same charge. Find the ratio of the diameters of the circular paths if the ions have the same speeds

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