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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”. Example:.

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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. Example: A positive charge q=3.2x10-19 C moves with a velocity v=(2i+3j-k) m/s through a region where both a uniform magnetic and electric field exist.Calculate the force on the charge if B=(2i+4j+k) T andE=(4i-j-2k) V/m.

  4. + 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. Let’s look at these two special cases:

  5. + 1) Uniform B, v perpendicular 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.

  6. 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.

  7. 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.

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

  9. + 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.

  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. 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

  12. + 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.

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

  14. 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

  15. 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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