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Chapter 8. Charges in Magnetic Fields. Introduction. In the previous chapter it was observed that a current carrying wire observed a force when in a magnetic field This force is experienced by any moving charge in a magnetic field. Introduction.

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chapter 8

Chapter 8

Charges in Magnetic Fields

introduction
Introduction
  • In the previous chapter it was observed that a current carrying wire observed a force when in a magnetic field
  • This force is experienced by any moving charge in a magnetic field
introduction1
Introduction
  • In applications where this interaction is used, the charges are moving through near vacuum so that relatively free motion can occur across that space (low electrical resistance)
factors affecting the force
Factors Affecting the Force

When a charged particle is in a magnetic field, the force on the charged particle depends on the following factors:

  • The magnitude and direction of the velocity of the particle
  • The magnitude and sign of the charge on the particle
  • The magnetic field strength
factors affecting the force1
Factors Affecting the Force
  • There is no interaction between a magnetic field and a stationary particle
  • Stationary charges do not generate a magnetic field to interact with the magnetic field they are in
  • The electric field created by the charged particle does not interact with the magnetic field
the force on a charged particle moving in a magnetic field
The force on a Charged Particle Moving in a Magnetic field
  • An electric current is a flow of electric charges
  • The magnitude of the current is defined as the rate of flow of electric charge:

I =

Where Δq is the charge and Δt is the time

the force on a charged particle moving in a magnetic field1
The force on a Charged Particle Moving in a Magnetic field
  • The rate of flow of charge is taken from a point:

e.g. if a current of 2A is flowing through a circuit, 2 coulombs of charge passes any point in the circuit each second

the force on a charged particle moving in a magnetic field2
The force on a Charged Particle Moving in a Magnetic field
  • This idea can be extended to point charges:
    • If one alpha particle (q = 3.2x10-19C) passes a point in one second, then the average current is 3.2x10-19A past that point
    • If one alpha particle (q = 3.2x10-19C) passes a point in two seconds, then the average current is 1.6x10-19A past that point
the force on a charged particle moving in a magnetic field3
The force on a Charged Particle Moving in a Magnetic field
  • The force on a current carrying wire in a magnetic field from the formula:

F = IΔlB sinθ

  • However to apply this to a charged particle, we need to consider how to define IΔl
the force on a charged particle moving in a magnetic field4
The force on a Charged Particle Moving in a Magnetic field
  • As discussed before, current is given by:

Iavg=

  • In this time, the particle has moved a distance of vtmetres, this can be taken as the length, Δl, of the current element
the force on a charged particle moving in a magnetic field5
The force on a Charged Particle Moving in a Magnetic field
  • Substituting the expressions for current and element length gives:
the force on a charged particle moving in a magnetic field6
The force on a Charged Particle Moving in a Magnetic field

As with a current carrying wire in a magnetic field

  • the force on a charge moving in a magnetic field is maximum when it is travelling perpendicular to the field
  • the force in a charge moving parallel or anti-parallel to the field is zero
the direction of the magnetic force
The direction of the magnetic force
  • The direction of magnetic force on a moving charge in a magnetic field can be found using the right-hand palm rule
  • However, the thumb points in the direction of conventional current (positive charge flow)
  • This means that the thumb points in the opposite direction to the motion of a negative charge
class problems
Class problems

Conceptual questions: 1-4

Descriptive questions: 2

Analytical questions: 2

motion at directions other than 90 to the magnetic field
Motion at Directions other than 90° to the Magnetic Field
  • Charged particles moving parallel to a magnetic field experience no magnetic force, and therefore move with constant velocity
  • Motion at angles θ to the magnetic field are more complex and are not included in the syllabus
  • Only charges moving perpendicular to the field are considered in this course
motion at directions other than 90 to the magnetic field1
Motion at Directions other than 90° to the Magnetic Field
  • Example of motion at an angle to the magnetic field

http://www.youtube.com/watch?v=a2_wUDBl-g8

motion of charged particles at right angles to the magnetic field
Motion of Charged Particles at Right Angles to the Magnetic Field
  • In the diagram shown, a charged particle enters a uniform magnetic field directed into the page
  • Using the right hand rule, the force is acting towards the top of the page
motion of charged particles at right angles to the magnetic field1
Motion of Charged Particles at Right Angles to the Magnetic Field
  • As the particle changes direction, so does the direction of the magnetic force acting on it
  • Since the magnetic force is always perpendicular to the velocity, the speed of the particle does not change
motion of charged particles at right angles to the magnetic field2
Motion of Charged Particles at Right Angles to the Magnetic Field
  • This motion is uniform circular motion
  • Charged particles moving at right angles to a magnetic field always follow a circular path
determination of the radius of the circular path
Determination of the Radius of the Circular Path
  • Centripetal acceleration is given by:
determination of the radius of the circular path1
Determination of the Radius of the Circular Path
  • The force is also given by:

Hence:

class problems1
Class problems

Conceptual questions: 4, 8, 10

Descriptive questions: 4

Analytical questions: 1, 3-4, 6-9

introduction2
Introduction
  • The acceleration of charged particles to very high speeds, and hence very high energies, is essential in many fields
  • It is particularly useful in atomic and nuclear physics, and in medical research, diagnosis and treatment
introduction3
Introduction
  • The most obvious way to do this is to pass the charged particle though a potential difference
  • Passing a proton through a potential difference of 1000V will result in a gain of 1000eV in kinetic energy
  • However we often require energies of MeV (106 eV) to GeV (109 eV)
introduction4
Introduction
  • We can accomplish higher energies by passing particles through a series of potential differences
  • Passing an electron 100 times in succession through 1000V is equivalent to passing it through 100,000V
introduction5
Introduction
  • To accelerate particles to energies in a linear accelerator to GeV energies requires a series of thousands of potential differences
  • This is impractical due to the sheer size of accelerator needed
  • Use of a cyclotron reduces the size of the accelerator considerably
components of a cyclotron
Components of a cyclotron

Ion source:

A source of protons to be accelerated

Semi-circular metal containers (Dees):

Two terminals of alternating potential difference between which the protons are accelerated

Ion source

components of a cyclotron1
Components of a cyclotron

Vacuum chamber:

The interior of the cyclotron is housed in an evacuated chamber

High frequency input:

The source of alternating potential difference

Ion source

components of a cyclotron2
Components of a cyclotron

Electromagnets:

The South pole of an electromagnet is below the Dees, and the North pole of another electromagnet is above, this generates a uniform magnetic field for the circular motion

principles of operation
Principles of Operation
  • The protons are accelerated towards the negatively charged Dee
  • Within the Dee they experience circular acceleration due to the magnetic field
principles of operation1
Principles of Operation
  • The electric field does not exist in the Dees because they are effectively hollow conductors
principles of operation2
Principles of Operation
  • When the proton leaves the Dee, the potential difference is reversed, accelerating the proton towards the other Dee
principles of operation3
Principles of Operation
  • This process repeats many times, each time the proton is accelerated across the gap between the Dees, the radius gets larger
principles of operation4
Principles of Operation
  • The proton is eventually removed from the cyclotron using electrodes
computational considerations
Computational Considerations
  • The radius of the proton’s circular orbit at any time in the Dees is given by:
computational considerations1
Computational Considerations
  • The period of the proton’s motion is independent of its speed:

Derivation on p. 172 of Key Ideas textbook

computational considerations2
Computational Considerations
  • Kinetic energy of the particle:

Derivation on p. 173 of Key Ideas textbook

some uses of cyclotrons
Some Uses of Cyclotrons
  • The plutonium used to make the first atomic bomb was made by bombarding Uranium 238 with deuterons
  • Production of isotopes to use in nuclear medicine
    • Injecting radioactive isotopes into organs and detecting them with gamma ray detectors
    • Positron decay from Nitrogen-13 used in Positron Emission Tomography (PET)
class problems2
Class problems

Conceptual questions: 10-13, 15

Descriptive questions: 12, 14, 18

Analytical questions: 8, 10, 11