Chapter 8Charges 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

• 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)

8.1 FORCES ON A CHARGED PARTICLE IN A MAGNETIC FIELD

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 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

• 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 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 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 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 field

• As discussed before, current is given by:

Iavg =

• In this time, the particle has moved a distance of vt metres, this can be taken as the length, Δl, of the current element

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 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 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

Conceptual questions: 1-4

Descriptive questions: 2

Analytical questions: 2

8.2MOTION AT RIGHT ANGLES TO THE 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 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

• 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 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 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

• Centripetal acceleration is given by:

Determination of the Radius of the Circular Path

• The force is also given by:

Hence:

Class problems

Conceptual questions: 4, 8, 10

Descriptive questions: 4

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

8.3APPLICATION:THE CYCLOTRON

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

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)

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

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 cyclotronIon 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 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 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• The protons are accelerated towards the negatively

charged Dee

• Within the Dee they experience circular acceleration due to the magnetic field

Principles of Operation

• The electric field does not exist in the Dees because they are effectively hollow conductors

Principles of Operation

• When the proton leaves the Dee, the potential difference is reversed, accelerating the proton towards the other Dee

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 Operation

• The proton is eventually removed from the cyclotron using electrodes

Computational Considerations

• The radius of the proton’s circular orbit at any time in the Dees is given by:

Computational Considerations• The period of the proton’s motion is independent of

its speed:

Derivation on p. 172 of Key Ideas textbook

Computational Considerations• Kinetic energy of the particle:

Derivation on p. 173 of Key Ideas textbook

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 problems

Conceptual questions: 10-13, 15

Descriptive questions: 12, 14, 18

Analytical questions: 8, 10, 11