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Learning Objectives. Book Reference : Pages 113-115. Charged Particles In Circular Orbits. To understand that the path of a charged particle in a magnetic field is circular To equate the force due to the magnetic field to the centripetal force

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charged particles in circular orbits

Learning Objectives

Book Reference : Pages 113-115

Charged Particles In Circular Orbits

To understand that the path of a charged particle in a magnetic field is circular

To equate the force due to the magnetic field to the centripetal force

To examine practical applications of circular particle displacement

path of a charged particle in a field 1

In the previous lesson we have seen that a moving charged particle is deflected in a magnetic field in accordance with Fleming’s left hand rule

Path of a Charged Particle in a Field 1

Magnetic field coming out of the page

Initial Path of the Electron

Conventional

Current

Conventional

Current

Conventional

Current

Force

Force

Force

Electron Gun

Electron Gun

Electron Gun

  • The force acts perpendicular to the velocity causing the path to change.... The force acts perpendicular to the velocity causing the path to change.... The force acts perpendicular to the velocity causing the path to change.... Circular motion is achieved
path of a charged particle in a field 2

The force on the moving charged particle is always at right angles to the current velocity... Circular motion!

  • As always with circular motion problems we are looking a the force to “equate” to the centripetal force
  • From the last lesson we saw that the force on a charged particle is F= BQv
      • BQv = mv2 / r
      • r = mv/BQ
  • The path becomes more curved (r reduced) if the flux density increases, the velocity is decreased or if particles with a larger specific charge (Q/m) are used

Path of a Charged Particle in a Field 2

BQv

Velocity

problems 1

A beam of electrons with a velocity of 3.2x107 m/s is fired into a uniform magnetic field which has a flux density of 8.5mT. The initial velocity is perpendicular to the field.

Explain why the electrons move in a circular orbit

Calculate the radius of the orbit

What must the flux density be adjusted to if the radius of the orbit is desired to be 65mm

[21mm, 2.8mT]

Problems 1

notes

The magic A2 crib sheet quotes the following :

Charge on an electron (e) = -1.6 x 10-19 C

Electron rest mass (me)= 9.11 x 10-31 kg

However it also quotes...

Electron Charge/Mass ratio(e/me) = 1.76x1011 C/kg

Two things.... Do not be phased by this since they could quote this for another particle... Simply inverting it (m/Q) for this sort of calculation saves one step in the calculator!

Notes

applications 1 crt revisited

Last lesson we saw the CRT (Cathode Ray Tube)

Applications 1 : CRT revisited

Such devices can also be called

Electron guns

Thermionic devices

The cathode is a heated filament with a negative potential which emits electrons, a nearby positive anode attracts these electrons which pass through a hole in the anode to form a beam. This is called Thermionic emission. The potential difference between the anode and cathode controls the speed of the electrons.

applications mass spectrometers 1

These are machines which can be used to analyse the types of atoms, (and isotopes) present in a sample

Applications : Mass Spectrometers 1

r = mv/BQ

The key to how it works is the effect that the mass has on the radius of the circular motion while keeping the velocity and flux density constant

applications mass spectrometers 2

First we need to ionise the atoms in the sample so that they become charged... Electrons are removed yielding a positive ion

Applications : Mass Spectrometers 2

applications mass spectrometers 3

A component known as a “velocity selector” is key to obtaining a constant velocity

Applications : Mass Spectrometers 3

Collimator with

slit

The +ve ions are acted upon by both an electric field & a magnetic field.

Only when they are equal & opposite do the ions pass through the slit

The electric field is given by F = QV/d & the magnetic field given by F = BQv Only ions with a particular velocity will allow QV/d = BQv & hence only ions with that particular velocity make it through the slit. Note that it is also independent of charge since the Qs cancel

applications cyclotrons 1

Cyclotrons are a method of producing high energy beams used for nuclear physics & radiation therapy

Applications : Cyclotrons 1

An alternating electric field is used to accelerate the particles while a magnetic field causes the particles to move in a circle, (actually a spiral since the velocity is increasing)

Compared to a linear accelerator, this arrangement allows a greater amount of acceleration in a more compact space

applications cyclotrons 2

Two hollow D shaped electrodes exist in a vacuum. A uniform magnetic field is applied perpendicular to the plane of the “Dees”

Applications : Cyclotrons 2

Charged particles are injected into a D, the magnetic field sets the particle on a circular path causing it to emerge from the other side of this D & to enter the next.

As the particle crosses the gap between the Ds the supplied current changes direction (high frequency AC) & the particle is accelerated, (causing a larger radius)

applications cyclotrons 3

Assuming Newtonian rather than relativistic velocities apply...

  • The particle leaves the cyclotron when the velocity causes the path radius to equal the radius R of the D
        • v = BQR/m
  • The period for one cycle of the AC must approximate to the time for one complete circle (2R) using s=d/t for t
  • T = 2Rm/BQR The frequency of the AC will be f=1/T
        • f=BQ/2m

Applications : Cyclotrons 3