Particles and Accelerators
Electric fields
Explain what is meant by the term Electric FieldAn Electric field is the space around any charged object where another charged object will experience a force.
Explain what is meant by the term Electric Field Strength
The Electric Field Strength is given the symbol E, and is defined as the force per unit charge.
The force acting on a small point charge (Q2) in this electric field is then:
Is Electric Field Strength, E, a scalar or vector?
Representing Electric Fields
Electric fields can be represented by field lines, as with magnetic fields.Direction - that of the force acting on a small positive test charge.
Strength - the number of field lines per unit cross-sectional area.
=> So the closer they are, the stronger the field.
If parallel and equally spaced, then the field is uniform.
Field lines cut surfaces of equal potential at right angles.
Electric Potential
Electric Potential, V, is defined as the electric potential energy gained per unit charge
Consider two parallel charged plates with a vacuum between them.
Since the force acting on a charge in an electric field is, F = Q × E,
The force acting on the electron is F = e.E
What will happen to the electron?
The total work done on the electron when it reaches the positive plate is
W = Force ×distance moved in the direction of the force
=> W = e.E.d
=> V = E.d and E = V/d in a uniform electric field.
Suggest alternative units for Electric Field strength.
If V = 1000V and d is 10-2m, show that the velocity of the electron as it hits the positive plate is ≈ 1.88 × 107ms-1
(e = 1.6 × 10-19C, me = 9.1 × 10-31kg)
Discuss what would happen if V were increased to 500kV.
Electrons in an old style television are accelerated through a potential difference of 20kV.
If 1% their energy is converted into X-rays when they hit the face of the cathode ray tube (crt),
estimate the wavelength of the x-rays produced.
with a potential difference of V between them.
There is no force acting horizontally.
What will be the path of the electron?
If v = 107ms-1, V = 1000V, d = 5×10-2m and the plates are 100mm long,
show that the electron collides ≈ 3.8cm along the positive plate.
Explain what would be different if the electron were replaced with an alpha particle.
Early particle accelerators
These consisted of long evacuated tubes, with a metal plate at each end.A large potential difference (p.d.) was applied between the metal plates.
The negative plate was called 'the cathode' and the positive plate, the anode.
What is the 'electron Volt', and explain why is it a convenient unit of energy in particle accelerators?
When accelerator p.d.s exceeded 500kV, the electrons were observed to arrive at the anode 'late'.
Explain why supporting your answer using a calculation.
Since air breaks down at ≈ 30kVcm-1, suggest why there is a practical limit that can be used for the maximum p.d.
To overcome this problem, 'linac' type particle accelerators are used.
The diagram below represents a 'linac' particle accelerator.
(b). Explain the principle of operation of a 'linac'.
(c). Explain why the first drift tubes are shorter than the remainder.
(d). Explain why the later drift tubes can be all the same length.
(e). Explain why the electron beam consists of pulses of electrons.
Assume that the RF generator produces a square wave output of amplitude 600V and a frequency of 500MHz.
(f). Calculate the velocity of the electrons when they reach the first drift tube.
(g). If the RF generator switches just as the electrons enter the first drift tube,
calculate the length of this tube so that the electrons will leave the tube just as the RF generator switches.
The 'linac' gives the electrons an energy of 50MeV.
(h). Calculate the length of the last few drift tubes.
Stamford linac is 2 miles long. †
Much smaller particle accelerators can be made but they required the use of magnetism.
Cyclotron accelerators
'Moving charged particles create a magnetic field'.
The region of the magnetic field is represented by magnetic field lines.
The magnitude, direction and sense of the magnetic field is represented by a quantity called
the magnetic flux density, which is given the symbol B
Magnetic flux density, B, is a vector.
The tangent to a magnetic field line at any point gives the direction of B at that point.
The number of magnetic field lines per unit cross sectional area is proportional to the magnitude of B.
If the magnetic field is uniform, the field lines are uniformly spaced.
By experiments:
- a charged particle, moving in a magnetic experiences a force, F,
- the force is at right angles to the particle velocity, v,
- the force is proportional to the magnitude of v,
- the force is proportional to the charge, Q,
- the force is also at right angles to the resolved component of the velocity
normal to the magnetic field.
=> F ∝ Q v sinθ
=> F = B Q v sinθ
(F = Q v × B)
The directions are given by Fleming's Left hand Rule (conventional current)
New units - Wb m-2 - consistent with flux density
The SI unit is Tesla - 1T = 1Wb m-2
Compare F = B Q v sinθ with F = Q E
Motion of a particle in a magnetic field
Cyclotron:
The diagram below shows a block diagram for a simple cyclotron particle accelerator.
D-shaped metal boxes, Magnetic field, Outlet, R.F generator, Particle source, Vacuum tube
(b). By referring to the diagram, explain the principle of operation of a 'cyclotron'.
(c). If the particles that are to be accelerated are protons, draw on a diagram
the direction of the magnetic field and the path of the protons.
Assume that the RF generator produces a square wave output of amplitude 5kV.
(d). If the protons have an initial speed of zero when first released by the source, calculate their velocity
after crossing the gap for the first time.
(e). If the magnetic field strength is 0.1T, calculate the radius of the orbit of the protons.
(f). Calculate the frequency of the RF generator, if the supply changes just as the protons enter the gap.
(g). How many rotations are necessary if the protons are to leave the cyclotron with an energy of 20MeV.?
(h). Calculate the radius of the cyclotron.
(i). What effect would doubling the strength of the magnet field have?
(j). What effect would ncreasing the output voltage of the RF generator have?
Particle collisions
1). - (a) Explain what is meant by the term momentum.- (b) State the principle of conservation of momentum.
2). - (a) Show that the kinetic energy of a particle is equal to its momentum squared divided by two times its mass. (Ek = p2/2m)
- (b) An electron has a momentum of 10-22N.s. Calculate its kinetic energy.
- (c) Calculate the potential difference through which the electrons have been accelerated to gain this momentum.
- (d) Sketch a graph of kinetic energy against momentum.
3). A proton of mass m and moving at a speed of 10v makes a head on collision with a stationary helium nucleus of mass 4m.
After the collision the helium nucleus moves forward with a velocity of 4v.
- (a) Calculate the velocity of the proton after the collision.
- (b) Calculate the kinetic energy of the proton before and after the collision.
- (c) Calculate the kinetic energy of the helium nucleus after the collision.
- (d) Explain whether the collision is elastic or inelastic.
When collisions occur 'head on' then the objects continue in one dimension.
When collisions occur 'off centre' then the objects will move in two dimensions after the collision.
4). An α particle travelling at 1.5 × 107ms-1 collides off centre
with a stationary helium nucleus resulting the in the motion in the diagram below.
- (a) By resolving momentum along and at right angles to the original direction of travel of the α particle,
calculate the velocity and direction of travel of the Helium nucleus.
- (b) By considering the kinetic energy of the particles, explain whether the collision is elastic.
- (c) Summarise the results for two equally massive particles in an off centre collision
in terms of their angles after the collision.
5). A proton travelling at 2.5 x 107ms-1 collides off centre
with a stationary Helium nucleus resulting the in the motion in the diagram below.
Using a scale diagram, calculate the velocity and direction of travel of the Helium nucleus after the collision.