Particles and Waves2. Forces on charged particles

2014Knox Academy, Haddington

Particles and Waves: Forces on charged particles

ContentsUnit Specification..................................................................................................................................2

2. Forces on Charged Particles..........................................................................................................2

Notes.................................................................................................................................................2

Contexts.............................................................................................................................................2

Electric Fields.........................................................................................................................................3

Comparing electric and gravitational fields.......................................................................................4

Gravitational fields.........................................................................................................................4

Electric fields..................................................................................................................................4

Moving charges in electric fields.......................................................................................................5

Question........................................................................................................................................5

Applications of electric fields.............................................................................................................5

Questions.......................................................................................................................................6

Magnetic fields......................................................................................................................................8

Permanent Magnets..........................................................................................................................8

Electromagnets..................................................................................................................................9

Magnetic Force on Moving Charges..................................................................................................9

Charge moving perpendicularly to the B-field...............................................................................9

Charge moving parallel or anti-parallel to the B- field.................................................................10

Charge moving at an angle to the B-Field....................................................................................10

Application: How an electrical motor works................................................................................10

Questions.....................................................................................................................................11

Particle accelerators............................................................................................................................13

Energy Units....................................................................................................................................13

Linear Accelerators..........................................................................................................................13

Cyclotron.........................................................................................................................................13

Synchrotron.....................................................................................................................................14

Questions.....................................................................................................................................14

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Unit Specification

2. Forces on Charged Particles a) Electric fields around charged particles and between parallel plates. b) Movement of charge in an electric field, p.d. and work, electrical energy.c) Charged particles in a magnetic field.d) Particle accelerators.

Notes a) Examples of electric field patterns include single point charges, systems of two point charges and the field

between parallel plates. No calculation of electric field strength required.b) The relationship between potential difference, work and charge gives the definition of the volt. Calculating

the speed of a charged particle accelerated in an electric field.c) A moving charge produces a magnetic field. The direction of the force on a charged particle moving in a

magnetic field should be described for negative and positive charges. No calculations required.d) Basic operation of particle accelerators in terms of acceleration, deflection and collision of charged particles.

Contexts a) Hazards, e.g. lightning, static electricity on microchips.b) Precipitators. Xerography. Paint spraying. Ink jet printing. Electrostatic propulsion.c) Accelerators include linear accelerator, cyclotron and synchroton. Medical applications of cyclotron.

Accelerators used to probe structure of matter.

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Electric FieldsIn physics, a field means a region where an object experiences a force without being touched. For example, there is a gravitational field around the Earth. This attracts masses towards the centre of the Earth. Magnets cause magnetic fields and electric charges have electric fields around them.

A charged particle in an electric field will experience a force. Lines of force show the strength and direction of the force. The closer the field lines the stronger the force. Field lines are continuous – they start on positive and finish on negative charges. The arrows show the direction of travel of a positive test charge placed in the field.

Electric field patterns for a point charge.

Positive point charge

+

Negative point charge

-

These are called radial fields. The lines are like the radii of a circle. The strength of the field decreases as we move away from the charge.

Electric field pattern between parallel plates.

The field lines are equally spaced between the parallel plates. This means the field strength is constant. This is called a uniform field.

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The electric field for two point charges close to one another.

Positive test charge experiences a force ‘outwards’

Positive test charge experiences a force ‘inwards’

Comparing electric and gravitational fieldsElectric fields have certain similarities with gravitational fields.

Gravitational fields

Gravity is always an attractive force. All masses attract all other masses.

If a mass is lifted or dropped through a height then work is done i.e. energy is changed from one form to another.

If the mass is dropped then EP → EK. (Work done by the field)

If the mass is lifted then EW → EP. (Work done against the field)

Change in gravitational potential energy = work done

Electric fieldsConsider a positive charge moved through a distance in an electric field. If the charge is allowed to move freely in the electric field (towards the negative plate), it will experience a force and accelerate towards the negative plate and gain kinetic energy. Work is being done on the charge by the electric field.

If a positive charge is moved against the direction of the electric field (towards the positive plate), work must be done and it will gain potential energy (c.f. a mass being rolled uphill). Work is being done against the electric field.

Change in electric potential energy = work done

If the charge moved is one coulomb, then the work done is the potential difference or voltage.

If one joule of work is done in moving one coulomb of charge between two points in an electric field, the potential difference (p.d.) between the two points is one volt (one volt = one joule per coulomb).

EW = QV

EW is the work done in joules. (J)(Work done is the energy transferred from one form to another.) Q is the charge in coulombs. (C) V is the potential difference in volts.(V)

Question

A positive charge of +3 µC is moved from A to B across a potential difference of 10 V.

a. Calculate the electric potential energy gained.b. If the charge is now released, state the energy change.c. How much kinetic energy will be gained on reaching the negative plate?

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Moving charges in electric fieldsFrom the previous example, when the positive charge is released at plate B then the electric potential energy is converted to kinetic energy.

QV = ½mv2

QuestionAn electron is accelerated from rest through a potential difference of 200 V. Calculate:

a. The kinetic energy gainedb. The final speed of the electron

(Mass of an electron=9.11 x 10-31 kg; charge on an electron = -1.6 x 10-19 C

Applications of electric fields

An old-style CRT (cathode ray tube) television uses electron guns. The electrons gain kinetic energy by accelerating through an electric field. Deflection of the electrons is usually done by electromagnetic coils.

A cathode ray oscilloscope (CRO) also depends on electric fields acting on electrons.

Electrostatic spraying makes use of electric fields. Paint or powder particles are blown from a nozzle, where they acquire a charge. The charged paint or powder particles follow the field lines and so reach the object, some reaching the back of the object as well as the front.

Other applications include photocopiers, ink jet and laser printers.

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QuestionsUse the values below for some of the following questions: Charge on electron (and proton) = 1.6 x 10-19 CMass of electron = 9.1 x 10-31 kgMass of Proton = 1.67 x 10-27 kg

1. Draw the electric field pattern for the following point charges and pair of charges:

a.

b.

c.

2. Describe the motion of the charge shown in each of the following field.

3. An electron volt (eV) is a unit of energy. It represents the change in potential energy of an electron that moves through a potential difference of 1 V. What is the equivalent energy of 1 eV in joules?

4. An electron has energy of 5 MeV. Calculate its energy in joules.

5. In the diagram, an electron accelerates from parallel conducting plate A to B.

The p.d. between the plates is 500V.a. Calculate the electrical work done in moving the electron from plate A to B.b. How much kinetic energy has the electron gained in moving from A to B?c. What is the speed of the electron just before it reaches plate B?

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6. Electrons are ‘fired’ from an electron gun at a screen. The electron gun and screen are in a vacuum. The p.d. across the electron gun is 2000 V. After leaving the positive plate the electrons travel at a constant speed to the screen. Calculate the speed of the electrons just before they hit the screen.

----

++++

- e

Electron gun Screen

7. A proton is accelerated from rest across a p.d. of 400 V. Calculate the increase in speed of the proton.

8. In an X-ray tube electrons forming a beam are accelerated from rest and strike a metal target. The metal then emits X-rays. The electrons are accelerated across a p.d. of 25 kV. The beam of electrons forms a current of 3·0 mA.

a. Calculatei. The kinetic energy of each electron just before it hits the target.

ii. The speed of an electron just before it hits the target.iii. the number of electrons hitting the target each second.

b. What happens to the kinetic energy of the electrons?

9. Sketch the paths whicha. an alpha-particleb. a beta-particlec. a neutron

would follow if each particle, with the same velocity, enters the electric fields below.

- - - -

Path of particle----

++++

Path of particle

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Magnetic fieldsModern electromagnetism began in 1819 with the discovery by the Danish scientist Hans Oersted that a current-carrying wire can deflect a compass needle. This is explained by the fact that when a charged particle moves a magnetic field is created. In other words, a magnetic field will exist around any current carrying conductor. (RH rule.)

Twelve years afterwards, Michael Faraday and Joseph Henry discovered (independently) that a momentary e.m.f. (potential difference) existed across a circuit when the current in a nearby circuit was changed. Also it was discovered that moving a magnet towards or away from a coil produced an e.m.f. across the ends of the coil. Thus the work of Oersted showed that magnetic effects could be produced by moving electric charges and the work of Faraday and Henry showed that an e.m.f. could be produced by moving magnets.

All magnetic phenomena arise from moving electric charges. Since electrons are in motion around atomic nuclei, individual atoms of all the elements exhibit magnetic effects. In some metals like iron and nickel these small contributions from atoms can be made to 'line up' and produce a detectable magnetic effect.

Permanent Magnets The field lines for some combinations of bar magnet are shown below..

The end of a magnet that points geographically north is the 'magnetic north' pole A magnetic north pole will point towards the magnetic south pole of a bar magnet. Thus a compass needle will show the direction of the magnetic field at any point in the field.

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Direction of Magnetic Field (B)

Direction of motion of charged particle (v)

Electromagnets A solenoid is made by winding a coil of wire around an iron core. The magnetic field of a solenoid or electromagnet is the same as a bar magnet.

Magnetic Force on Moving Charges A magnetic field surrounds a magnet. When two magnets interact, they attract or repel each other due to the interaction between the magnetic fields surrounding each magnet.

A moving electric charge will behave like a mini-magnet as it creates its own magnetic field. This means it will experience a force if it moves through an external magnetic field

Charge moving perpendicularly to the B-fieldConsider a charge +q moving with velocity v, perpendicular to a magnetic field of magnetic induction B.

The direction of the force on a positively charged particle is given by the Fleming’s left hand rule:

Point the First finger of the left hand in the direction of the Field (to the left for this example).

Point the seCond or Centre finger in the direction the Current i.e. the direction of motion of the positive charge.

The THumb at right angles to the first finger (as if shooting a pretend gun) will point in the direction of the THrust or force (out of the page for this example).

(A moving negative charge or electron will constitute a negative current. So the centre finger will point in the opposite direction to the motion of an electron.)

Change of direction due to the magnetic force on a charged particleThe direction of the force F is perpendicular to the plane containing v and B.

A particle travelling at constant speed under the action of a force at right angles to its path will move in a circle.

The sketch below shows this situation for a negatively charged particle. (The X indicates that the direction of the field is 'going away' from you 'into the paper'; motion ‘out of the paper’ is shown by a dot).

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If you apply the left-hand rule at point P you will find that the force is to the left.

At point R the particle has begun to turn to the left, but by applying the left-hand rule again we see that the force is now diagonally towards the top left of the paper.

So the force causes the particle to change direction, but the change of direction of motion causes the direction of the force to change so that it remains perpendicular to the direction of motion.

The result is that the charged particle will move in a circle, of radius r. The magnetic force supplies the central acceleration, and maintains the circular motion.

Charge moving parallel or anti-parallel to the B- field If the angle θ, between the direction of travel of the particle and the B-field, is zero then the particle never crosses any field lines. It does not experience any force and the direction of the charged particle is not altered.

Charge moving at an angle to the B-FieldIf the charge q is moving at an angle θ to the field, where 0 < θ < 90o, then only the component of the velocity v that is perpendicular to the field (v sinθ) produces the force. At Higher you will only be asked about charges which are perpendicular or parallel to the B-field.

Application: How an electrical motor worksIn a simple electrical motor, a coil of wire is placed in a magnetic field between two magnets. The coil of wire is attached to a thin metal rod which is free to rotate about a central axis. This is shown in the image below.

Initially the circuit is switched off and no current flows through the wire. There is no movement of the motor.

When the circuit is switched on, electric charges flow in the coil of wire.

The conventional current flow is away from the cell on the right hand side of the coil and back to the cell on the left hand side of the coil.

The direction of the magnetic field is from left to right (the left hand magnet is marked N and the field direction goes from N to S).

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The right hand side of the coil, where the current is going in to the page (away from the reader) experiences a downwards force. In contrast, the left hand side of the coil (where the current is coming out of the page) experiences an upwards force.

This torque causes the coil of wire to rotate around a central axis. Almost all commercial electric motors work on this simple principle. (Commercial motors have multiple coils which are arranged at different angles to each other to ensure that the motor turns smoothly.)

This section showed how understanding the force on charged particles led to the development of electric motors. The next section shows how understanding the forces on charged particles led to the development of powerful particle accelerators that have dramatically changed our understanding of the world around us.

Questions1. An electron travelling with a constant velocity enters a region where there is a uniform magnetic field. There

is no change in the velocity of the electron. What information does this give about the magnetic field?

2. The diagram shows a beam of electrons as it enters the magnetic field between two magnets.

The electrons will:

A. be deflected to the left (towards the N pole) B. be deflected to the right (towards the S pole) C. be deflected upwards D. be deflected downwards E. have their speed increased without any change in direction.

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NS SN

beam ofelectrons

3. The diagrams show particles entering a region where there is a uniform magnetic field. Use the terms: up, down, into the paper, out of the paper, left, right, no change in direction to describe the deflection of the particles in the magnetic field

4. An electron enters a region of space where there is a uniform magnetic field. As it enters the field the velocity of the electron is at right angles to the magnetic field lines. The energy of the electron does not change although it accelerates in the field. Use your knowledge of physics to explain this effect.

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(h)magnetic field

xx xxxxx xxxxx xxxxx xxxxx xxx

proton

(g)magnetic field

xx xxxxx xxxxx xxxxx xxxxx xxx

electron

(f)

magnetic field

proton

(e

magnetic field

electron

magnetic field

proton

(d

magnetic field

neutron

(c)

alpha particle

magnetic field

(b)

magnetic field

electron

Particle accelerators

Energy UnitsIt is usually more useful to talk about the kinetic energy of a particle, than to talk about its speed. However the Joule is too large a unit to be conveniently used when discussing the energy of a single particle.

Imagine a single electron accelerated across a potential difference of one volt.

EW = Q V so the electron gains E = 1.6 x 10-19 x 1 = 1.6 x 10-19 J of energy.

This quantity is called one electron-volt: 1 eV = 1·6 x 10-19 J

Linear AcceleratorsIn a CRT the electrons accelerate because they are attracted to an anode (positive). We could accelerate them to higher speeds if this became a cathode (negative) as the electrons passed it, repelling them in their existing direction of travel. We could accelerate them still further with a whole sequence of anodes and cathodes, constantly switching sign, each of which would attract the approaching electrons and repel the departing ones. Virtual Physics has an excellent animation explaining this effect.

This is exactly what a linear accelerator does, except that particles other than electrons can also be accelerated. By causing high-velocity (therefore high kinetic energy) beams of charged particles to collide, we can study their interactions. The high-energy particles are also useful in themselves, for example in the treatment of some cancers.

Linear accelerators can be several kilometres long. The energies that can be obtained are limited by the available length of the accelerator and by the maximum switching frequency of the electric fields (since these have to switch faster and faster as the speed of the particles increases).

When a beam of particles has been accelerated, it can be deflected to a target using an electric field. Collisions can be caused and studied by deflecting different beams to the same point.

Could we reach higher speeds by somehow reflecting the particles back along the accelerator?

CyclotronA cyclotron is a compact type of particle accelerator, often used as the first stage in larger installations. It uses magnetic fields to change the direction of the particles and electric fields to accelerate them.

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The two-semi-circular areas are called dees due to their shape. A magnetic field is normal to the dees.

A particle in a magnetic field like this would travel in circles. However in the cyclotron, there is a gap between the two dees. An alternating electric field accelerates the electrons across the gap. As they accelerate, the circle they describe becomes bigger. The result of combining these two effects is that the electrons travel in an expanding spiral, gaining speed and energy until they finally leave the dees and hit the target.

Each time an electron travels through a dee, it travels a greater distance, but the time for the journey remains the same. So the electric field can alternate at a steady rate which does not depend on the speed of the electrons. However as the speed of the electrons approaches 0.1c, relativistic corrections have to be made and the frequency is no longer constant. One of the factors limiting the energies that can be reached with a cyclotron, is that the required frequency of the electric field becomes too high to be achievable. Other practical limitations include the size of the dees and the size of magnet needed.

A good animation can be found at: http://webphysics.davidson.edu/physlet_resources/bu_semester2/c13_cyclotron.html

SynchrotronThe Large Hadron Collider at CERN is a synchrotron. The particles travel around a large hollow ring. Alternating electric fields increase the speed of the particles, while magnetic fields keep them in a circular path. The strength of the magnetic fields change as the particles gain kinetic energy, in order to provide sufficient force to keep them on their path. Particles travel around the ring many times in order to gain more energy.

The larger the ring and the stronger the magnets, the greater the particle energy. Superconducting magnets can be used to overcome the limits of ordinary magnets. A limitation of the energies that can be reached by this design is the phenomenon of synchrotron radiation – as they travel in a circle, the particles emit radiation. Since energy must be conserved, this leads to a loss of kinetic energy. The LHC is currently the world’s most powerful particle accelerator. Protons can reach energies of up to 7 TeV (7 x 1012 eV). By sending two beams round in opposite directions, collisions can have energies of 14 TeV.

Linear Accelerator questionIn order to send the particles back the way they had come, we would need to bring them to a halt – so the kinetic energy from the first transit would all be dissipated. We would start the return journey back at a speed of zero.

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QuestionsIn the following questions, when required, use the following data:

Charge on electron = –1·60 × 10-19 C Mass of electron = 9·11 × 10-31 kg

Charge on proton = 1·60 × 10-19 C Mass of proton = 1·67 × 10-27 kg

1 In an evacuated tube, an electron initially at rest is accelerated through a p.d. of 500 V.a. Calculate, in joules, the amount of work done in accelerating the electron.b. How much kinetic energy has the electron gained?c. Calculate the final speed of the electron.

2 In an electron gun, electrons in an evacuated tube are accelerated from rest through a potential difference of 250 V.

a. Calculate the energy gained by an electron.b. Calculate the final speed of the electron.

3 Electrons in an evacuated tube are ‘fired’ from an electron gun at a screen. The p.d. between the cathode and the anode of the gun is 2000 V. After leaving the anode, the electrons travel at a constant speed to the screen. Calculate the maximum speed at which the electrons will hit the screen.

4 A proton, initially at rest, in an evacuated tube is accelerated between two charged plates A and B. It moves from A, where the potential is 10 kV, to B, where the potential is zero. Calculate the speed of the proton at B.

5 A linear accelerator is used to accelerate a beam of electrons, initially at rest, to high speed in an evacuated container. The high- speed electrons then collide with a stationary target. The accelerator operates at 2·5 kV and the electron beam current is 3 mA.

a. Calculate the gain in kinetic energy of each electron.b. Calculate the speed of impact of each electron as it hits the target.c. Calculate the number of electrons arriving at the target each second.d. Give a reason for accelerating particles to high speed and allowing them to collide with a target.

6 The power output of an oscilloscope (cathode-ray tube) is estimated to be 30 W. The potential difference between the cathode and the anode in the evacuated tube is 15 kV.

a. Estimate the number of electrons striking the screen per second.b. Calculate the speed of an electron just before it strikes the screen, assuming that it starts from rest

and that its mass remains constant.

7 In an oscilloscope electrons are accelerated between a cathode and an anode and then travel at constant speed towards a screen. A p.d. of 1000 V is maintained between the cathode and anode. The distance between the cathode and anode is 5·0 × 10-2 m. The electrons are at rest at the cathode and attain a speed of 1·87 × 107 m s-1 on reaching the anode. The tube is evacuated.

a.i. Calculate the work done in accelerating an electron from the cathode to the anode.

ii. Show that the average force on the electron in the electric field is 3·20 × 10-15 N.

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iii. Calculate the average acceleration of an electron while travelling from the cathode to the anode.

iv. Calculate the time taken for an electron to travel from cathode to anode.v. The anode to screen distance is 0·12 m. Calculate the time taken for an electron to travel

from the anode to the screen.b. Another oscilloscope has the same voltage but a greater distance between cathode and anode.

i. Would the speed of the electrons be higher, lower or remain at 1·87 × 107 m s-1?Explain your answer.

ii. Would the time taken for an electron to travel from cathode to anode be increased, decreased or stay the same as in (a) (iv)? Explain your answer.

8 In an X-ray tube a beam of electrons, initially at rest, is accelerated through a potential difference of 25 kV. The electron beam then collides with a stationary target. The electron beam current is 5 mA.

a. Calculate the kinetic energy of each electron as it hits the target.b. Calculate the speed of the electrons at the moment of impact with the target assuming that the

electron mass remains constant.c. Calculate the number of electrons hitting the target each second.d. What happens to the kinetic energy of the electrons?

9 On the same diagram shown below sketch the paths that (a) an electron, (b) a proton and (c) a neutron would follow if each particle entered the given electric fields with the same velocity.

10 In the following examples identify the charge of particle (positive or negative) which is rotating in a uniform magnetic field. (X denotes magnetic field into page and • denotes magnetic field out of page.)

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+ + + + + + + + + + +

Path of particles

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XXXX

XXXX

XXXX

XXXX

(d)(c)

(b)(a)

XXXX

XXXX

XXXX

XXXX

(a)

11 In the following descriptions of particle accelerators, some words and phrases have been replaced by the letters A to R.In a linear accelerator bunches of charged particles are accelerated by a series of ____A____. The final energy of the particles is limited by the length of the accelerator.This type of accelerator is used in ____B____ experiments.In a cyclotron the charged particles are accelerated by ____C____. The particles travel in a ____D____ as a result of a ____E____, which is ____F____ to the spiral. The radius of the spiral increases as the energy of the particles ____G____. The diameter of the cyclotron is limited by the ____H____ of the magnet. The resultant energy of the particles is limited by the diameter of the cyclotron and by ____I____. This type of accelerator is used in ____J____ experiments.In a synchrotron bunches of charged particles travel in a ____K____ as a result of C shaped magnets whose strength ____L____. The particles are accelerated by ____M____. As the energy of the particles increases the strength of the magnetic field is ____N____ to maintain the radius of the path of the particles. In synchrotron accelerators the particles can have, in theory, an unlimited series of accelerations as the particles can transit indefinitely around the ring. There will be a limit caused by ____O____.In this type of accelerator particles with ____P____ mass and ____Q____ charge can circulate in opposite directions at the same time before colliding. This increases the energy of impact. This type of accelerator is used in ____R____ experiments.

From the table below choose the correct words or phrases to replace the letters.

Letter List of replacement word or phrase

A, C, E, M constant magnetic field, alternating magnetic fields, alternating electric fields, constant electric fields

B, J, R colliding-beam, fixed-target

D, K spiral of decreasing radius, spiral of increasing radius, circular path of fixed radius

F perpendicular, parallel

G decreases, increases

H physical size, strength

I, O gravitational effects, relativistic effects

L can be varied, is constant

N decreased, increased

P, Q the same, different

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