The Motor Effect

… The Force on a current carrying conductor in a magnetic field can be found by using Flemings Left Hand “Motor” Rule (“Motor cars drive on the left”). Remember to start with “Fi Fi Fi”, First Finger Field.

… Magnetic field lines are called magnetic ‘Flux’ measured in Webers (Wb).

… Magnetic flux density, B = Flux / Area

… Magnetic force on a moving charge or current:

Force = B I L = B e v (e is the charge on an electron, sometimes this letter is ‘q’ for charge on a general object)

… If the conductor, coil or charged particle is at an angle A, other than 90° to the magnetic field, then:

F = B I L sinA

or

F = B q v sinA

… The force on a charge moving in a magnetic field can provide the centripetal force necessary for it to move in a circle. This is a favourite question in magnetic fields! This means that:

Bqv = (mv²)/r

The Cyclotron

… The cyclotron is a particle accelerator made up of two D-shaped hollow electrodes. Charged particles are accelerated across the gap between the two back-to-back electrodes by an alternating potential difference with a constant frequency.

A magnetic field acts through the D-shapes perpendicular to the direction of motion of the particles, so this makes the charged particles move in circular motion where:

Bqv = (mv²)/r

And so

r = mv/Bq

As the particle’s velocity increases, the radius of circular motion increases and so the particle describes larger and larger circles within each D-electrode.

The time taken for the particle to describe a semicircle in one of the Ds is given by:

t = πr/v (half a circumference ÷ speed)

and so

v = πr/t

Substituting this into the above equation gives:

Bqr = mπr/t

and so

t = mπ/Bq , which means that the time period for a compete revolution through both D section is constant:

T = 2mπ/Bq

The period is completely independent of the velocity of the particle and is constant for EVERY revolution. This means that the frequency of the alternating voltage applied to the D sections can also be constant, given by:

f = 1/T = Bq/2mπ

Finally, when the radius of curvature is large enough, the particles escape the D sections travelling at high velocity.

Note that electromagnetic radiation is given off by the constantly accelerating charged particles. A microwave oven, however, uses a different approach to generating microwave energy called a magnetron.

The Synchrotron

… A synchrotron is also a particle accelerator in which the accelerating particle beam travels around a fixed closed-loop path (a large, thin torus).

- There are two main types of synchrotrons: high energy physics machines (e.g. the Large Hadron Collider at CERN) and sources of synchrotron radiation.
- The magnetic field, which keeps the particles in circular motion, increases with time during the accelerating process (it is synchronised to the increasing kinetic energy of the particles).
- Acceleration of the particles is achieved via application of radio frequency fields at cavities along the circumference of the ring.
- A synchrotron structure is often that of a many-sided polygon, rather than a continuous circle.
- the maximum energy that a cyclic accelerator can give the particles is limited by the maximum magnetic field strength to keep the particles in circular motion.
- pre-accelerated particle beams are injected into a synchrotron (eg. from a cyclotron)
- As the charged particles are constantly accelerated towards the centre of their circular motion, they emit radiation (which happens when any charged particle is accelerated). This Synchrotron X-ray Radiation is very intense – millions of times ‘brighter’ than convention sources and is often used to study atomic structures of materials, as well as in aiding drug development and biological investigations of proteins, viruses and vaccines.

Linear Accelerators (‘Linac’)

- Charged particles are accelerated along a straight evaculated tunnel by a set of switched anodes which provide a moving accelerating field.
- Cheaper than circular accelerators because no magnetic field is necessary to bend the particles.
- The Stanford Linear Accelerator in California is 3km long and accelerates electrons up to energies of 50GeV.
- Electrons are provided from a heated filament.
- A high frequency radio signal is fed to each of 100,000 electrodes at a continuously changing frequency. In effect, the electrons are surfing along the switching voltage ‘wave’.
- There are more efficient at feeding energy to electrons compared to circular accelerators because no synchrotron radiation is emitted.

Velocity Selector

… If a charged particle (such as an ion in a mass spectrometer) is made to move through a perpendicular magnetic field AND a perpendicular electric field, then there will be two forces acting on it – one upwards and one downwards. If the velocity is just right, then the two forces will cancel each other out and the particle will travel in a straight line.

A collimator plate then allows the straight-line particles to go through a hole while stopping any other particles that have deviated off course. We have effectively filtered only those particles which are travelling at a particular velocity.

Faraday’s law of magnetic induction

… An electromotive force (EMF, or potential difference) is induced when a conductor cuts through lines of magnetic flux (or vice versa).

… The right hand rule can be used to discover the direction of the induced (generated) current (remember ‘geneRIGHTor’!)

… A coil has an associated ‘Flux Linkage’ with an external magnetic field.

Flux Linkage = NΦ = BAN

If the normal to the coil’s plane is at an angle θ to the magnetic field passing through it, then:

Flux Linkage = NΦ = BANcosθ, because Φ = BAcosθ (the effective flux passing through the component of the area at angle θ)

This means that if θ = 90° then the Flux Linkage = 0

… The RATE of change of flux linkage of a coil is the emf induced in the coil. This can be written:

ε = Δ(NΦ) / Δt

So Faraday’s law is telling us about the MAGNITUDE of the induced EMF.

Another useful form of this equation for a coil rotating in a magnetic field is:

ε = BANωsin(ωt)

… When a coil rotates within a magnetic field at a constant angular velocity:

- The rate of change of flux linkage is ZERO when the coil is perpendicular to the B-field (flux linkage is maximum). This occurs when θ = 0°.
- The rate of change of flux linkage is MAXIMUM when the coil is parallel to the B-field (flux linkage is zero but changing rapidly). This occurs when θ = 90°

… So if the flux linkage of a coil in an external magnetic field is BAN, and if the magnetic field is switched off in time t (e.g. In a solenoid) then the emf induced in the coil = BAN/t

… Lenz’s law states that “The direction of current induced in a conductor by a changing magnetic field due to Faraday’s law of induction will be such that it will create a field that opposes the change that produced it.”

So Lenz’s law is really telling us about the DIRECTION of the induced EMF.

… When the coil of a motor turns due to the interaction of the coil’s magnetic field and the external magnetic field (the motor effect), the coil also starts to CUT through the external magnetic field lines.

This ‘cutting’ action induces an EMF which “opposes the change that caused it” (i.e. the coil’s motion). This is Lenz’s Law in action.

In practice, this means that the induced EMF opposes the motor’s power supply EMF, and so reduces the current flowing in the coil. The coil therefore reaches a maximum speed at which there is only a small current flowing.

When a load is applied to the motor:

the coil will slow down

The induced back EMF will decrease

The coil current will increase

A greater motor effect force will act on the coil to try and keep it turning against the load.

… ε = BLv can also be a useful equation to have in your magnetic fields ‘toolkit’, where:

L is the Conductor length

B is the flux density

v is the relative velocity of the conductor cutting through the field (or field cutting through the conductor)

Note that ε = BLv assumes that the conductor and field lines are cutting each other ‘head on’ at 90°, otherwise we would need to use trigonometry to find the effective component of the length at an angle θ.

… The magnetic field flux density, B, within a solenoid of length L, turns N, and which has a current I flowing in it is given by:

B = (μo)NI/L

μo is a constant called the permeability of free space (how easy it is for a magnetic field to establish in a vacuum or air)

Transformers

… The alternating primary voltage creates an alternating magnetic field (a changing flux, ΔΦ/Δt) which circulates in the transformer core, linking to the secondary coil:

Vp = Np ΔΦ/Δt

The changing flux induces an alternating EMF in the secondary coil:

Vs = Ns ΔΦ/Δt

As the rate of change of flux is the same for both coils, we can combine these two equation to give:

Vp/Vs = Np/Ns

… If a transformer is assumed to be 100% efficient, then:

Power In = Power Out

Vp Ip = Vs Is

… Induced eddy currents can cause heating and therefore energy losses (I²R losses) in the transformer core. This is reduced by laminating the core.

… Step up transformers are used to transmit electricity across the national grid at high voltage (400kV). The lower current in the overhead lines means that less energy is lost due to resistive heating.

… Do have your formulae sheet with you when you answer questions on B-fields :).