A2 Physics


Magnetic Fields and Induction

Picture
Iron filings around a permanent magnet.

Contents

    Magnetic field lines
    Magnetic flux density, B
    The motor effect
    The electric motor
    Moving charges in magnetic field
    Mass spectrometry
    The cyclotron
    Electromagnetic induction
    Faraday and Lenz's law
    Magnetic flux and linkage
    AC generator
    Transformers

Magnetic Field Lines

A magnetic field is a region surrounding a magnet or current carrying wire which acts on any other magnet or current carrying wire placed in the field. Magnetic field lines always form loops. Some diagrams depict open-ended field lines, however, these always connect up if the diagrams was drawn fully.
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Field lines always point from north to south; even within the magnet itself.
Picture
Iron filings lining up along magnetic field lines in a bar magnet.
Below are 2 diagrams of the field lines of magnets attracting and repelling.
Picture
Field lines for magnets attracting.
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Field lines for magnets repelling.

Magnetic Flux Density, B

"Magnetic flux density, B, is the force, F, per unit length, l, per unit current, I, on a current carrying conductor at right angles to the magnetic field. It is otherwise known as the magnetic field strength. The unit of measurement is the tesla (T)". 

The Motor Effect

When a current passes along a wire in a magnetic field, a force is exerted on the wire. This is called the motor effect. The force on the wire depends on:
        • The current in the wire and its direction
        • Strength of the magnet
        • Length of the wire
        • Angle of the wire/current relative to the magnetic field lines  

The arrows in the diagram above show the conventional current. In truth, electrons move from negative to positive, however, by convention, the current is said to go from positive to negative (this is the way its always been done and this is the way it (probably) always shall be, there is a lot at stake if this changes).

The direction of the force can be found using Fleming's left hand rule. The left hand is positioned such that the thumb, first and second finger are all perpendicular to one another:
Picture
A very famous hand...
To increase the force on the wire you can:
  • increase the current
  • increase the magnetic field strength
  • ensure the current is exactly perpendicular to the magnetic field lines.

The direction of the force is reversed if either the direction of the current or magnetic field is reversed.

The Electric Motor

The electric motor works on the principles of the motor effect. A current carrying coil is placed in a magnetic field. The current enters the field going in one direction and returns through the field in the opposite direction. This change in direction means that the force on the two opposite sides of the coil are in opposite direction; this makes the coil spin. The slip ring commutators (SLCs) allow the coil to spin and slip past the circuit contacts without the wires twisting up and tangling. The SLCs are normally made of conducting brushes that allow the wires to be in contact for as long as possible.
When using a coil of wire the effect of the force is magnified by the number of coils, n, there are:
F (N) = force on the wire
B (T) = magnetic field strength
l (m) = length of wire within the magnetic field
n = number of turns in the coil

Moving Charges in Magnetic Fields

The force, F, on a charge, q, moving in an, electric field, E, and a magnetic field, B, with a speed, v, is given by the Lorentz force equation:
You do not need to know the meaning of this equation for A-level. F, E, v and B are all underlined meaning that they are all vectors because they are all quantities that have direction. vxB is know as a 'cross product' which accounts for charges moving through a magnetic field that aren't perpendicular. 

Often we either only deal with a magnetic field or an electric field. You can see that when the magnetic field is zero, we have the definition of an electric field F=qE. When there is only a magnetic field and the direction in which the charge is moving is at right angles to the magnetic field then the force, F, on that charge is given by the equation: 

Charges in Circular Motion

Magnetic force is always perpendicular to the direction of motion of a charge (this is the vxB from the Lorentz force equation). This means that the force on a moving charged particle in a magnetic field is centripetal. If the force on a moving charge in a magnetic field is F=Bqv, and the centripetal force is:
From this we can rearrange to find an expression for the radius of curvature, r, of the circling charge:
From this we can see that 2 charges moving at the same speed through the same magnetic field; if one of them has more mass it will have a larger radius of curvature i.e. because it has a greater mass its inertia is greater and thus its direction is 'more difficult' to change.
Picture
Opposite charges deflect in opposite directions. Same mass, same radius of curvature.
Picture
Larger mass with equal and opposite charge, larger deflected less.

With a strong enough magnetic field electrons can be made to move in a perfect circle. In the photo below tesla coils provide the uniform magnetic field in within which the electrons in the beam are undergoing circular motion.

Picture
A Thomson tube.

Mass Spectrometry

A mass spectrometer uses the principles mentioned above. A mixture of isotopes of differing molecular weights can be separated due to their differing radius of curvature. The diagram below shows
The diagram shows that CO2 molecules can have different masses due to varying isotopes of carbon i.e.  C-12, C-13 and C-14 will mean that their respective CO2 molecules have an atomic mass of 44, 45 and 46 respectively. This is due to the radius of curvature of a charged particle in a magnetic field being directly proportional to the mass.

The Cyclotron

The cyclotron was one of the earliest types of particle accelerators. It makes use of the magnetic force on a moving charge to bend moving charges into a semicircular path between accelerations by an applied electric field. The applied electric field accelerates electrons between the 'dees' of the magnetic field region. The field is reversed at the cyclotron frequency to accelerate the electrons back across the gap.

Each time the particle crosses the gap it gains speed and increases its radius of orbit. It gains speed because the electric field. The particle's speed only increases across the gap. The time, t, spend in the region of one of the dees is:
Using the radius equation from above for a charged particle in a magnetic field the equation becomes:
Therefore, the time it takes to complete one whole revolution is 2t which we can call simply the period, T. The cyclotron frequency, f, is therefore:

Electromagnetic Induction

Electromagnetic (EM) induction = when an emf is induced in a wire when a complete loop of wire cuts across lines of a magnetic field. When the wire is part of a complete circuit/loop a current will flow. The emf can be increased by moving the wire faster, using a stronger magnet, making wires into a coil and ensuring that the wires cut across the field at 90 degrees.

Some More Magnetic Fields...

Picture
Right hand 'screw' rule for B field around a current carrying wire.
Picture
Magnetic field lines around a solenoid. The crosses represent wires going into the page, the dots out of the page.


​The poles of a solenoid can be determined using the right hand screw rule and knowing that field lines always come out of the north pole and into the south.
Electron beam fired perpendicularly into the magnetic field. Fleming's left hand rule can be used to show that the force on the electron in the field is downwards. Remember: the conventional current is actually in the opposite direction to the direction that the electron is moving.

If it was a positive charge moving, the conventional current is in the direction of the flowing positive charge.




When confined to a wire a p.d is induced and electrons will flow in a complete circuit.
 

Faraday and Lenz's Law

Faraday’s law = “the induced emf in a circuit is equal to the rate of change of magnetic flux linkage through the circuit”.
 
Lenz’s law = “the direction of the induced current is always such that it opposes the change that causes the current”.

Lenz's Law

When a bar magnet is pushed into a coil connected to an ammeter the meter deflects. Pulled out of the coil the meter deflects in the opposite direction. The induced current passing round the circuit creates a B field around the coil. The coil field must act against the incoming north pole otherwise it would pull the north pole in faster = not allowed (conservation). 
 



As magnet enters coils a current is induced producing a B field in the coil. The top of the coil becomes a north pole.

This B field opposes the motion of the bar magnet.

As the magnet leaves its south pole induces a north pole on the bottom of the coil

1) As the magnet enters it generates a current in the loop  that sets up a magnetic field to oppose the entry of the magnet.

2) When the magnet is in the middle of the coil the it is at the point where the magnetic poles will switch. At this point there is no current flowing in the coil since the p.d is zero. 

3) When exiting the coil the magnet is moving faster (because of its acceleration due to gravity) and it induces a current in the loop that sets up a magnetic field to oppose the magnet moving away i.e. the magnetic poles of the coil change.
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The p.d. as the magnet enters the coil is lower than when it exits the coil. This i because the magnet is moving faster as it exits and thus the rate of change in magnetic flux linkage (i.e. and hence the induced emf) is greater when it exits. The p.d. changes direction because the current and the magnetic fields both switch to oppose the changes occuring (as explained above).
Picture
From Feynman's lectures on physics
Picture
From Feynman's lectures on physics
In the demonstration above the left pendulum is made of a complete sheet of copper. This will allow eddy currents to flow through the sheet generating a magnetic field that will oppose the motion of the pendulum, slowing it down. On the other hand, the diagram on the right  has slits in it, considerably reducing the possibility of strong eddy currents thus generating only a weak magnetic field which opposes its motion. The pendulum with the slits will therefore swing for longer.
This setup is synonymous with the one directly above.

The first magnet induces a current and therefore an opposing magnetic field, slowing its motion as it drops through the complete loop.

The ring that does not form a complete loop will generate no current flow. In which case there is no opposing magnetic field that is set up by the dropping bar magnet.

The second magnet will therefore drop faster than the first one since there are no magnetic fields opposing its motion downwards.

Faraday's Law

A length of wire, l, is part of a complete circuit cutting through a magnetic field of flux density, B.

The conductor experiences a force of F=BIl. The force opposes the motion (Lenz's law).

An equal and opposite force is needed for constant speed in the field.
Work done, W, by the applied force to move the wire a distance, Δs, is:
Charge transferred along the conductor in this time is:
So the induced emf is:
therefore, in a conductor being moved  perpendicularly to a B field at constant speed:

Magnetic Flux and Linkage

Magnetic flux, Φ, is defined as:
The unit of magnetic flux is the Weber. Where 'A' is the area sweeped out = lΔs.

The magnetic flux linkage through a coil of N turns is:
To reiterate Faraday and Lenz's law. The induced emf in a circuit is equal to the rate of change of magnetic flux linkage through the circuit (Faraday). The direction of the induced current is always such that it opposes the change that causes the current (Lenz). The opposing change is represented by the negative sign in the equation:

AC Generator

A simple AC generator consists of a rectangular coil which is forced to spin in a magnetic field. The creates a change of magnetic flux through the coil, which generates an emf, which drives a current. The faster the rate of change of flux (i.e. spinning faster) or the greater the number of turns in the coil the larger the emf induced.

When rotating faster the peak amplitude for the current is maximum and the alternating current has the highest frequency.
As the coil spins, the flux linkage changes continuously (due to the angle at which the coil is at). Flux linkage through the coil is given by:
The area is the 'flat area' to the field. When the loop lies 'flat' (i.e. parallel to the magnetic field) then the magnetic flux linkage is maximum because sin(90)=1.When the loop is flat the coil cuts across the magnetic field lines and moves at maximum relative speed doing this. When the coil is upright there is no change in magnetic flux (i.e. emf = 0) because the coil isn't 'cutting across' the field lines.
 
If the flux linkage changes then the emf will change:
Where t is the time, f is the frequency and w is the angular velocity.
The induced emf is zero when the coils are perpendicular to the field lines and maximum when they are parallel. Remember, the induced emf is that rate of change in magnetic flux linkage. As the coil cuts perpendicularly through the magnetic field lines this will generate the maximum rate of change of flux linkage i.e. when the coils are flat.

Transformers

The iron core serves to magnify the effect of the field produced around the primary coil since iron is easily magnetised and its poles easily switched.

Since the direction of the current alternates the magnetic field changes direction (i.e. clockwise-anticlockwise-clockwise-anticlockwise, etc.)

This alternating field passes through the secondary coil. Since the B field alternates then an emf will be produced in the secondary coil. This is because an emf is produced when there is a rate of change in magnetic flux, which there is!
Picture
Increased number of turns the secondary coil = step up transformer
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Decrease number of turns the secondary coil = step down transformer
The flux linkage in the secondary coil = NSΦ and the flux linkage in the primary coil = NPΦ. From Faraday's law the induced emf in the primary and secondary coils are:
Dividing the two equations:

Efficiency

Transformers are highly efficient because:

        1. Low resistance windings to reduce energy loss through heat. 
        2. Laminated core. Produces internal eddy currents aimed at increasing magnetic flux. Energy loss through heat reduced. 
        3. Soft iron core used. Easily magnetically ‘switched’. Reduces power (and therefore energy) loss.

The efficiency of a transformer is the ratio of power output to power input: