# AQA A-Level Physics/Magnetic Fields

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 diagram is fully drawn.

Magnetic Flux Density (B): Magnet 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 the 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
• The strength of the magnet
• The length of the wire within the magnetic field
• Angle of the wire/current relative to the magnetic field lines

The arrows in the diagram 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 it has always been done). The direction of the force can be found using Fleming’s left-hand rule.

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 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 forces on the two opposite sides of the coil are in opposite directions; meaning the coil begins to spin.

When using a coil of wire the effect of the force is magnified by the number of coils, n, that there are:

• $F$ = force on the wire
• $B$ = magnetic field strength
• $I$ = current flowing through the wire
• $l$ = length of wire within the magnetic field
• $n$ = number of turns

## 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:

This does not need to be known for the A level syllabus. F, E, v and B are all underlined meaning they’re vector quantities (University level equation. Won’t need this at all)

Most often, we deal with either 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

However, it must be remembered that this equation only works when the charge is travelling perpendicular to the magnetic field.

## Charges in Circular Motion

Magnetic force is always perpendicular to the direction of motion of a charge. This means that the force on a moving charged particle within a magnetic field is centripetal. If the force on a charge in a magnetic field is $F=Bqv$ , and centripetal force is ${\frac {mv^{2}}{r}}$ then we know that

From this, we can rearrange to find an expression to find the radius of curvature, r, of the circling charge

From this, we can learn that velocity and mass are directly proportional to the size of the radius and that the strength of the magnetic field and the size of the charge are inversely proportional to the radius. So, if the mass or the velocity is increased, then the radius will increase. Whereas if the charge or the strength of the magnetic field are increased, then the radius decreases as these variables are inversely proportional.

## The Cyclotron

The cyclotron was one of the earliest types of particle accelerators, making use of the magnetic force on a moving charge to bend moving charges into a semi-circular 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 orbital radius. It gains speed because of the electric field. The particle’s speed only increases across the gap. The time, t, spent 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 taken to complete one whole revolution is 2t which we can simply call the period, T. The cyclotron frequency, f, is therefore

## Electromagnetic Induction

Electromagnetic (EM) induction is when an emf is induced in a wire when a complete loop of wire cuts across lines of a magnetic field (the magnetic flux lines). When the wire is part of a complete circuit/loop a current will flow. The emf can be increased by moving the wire faster or using a stronger magnet. This is because the emf is induced by the rate of change of the flux linkage with the wire, and so if the time taken through the field decreases or the number of flux lines increase, this will directly cause an increase in the induced emf. However, it’s important to make sure the wire/solenoid cut across the field at 90 degrees.

Important to remember

- 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 a magnetic field. Conventional current is actually in the opposite direction to the direction that the electron is travelling in. If it was a positive charge moving, the conventional current is the direction of the flowing positive charge.

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 magnetic field around the coil. The coil field must act against the incoming North Pole other it would pull the North Pole in faster, which cannot occur otherwise energy would not be conserved, and infinite energy could be created.

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 then 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.

Also, the second peak as the magnet leaves should be larger, as due to the acceleration due to gravity, the magnet will be moving faster and so the amount of emf induced would be greater. Also, the direction of the p.d changes because the current and the magnetic field both switch to oppose the changes occurring, causing the p.d to flip.

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. An equal and opposite force is needed for a constant speed in the field

Work done, W, by the applied force to move the wire distance, Δs, is:

The charge transferred along the conductor in this time is:

So the induced emf is:

Magnetic flux, φ, is defined by:

The unit of magnetic flux is the Weber. Where A is the area swept out=lΔs

The magnetic flux linkage through a coil of N turns is:

## AC Generator

A simple AC generator consists of a rectangular coil which is forced to spin in a magnetic field. This creates a change of magnetic flux through the coil, which generates emf, driving a current. The faster the rate of change of flux 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 at its maximum and the alternating current has a higher frequency. As the coil spins, the flux linkage changes continuously (due to the angle at which the coil is). 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 cos (0) =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 the emf will change:

Where $t$ is time, $f$ is the frequency and omega is angular velocity

The induced emf is zero when the coils are perpendicular to the field lines and maximum when they’re parallel. Remember, the induced emf is the rate of change in magnetic flux linkage. As the coil cuts perpendicularly through the magnetic field lines, this will generate the maximum 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, e.t.c)

This alternating field passes through the secondary coil. Since the magnetic 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 is due to the constantly alternating magnetic field.

Efficiency:

Transformers are highly efficient because:

1. Low resistance windings reduced 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: