Magnetic fields (3.7.5)

A magnetic field is an area where a magnet or magnetically susceptible material will experience a non-contact force.

Magnetic field lines represent the size, shape and strength of a magnetic field.

It is important to note that magnetic field lines originate from the north pole of a magnet and enter into the south pole.

The direction of the flux lines shows the direction of the force that would be experienced by a free magnetic north pole at that point in the field. Another north pole would be:

• attracted towards the south pole, and
• repelled away from the north pole.

A current passing through a wire creates a temporary magnetic field around it due to the moving charges. The moment the current stops flowing, the magnetic field dissipates.

The diagram above shows a solenoid. A current is passed through a coil of wire, creating a temporary magnetic field around it. This field can be strengthened or weakened by altering the current.

Magnetic field lines (flux lines) are observed around permanent magnets and current-carrying wires. The shape of the magnetic field depends on the arrangement of the wire.

Some examples of this include a long straight wire and a coil of wire, referred to as a solenoid.

A straight current-carrying wire will produce a circular magnetic field around it at 90 degrees to the direction of the current.

The field is strongest and closest to the wire, but it gets weaker with distance. This is shown by the field lines being denser closer to the wire and becoming increasingly spaced as the distance from the wire increases.

The direction of the magnetic field can be determined by using Maxwell’s right-hand rule.

To apply Maxwell’s right-hand rule:

• The thumb should point in the direction of the conventional current flowing through the wire. In an electrical circuit with a cell, the conventional current flows from the positive end of the cell to the negative end of the cell.
• The fingers curl in the direction of rotation of the circular magnetic field.

Current flowing through a coil of wire will produce a magnetic field similar to that found around a permanent bar magnet. This arrangement is known as a solenoid. It will form a north pole at one end (where the field lines originate from) and a south pole at the other (where the lines return to).

The right-hand rule can be used to determine which end is the north pole and which is the south pole.

• Grab the coil so that the fingers follow the direction of the conventional current
• The thumb will point towards the north pole.

The strength of the magnetic field can be increased by using more turns in the coil or by adding a ferrous core, such as an iron nail.

In a flat coil of wire carrying a current, the magnetic field lines will leave from one side of the coil (the north pole) and enter the other side of the coil (the south pole).

The direction of the field lines can be found using the right-hand rule. The thumb points in the direction of the conventional current. The fingers curl in the direction of rotation of the circular magnetic field.

Fleming’s left-hand rule can be used to determine the direction of the force on moving charged particles in a magnetic field.

• The middle finger points in the direction of conventional current, so it is important to note that this is the direction in which positive charges are travelling.
• The index finger represents the direction of the magnetic field.
• The thumb points in the direction of the force felt by the moving charge.

A common memory tool to remember the order is FBI. Where:

• F represents force,
• B represents magnetic field, and
• I represents current.

The direction of the magnetic field can be represented in three different ways depending on the observer’s viewpoint. If the magnetic field is:

• at right angles to the observer, it is represented by arrows pointing in the direction of the field.
• pointed towards the observer, or out of the page, it is represented by circles or dots.
• pointed away from the observer, or into the page, it is represented by crosses.

A common memory tool is to picture an archery arrow. If the arrow travels away from you, you see the cross shape of the feathers on the back, whereas if it travels towards you, you will see the round shape of the arrowhead.

When a charged particle enters a uniform magnetic field, it will experience a force. Using Fleming’s left-hand rule, you can determine the direction of that force.

In the diagram above, the magnetic field is acting into the page. The conventional current is the direction in which positive charges are travelling. This causes the positive charge to experience an upward force. The negative charge will experience a force in the opposite direction.

The strength of a magnetic field (magnetic flux) can be investigated using a current-carrying wire and a digital balance. The experiment is set up as in the diagram below, with a magnet resting on a digital balance. The balance should be zeroed.

1. A taught wire should be passed between the two poles of the magnets and clamped in place so it cannot move. Ensure the wire is perpendicular to the magnetic field, as this will generate the largest forces and help reduce uncertainty in the measurements.
2. The length of the wire within the field should be measured.
3. The wire is connected to a series circuit that includes a way to alter the current, such as a variable resistor or a variable power supply, and a way to measure the current, such as an ammeter.
4. Slowly increase the current in equal increments and measure the force detected by the balance. The balance will typically be set to read in grams, which can be converted into newtons by first dividing by 1000 and then multiplying by 9.81.
5. This should be done at least seven times, and each result should be repeated thrice.
6. Then, a graph should be plotted, with force on the –axis and current multiplied by length on the -axis. The gradient will equal the strength of the magnetic field.

The strength of a magnetic field is referred to as the magnetic flux density and is measured in Teslas (T). It can be thought of as the density of magnetic flux lines within a area.

A standard magnet found in a school laboratory will have a magnetic flux density of approximately . The magnets in MRI machines are much stronger, reaching or more. When activated, the electromagnets in the Large Hadron Collider can reach up to .

When the magnetic field lines and the direction of motion of the charge are perpendicular, the magnetic force is maximal and is equal to:

When the magnetic field lines and the direction of motion of the charge are parallel, the magnetic force is equal to zero:

A particle of charge travelling in a magnetic field experiences a magnetic force .

When the velocity vector of the particle is perpendicular to the magnetic field vector , the magnetic force acts on the particle perpendicularly to both and .

It is important to note that the particle will follow a circular trajectory because the magnetic force is perpendicular to the velocity. The magnetic force will always point toward the centre of the circle, acting as a centripetal force.

A variation of Fleming’s left-hand rule can be used on charged particles to determine the direction of the magnetic force acting on that particle.

For a positive charge, the index finger of the left hand points in the direction of the magnetic field, the middle finger in the direction of motion, and the thumb points in the direction of the magnetic force.

Follow the same process for a negative charge, but flip the direction of the magnetic force at the end.

When a coil rotates within a magnetic field, the amount of flux passing through the coil will vary.

The diagram below shows the front and side views of a rotating coil. Blue field lines pass through the area within the coil, while red field lines do not.

The crosses in the diagram above represent magnetic field lines coming from the page.

In this arrangement, the field lines are perpendicular to the plane of the coil. This means that at this moment in time, the magnetic flux is at its maximum as the area that the flux lines can pass through is greatest. The moment the coil rotates so the lines are no longer perpendicular, the magnetic flux will no longer be at its maximum.

The diagram below shows the front and side views of a coil resting in parallel to the magnetic field lines. This means that the magnetic flux will be zero, as the area through the coil that the flux lines can pass through is zero.

The crosses in the diagram above represent magnetic field lines coming from the page.

If the magnetic field lines are passing through an area at an angle between and , the magnetic flux can be calculated using:

Where:

• is the magnetic flux, it is measured in webers ,
• is the magnetic flux density, it is measured in teslas ,
• is the area, it is measured in meters squared and
• is the angle between the field lines and a line normal to the plane of the area, it is measured in degrees .

The diagram above shows a rotating coil in three different positions. The dashed black line represents the normal to the coil.

It is important to remember to use the angle between the field lines and this normal.

It is important to note that the magnetic flux linkage equals the magnetic flux passing through a coil multiplied by the coil’s number of turns. It can be found using the formula below:

Where:

• is the number of turns of the coil,
• is the magnetic flux, it is measured in webers ,
• is the magnetic flux density, it is measured in teslas , and
• is the area perpendicular to the magnetic field lines, it is measured in meters squared .

Flux linkage has units of weber turns turns.

The magnetic flux lines passing through the coil may not be perpendicular to it. In this case, the following formula must be used to determine the magnetic flux linkage:

Where;

• is the number of turns of the coil.
• is the magnetic flux, which is measured in webers ,
• is the magnetic flux density, which is measured in teslas ,
• is the area, which is measured in meters squared and
• is the angle between the field lines and a line normal to the plane of the area, it is measured in degrees .

The diagram above shows a rotating coil in three different positions. The dashed black line represents the normal to the coil.

It is important to note that you should use the angle between the field lines and this normal in the magnetic flux linkage calculation.

The graph below shows how the magnetic flux linkage changes with time for a coil rotating within a magnetic field.

The flux lines pass from the magnet’s north pole to the magnet’s south pole. When the coil is upright (as shown on the far left), the area through which the flux lines can pass is at a maximum. This leads to the maximum flux linkage.

The graph follows a cosine curve to show how the flux linkage varies with time for a full rotation.

In a neutral wire, the charges are initially balanced. When the wire moves through a magnetic field, the Lorentz force acts differently on the free electrons (negative charges) than on the positive lattice ions. This force pushes the electrons to one side of the wire, creating a separation of charges. This separation sets up an electric field across the wire.

According to Fleming’s left-hand rule, the magnetic forces act differently on each end of the wire. The positively charged end experiences an upward force, while the negative charges in the opposite end are pushed downward.

Since the conventional current is defined as the direction in which positive charges would flow, we say that the induced current flows upward in the wire. When we apply Fleming’s left-hand rule to this induced current, we see that it generates a magnetic force that opposes the original motion of the wire. This opposition illustrates Lenz’s law, which states that the induced EMF always acts to counter the change that produced it.

A scenario that demonstrates Lenz’s law is the movement of a permanent magnet towards a coil of wire connected to an ammeter. When the magnet is moved towards the coil, its magnetic flux lines move through the coil, causing a change in the magnetic flux. This, in turn, induces an EMF and, therefore, a current in the coil.

According to Lenz’s law, the induced EMF acts in such a direction as to oppose the change that caused it. In this scenario, as a magnet is moved into the coil, an EMF is induced, which induces a current.

The induced current in the coil will then induce a new magnetic field around it (according to Fleming’s right-hand rule). The magnetic field induced will be in a direction that opposes the magnet’s motion as it is moved towards it.

The EMF can be found from the gradient of a flux linkage against the time graph, as shown below. Please note that:

• if the gradient is positive, then the EMF will be negative,
• if the gradient is negative, then the EMF will be positive.

This is due to the inclusion of a negative sign in the Faraday equation due to Lenz’s law.

The change in flux linkage can be found from the area under an EMF against time graph.

As a coil rotates in a magnetic field, the flux linkage constantly changes due to the varying area within the coil that the flux can pass through.

When the coil is perpendicular to the magnetic field lines, as shown on the far left of the graph above, the area within the coil that flux can pass through is at its greatest. Therefore, the flux linkage will be at its maximum.

When the coil is parallel with the magnetic field lines, the area within the coil which flux can pass is zero. Therefore, the flux linkage will also be zero and at its minimum.

Once the coil is rotated 180 degrees, as shown in the middle of the diagram, the area will once again be maximum, but as the coil is facing in the opposite direction, the polarity of the magnet has been reversed. Hence, the flux linkage is at a maximum but has a negative value.

The gradient of a flux linkage against time graph for a rotating coil in a magnetic field can be used to determine how EMF changes with time.

The graph of flux linkage against time and also the graph of EMF against time are both cosine, but with a phase difference of

It is important to note that for a rotating coil in a magnetic field, it might seem counterintuitive that the induced EMF is highest when the coil’s effective area – and thus its flux linkage – is smallest. However, EMF depends on the rate of change of magnetic flux, not just its magnitude.

Only the top and bottom segments move when the coil is upright, travelling nearly parallel to the field lines, resulting in a slow change in flux and a low EMF. When the coil is flat, even a slight rotation causes these segments to rapidly cut through many field lines, producing a higher EMF according to Faraday’s law.

Thus, the induced EMF is more sensitive to the rate of flux change than to the absolute area exposed to the magnetic field.

A search coil can measure the alternating EMF created by a change in the magnetic flux.

In the diagram below, a coil of wire is connected to a signal generator, which produces an alternating current. This means that the coil is producing an alternating magnetic field.

A second coil, known as a search coil, is placed within the first coil of wire. This coil will experience a continually changing magnetic flux, which will induce an EMF within it.

The alternating EMF can be measured directly if this search coil is connected to an oscilloscope.

An example of using a search coil is to carry out an experiment to investigate how the angle of the search coil affects the maximum EMF induced.

The search coil, a small coil of wire, is placed within a larger coil of wire, generating an alternating magnetic field. This alternating magnetic field will cause a change of flux linkage within the smaller coil, which in turn induces an EMF that can be measured.

The angle of this smaller coil can be adjusted, which will change the amount of flux linkage and the size of the maximum induced EMF.

When the search coil is parallel to the plane of the first coil, it presents the greatest area for the changing magnetic flux to pass through. This means the greatest EMF will be induced.

When the search coil is rotated 90 degrees to be perpendicular to the plane of the first coil, it presents no area for the changing magnetic flux to pass through. This means no EMF will be induced.

To investigate how the angle of a search coil affects the maximum induced EMF, a small coil is placed inside a larger coil generating an alternating magnetic field. Changing the angle of the smaller coil alters the flux linkage, thereby affecting the maximum induced EMF.

Then measure the max EMF induced for a range of angles between 0^o and 90^o and then plot a graph of the cosine of the angle between the planes of the two coils against max EMF. The result should be a straight line.

An oscilloscope is an instrument used to display and analyse the waveforms of electrical signals. It can be used as a DC or AC voltmeter.

An oscilloscope will typically show:

• Time on the X axis (called the time-base), usually in units of milliseconds per division (ms div⁻¹), where each line corresponds to a millisecond.
• Voltage on the Y axis, representing the amplitude of the wave.

Oscilloscope readings are often used to calculate the frequency of a wave. To do this:

1. Determine the period by reading the time for one complete wave cycle from the time-base setting.
2. Convert the period into seconds.
3. Use the relationship to calculate the frequency.

When dealing with wave graphs, be careful to read the axes labels, as the type of axis (time or distance) will determine what parameters can be interpreted from the graph.

A wave on a string might be represented by a distance–time graph, whereas oscilloscopes plot the voltage of an electrical signal against time.

Transformers are devices designed to raise or lower the voltage in a circuit:

• A step-up transformer will raise the voltage and, at the same time, reduce the current.
• A step-down transformer will lower the voltage and increase the current.

The national grid uses transformers to reduce the energy lost as heat in power lines. Since power is proportional to the current squared reducing the current by increasing the voltage helps reduce power loss and improve efficiency.

The transformer itself consists of three main components. A primary coil, a secondary coil and a soft iron core.

An alternating electric current is passed through the primary coil, creating an alternating magnetic field. The iron core helps to guide this magnetic field towards the secondary coil. The term soft describes how easily the iron core can be magnetized and demagnetized.

The secondary coil, therefore, experiences a continually changing magnetic flux linkage. According to Faraday’s law, this will induce an alternating EMF and current.

The magnitude of the EMF induced depends on the number of coils on the secondary coil. A step-up transformer will have more coils on the secondary coil to increase the voltage, whereas a step-down transformer will have fewer coils on the secondary coil to reduce the voltage.

An investigation can be conducted to prove the transformer equation experimentally and test the relationship between the number of coils and the current.

Start by wrapping a coil of wire around one end of an iron bar. Connect both ends of the wire to a power supply that can provide an alternating current. This is the primary coil. Use an ammeter in series with the power supply to accurately measure the generated current.

Wrap another coil of wire around the other end of the rod and attach another ammeter. This is your secondary coil.

1. Count the number of coils in both the primary and secondary coils and record them in a table.
2. Turn on the power supply to provide an alternating EMF and record the current in the primary coil.
3. Record the current through the secondary coil and see whether the results collected obey the transformer equation:

Continue to test different combinations of current and coils and determine whether the equation continues to be obeyed. This equation assumes that the transformer is ideal and perfectly efficient and that 100% of the electrical power is transmitted from the primary to the secondary coil.

In reality, no transformer is 100% efficient. So, the results collected experimentally will not match what the equation predicts.