Magnetic fields (Topic 7B)
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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 .
The strength of the magnetic flux density is measured in Teslas (T).
We can determine what 1 Tesla is equivalent to by referring to the equation for the force on a current-carrying wire in a magnetic field:
Then, rearranging to make the subject:
A magnetic field with a strength of will cause a length of wire perpendicular to a magnetic field, carrying of current, to experience of force.
It is important to note that when the direction of the current is perpendicular to the magnetic field lines, the force will be at its maximum value, . However, when the current is parallel to the magnetic field lines, the force will be zero, as .
Question walkthrough
Finding Flux Density from Force Balance
Balances gravitational and magnetic forces on a current-carrying wire suspended in a field to calculate the magnetic flux density.
Magnetic flux refers to the amount of magnetic field passing through a given area.
It is different to the magnetic flux density, which is the amount of magnetic field passing perpendicularly through exactly
Therefore, the magnetic flux is the magnetic flux density multiplied by the size of the area :
Where:
- 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
It is important to note that this formula is only valid when the field lines are perpendicular to the plane of the given area.
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.
Question walkthrough
Magnetic Flux Through a Rotating Coil
Calculate the magnetic flux through a single rectangular coil at a given angle to a magnetic field, using its dimensions and the field's flux density.
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.
Question walkthrough
Flux Linkage in a Rotating Coil
Calculate the flux linkage of a multi-turn rectangular coil rotating in a magnetic field, given its dimensions, flux density, and angle to the field lines.
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.
Question walkthrough
Current Direction for a Floating Wire
Applies Fleming's left-hand rule to determine the direction of current in a wire that remains motionless when released in a magnetic field.
Four variables determine the force on a current-carrying conductor in a magnetic field.
- The strength of the magnetic field (the magnetic flux density).
- The size of the current flowing through the conductor.
- The length of the wire within the magnetic field.
- The angle between the conventional current’s direction and the magnetic field’s direction.
These variables are represented in the equation below:
Where:
- represents force measured in newtons (N),
- represents the magnetic flux density measured in teslas (T),
- represents current measured in amperes (A),
- represents the length of the conductor measured in meters (m), and
- represents the angle between the direction of conventional current and the direction of the magnetic field measured in degrees (o) or radians (rad).
Question walkthrough
Finding Wire Length at an Angle
Uses the force equation for a current-carrying wire at an angle to a magnetic field to calculate the length of wire, given the balancing scale reading.
When a charged particle is in motion within a uniform magnetic field, the particle experiences a magnetic force of magnitude:
Where:
- if the magnetic force in newtons ,
- is the magnetic flux denity in teslas ,
- is the charge of the particle expressed in coulombs ,
- is the speed of the particle in metres per second , and
- is the angle between the particle’s direction of motion and the magnetic field direction.
For an electron or a proton, the charge is .
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:
Question walkthrough
Force on a Charged Particle in a Field
Calculate the magnetic force on a moving electron using F = QvB, given the field strength, particle speed, and perpendicular field orientation.
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.
It is important to recall and understand the proper use of Fleming’s left-hand rule and its variation.
The magnetic force, , is a centripetal force that acts on the charged particle. Hence, we can apply the formula for a centripetal force:
However, centripetal acceleration can be expressed in terms of speed, , and radius, , of circular motion as:
Finally, combining the previous two equations, we get:
Where:
- is the mass of the particle in ,
- its speed in and
- is the radius of the circle traversed by the particle, in .
Knowing that the magnetic force is also equal to:
The equation becomes:
Thus, the radius of the circle completed by the charged particle is:
The radius of the circle can be increased by:
- increasing the mass of the charge ,
- increasing the speed of the charge ,
- reducing the magnetic flux density , and
- reducing the quantity of charge .
Question walkthrough
Finding Radius of a Proton's Circular Path in a Field
Calculate the radius of the circular path of a proton moving at right angles to a uniform magnetic field, given its charge-to-mass ratio and speed.
Faraday’s law links how a change in the magnetic flux (for example, by rotating a coil in a magnetic field) to the size of the EMF induced in the coil.
The law states that the magnitude of the EMF induced is directly proportional to the rate of change of magnetic flux linkage. Faraday’s law is represented by the equation below:
Where:
- represents the induced EMF and is measured in volts ,
- represents the number of turns in the coil,
- represents the change in the magnetic flux and is measured in webers , and
- represents the time taken for the change in magnetic flux to occur and is measured in seconds .
If the EMF is induced in a conductor that is part of a complete circuit, then a current will also be induced.
Lenz’s law determines which direction the induced EMF will act in when there is a change in the magnetic flux (for example, when a conductor is moved through a magnetic field.
The law states that the EMF induced acts in such a direction as to oppose the change that caused it. This is represented in Faraday’s law equation below by the inclusion of a minus sign :
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.
Question walkthrough
Ammeter Deflection During Induction
Describe how the direction and presence of induced current changes as a magnet is pushed into a coil, held stationary, and then withdrawn.
Lenz’s law and Faraday’s law can be combined to form an equation that can both be used to work out the size of the electromotive force EMF induced by the rate of change of magnetic flux linkage (for example, by rotating a coil in a magnetic field) and also the direction in which the EMF acts.
The negative sign represents the effect of Lenz’s law, which states that the direction of the induced EMF acts to oppose the change that caused it.
- represents the induced EMF and is measured in volts ,
- represents the number of turns in the coil,
- represents the change in the magnetic flux and is measured in webers , and
- represents the time taken for the change in magnetic flux to occur and is measured in seconds .
Several factors affect the induced e.m.f in linked coils. When the current in one coil changes, it creates a changing magnetic flux. If a second coil is linked to the first (sharing its flux), this changing flux induces an e.m.f. in the second coil. Only a changing current does this. A steady current gives constant flux and induces no e.m.f.
Because , the induced e.m.f. is larger when:
- the current changes more rapidly,
- the second coil has more turns (larger ), and
- the coils are more strongly linked. They are closer together, have more turns on the first coil, or share a soft iron core.
A transformer is a device that raises or lowers the voltage in a circuit. An alternating current in one coil induces an e.m.f. in a second coil linked to it through a shared soft iron core. It is a neat real-world picture of how a changing current in one coil drives an e.m.f. in another.
For your Edexcel exam, though, knowledge of transformers are not required. You don’t need the turns-ratio equation or how they step voltages up or down. Keep your focus on the factors affecting the induced e.m.f., and treat transformers only as an aid to understanding.
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.
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.