## Module 6: Electromagnetism

Contents

## Charged Particles, Conductors and Electric and Magnetic Fields

### Electric fields

An **electric field** is a vector quantity.

The existence of field lines will indicate areas where a charged particle will experience a non-contact electric force.

The field lines point in the **direction of the force that a small positive test charge would experience if it were placed in the electric field.**

The denser the field lines, the stronger the electric field.

Each positive or negative charge will produce its own electric field. Electric fields from different sources may interact with each other. See diagram below:

### Acceleration of a charged particle by the electric field

The electric field strength can be thought of as the force applied per coulomb of charge. Therefore, the magnitude of force applied to a charge in an electric field is:

Combining this equation with Newton’s will allow us to find the acceleration of a charged particle in an electric field.

### Electric field between parallel plates

The electric field produced by two parallel charged plates can be visualised as follows:

A positive and negative test charge (which have the same magnitude of charge) will experience forces of the same magnitude but opposite direction in such a field. This is illustrated by the following figure:

There is an electric potential (V) which exists between these two charged plates. This is commonly measured in volts. The difference in potential between the two plates is also known as the potential difference.

The electric field strength can be calculated as follows:

Using the above equation and combined with what we already know we can then find the acceleration of a charged particle.

### Work done on the charge

Electric potential energy is a form of energy that is stored in an electric field.

Electric potential energy is transformed to other forms of energy when the field does positive work on the charged particle. That is, the direction of movement of the particle is in the same direction as the field’s force on the particle. This is also known as doing work on the field.

Conversely, other forms of energy are transformed into electric potential energy when the field does **negative **work on the charged particle. That is, the direction of movement of the particle is not in the same direction as the field’s force on the particle. This is also known as work being done by the field on the **positive **charge.

**Electric potential** (V) is defined as the work required per unit charge to move a positive point charge from infinity to a place in the electric field. Therefore, its formula is given by:

Using the formula from before concerning electric field strength, we can find the work done on a point test charge to move it a distance across a potential difference.

When work is done by a charged object on an electric field, the object is forced to move against the direction it would naturally go. Therefore, in this case the field would do negative work on the object.

When work is done on a charged object by a field, the object is moving in the direction it would naturally go. Therefore, in this case the field would do positive work on the object.

If a charged object moves horizontally through the electric field, no work is done by or on the field.

### Comparison between electric fields and gravitational fields

In both electric fields and gravitational fields, the fields exert a non-contact force on objects with particular characteristics.

- In a gravitational field, a gravitational force will act on any object with mass.
- In an electric field, an electrical force will act on any object with charge.

In both uniform electric fields and gravitational fields, the field strength is constant and the direction of acceleration is always in one direction.

- Therefore, the trajectory of any charged particles moving through a uniform electric field between 2 parallel plates will look exactly the same as the motion of a projectile in a gravitational field.

The similarities between gravitational and electric fields means we can use all of our equations of motion from Advanced Mechanics to perform calculations on the movement of charged particles in electric fields.

### Magnetic fields

Magnetic fields are similar to electric fields in that they are both fields that exert a non-contact force on charged particles.

The force exerted by magnetic fields on charged particles are known as **the magnetic force and** the direction (and magnitude) of such a force is completely different to that exerted by an electric field.

The unit of magnetic field strength is the **Tesla (T)**.

The direction of the magnetic force acting on a charged particle can be found using the right-hand palm rule.

- Point all your fingers except for your thumb in the direction of the magnetic field.
- Point your thumb in the direction of travel of a positive charge. (NOTE If the question involves a negative charge, then point your thumb in the opposite direction to the direction of travel of the negative charge).
- The direction your palm faces is the direction in which the magnetic force acts.

The magnitude of the force acting on the charged particle is given by:

### Uniform circular motion

The convention for drawing magnetic field lines is:

Since the magnetic force always acts perpendicular to the velocity vector of the moving charged object, therefore it is possible to recreate uniform circular motion by orienting the magnetic field so that it is always perpendicular to the velocity vector. An example can be seen below:

From the formula for the magnitude of the magnetic force, it can be seen that the magnitude of the magnetic force stays constant.

- The force also continuously changes direction to point perpendicular to the velocity vector.
- Therefore, the charged particle will undergo uniform circular motion.

Note that the charged particle will only continue to move in uniform circular motion so long as the charged particle remains **inside the magnetic field.**

Once we have deduced that the particle undergoes uniform circular motion, we can use our formulas from Advanced Mechanics to calculate other quantities related to the moving particle such as its speed and the relationship between the radius of the circular trajectory and the magnetic field strength.

Equating and will give:

- From there it can be seen that there is an inversely proportional relationship between the radius of the track and the charge and magnetic field strength. There is a proportional relationship between the radius of the path and the mass and velocity of the object.

## The Motor Effect

### Current-carrying conductor in magnetic field

The magnitude of the force is quantified by:

- Where F is force (N).
- N is the number of wires together (if applicable).
- B is magnetic field strength (T or Tesla).
- L is the length of the wire (m) in magnetic field.
- I is the current passing through the wire (A).
- θ is the angle between the wire and the magnetic field.
- This angle is the angle located between the thumb and the four fingers in the right hand palm rule.

The direction of the force on the wire is determined by the **right hand palm rule:**

- Four fingers point in direction of magnetic field.
- Thumb points in direction of current.
- Palm will point in direction of force.

For direction of force on negative charges, simply reverse direction of conventional current and apply right hand palm rule.

A force exists for a current-carrying wire because the magnetic field of the wire interacts with the external magnetic field such that one side cancels while the other side add constructively. Thus, a force results due to imbalance in magnetic fields.

Maximum force is produced when the current-carrying conductor is perpendicular to the magnetic field.

No force is produced when the current-carrying conductor is parallel to the magnetic field.

### 2 parallel current-carrying conductors

Each current-carrying conductor will produce its own magnetic field. The direction of this magnetic field can be determined by the right-hand grip rule.

- The thumb should point in the direction of travel of the positive current.
- Then your four fingers will curl in the direction of the magnetic field lines.
- NOTE This is different to the other right-hand grip rule for solenoids.

See below for a demonstration: - F is force (N).
- L is length of section of wire that is parallel to the other wire (m).
- I1 and I2 are current in wires.
- R is distance of separation between wires (m).
- K is constant and k

Strength of magnetic field:

- B is magnetic field strength (T).
- I is current (A),
- r is distance between the wires.
- K is constant ()

Alternatively the quantity k can be given by:

Direction of force can be determined qualitatively by seeing whether wires attract or not.

From the diagram it can be seen that there is a force of attraction between the 2 wires if their currents travel in the same direction.

There is a force of repulsion between the 2 wires if their current travels in the opposite direction.

## Electromagnetic Induction

### Magnetic Flux

**Magnetic flux** – the amount of magnetic field passing through a given area.

- Where B is magnetic field strength (T)
- A is area (m2)
- φ is flux in webers (Wb).
- θ is the angle between magnetic field lines and conductor. When magnetic field is perpendicular to area, θ = 90.

**Magnetic field** – The amount of magnetic flux per unit area or magnetic flux density.

Rearranging the equation gives

- Hence unit for magnetic field can be Tesla (T) or Weber per metre squared (Wbm-2).

Strength of a magnetic field depends on density of magnetic flux.

Therefore it can be seen that flux can change if we vary the angle between the magnetic field and the area of the plane.

### Lenz’s law and Faraday’s law

Faraday’s Law:

- Where n is the number of turns or coils.
- Note: Unit of EMF is Volts.
- is the rate of change in flux over time.
- Minus sign indicates direction and is explained by Lenz’s Law.
- Can be rewritten as:
- If asked to calculate EMF, keep minus sign and then at the end say “therefore magnitude of EMF induced is …”

In order to induce EMF, a magnetic flux change is necessary created by relative motion between conductor and magnetic field.

As the speed of the conductor through the magnetic field increases, the rate of flux change increases and so induced EMF increases.

The greater the change in area as the conductor moves, the greater change in flux and greater induced EMF.

Induced EMF gives rise to current only in closed circuits. Only in closed circuits will motor effect take place. Only EMF will not results in force on conductor.

If asked for why current flows, no need to mention Lenz’s Law. Just say induced EMF induced current because of closed circuit.

Lenz’s law tells us that:

- Whenever an EMF is being induced in a conductor as a result of changing magnetic flux, the direction of the induced EMF will be such that the current it produces will give rise to a magnetic field that always opposes the change and hence opposes the cause of induction.
- This results in an induced magnetic field in the conductor which results in an induced force that opposes the change in flux experienced by the conductor.
- Must cite Faraday’s Law to explain induced EMF before current. Must say relative motion between magnetic field and conductor.
- When applying Lenz’s Law to moving coils use right hand grip rule on region of coil inside magnetic field to work out the current direction that opposes change. Note: Must consider flux inside loop only.

The technique to applying Lenz’s law is always:

- Consider the direction of change in magnetic flux
- What will oppose the change in flux for each case?
- Determine the direction of the induced current required to oppose the change for each case.

### Ideal transformers

**Transformers** – devices that increase or decrease the size of the AC voltage as it passes through them.

Simplest transformers consist of:

- Primary coil – AC input.
- Secondary coil – AC output which is connected to load.

Primary coil and secondary coils will have different number of turns of wires.

Both primary and secondary coils are wound around a soft iron core.

As AC power is input into primary coil, it produces its own magnetic field which results in changing magnetic flux.

The changing magnetic flux is passed through the soft iron core to the secondary coil.

This induces an EMF in the secondary coil.

The size of the magnetic flux developed by the primary coil is related to the number of coils and the EMF induced in the secondary coil depends on the number of turns in the secondary coil.

By varying the number of turns in each coil, the voltage of the AC power can modified.

Since energy is conserved, therefore decreasing or increasing the voltage will increase or decrease the current.

Role of soft iron core:

- Acts as medium through which magnetic flux can pass through. Links flux between primary and secondary coil and prevent flux leakage.
- Intensifies/concentrates magnetic flux so mutual induction is more efficient.

Soft iron core magnifies magnetic flux by temporarily aligning its domains with external magnetic field, becoming a temporary ferromagnet that has its own magnetic field which adds to the magnetic flux. (partial magnetisation)

Transformers are dependent on AC because for a current and voltage to be induced in secondary coil, the secondary coil needs to experience a constant magnetic flux change. A Δφ is needed for electromagnetic induction. DC would not induce a current in 2nd loop because φ through secondary coil would remain constant.

**Step-up transformers** – devices used to increase the voltage of AC electricity.

- Voltage output from secondary coil greater than voltage input of primary coil.
- Doing so decreases the current so minimal power loss during transmission.
- Primary wires thinner than secondary as secondary needs to carry more current.

**Step-down transformers** – devices used to decrease the voltage of AC electricity.

- Voltage output from secondary coil less than voltage input from primary coil.
- Doing so increase the current so electricity can be used in appliances.
- Primary wires thicker as primary carries more current

There are two main formulas associated with transformers:

However, the ideal transformer model makes some assumptions such as no flux leakage and no power loss which are not applicable to real life.

The no flux leakage assumption is violated because it is impossible to fully direct all magnetic flux from one coil to another.

- This is partially addressed through the use of a soft iron core which attempts to capture as much magnetic flux as possible.

The no power loss assumption is violated because of the existence of eddy currents and resistive heat production.

Eddy currents

Caused by the changing magnetic flux as the soft iron core is a solid conductor.

Circulation of eddy currents generates heat much like the induction cooktops.

However, eddy currents may be reduced by disrupting the plane surface of the conductor by having slots on the surface or lamination.

i.e. the surface of the conductor is no longer uniform, it is marked with holes and insulation material to prevent effective circulation.

This reduces magnitude of eddy currents.

Lamination:

- Soft iron core constructed with stacks of iron sheets each coated with insulation materials.
- Restricts circulation of large eddy currents as surface of conductor is disrupted by insulation.
- Decreases heat generation by eddy currents and thus increases efficiency.

Ferrites

- Iron core sometimes made of ferrites which discourage eddy current formation as they are poor conductors.
- Ferrites are oxides of iron.

Resistance in coils

Coolants used to cool down wire coils in transformer.

The iron core with the coils is submerged in a container with a cooling oil.

The cooling oil helps to cool down the coils by drawing heat away from them.

Cooling oil is circulated by pumps to repeatedly draw heat away from transformer.

### Use of transformers in high-voltage power transmission

Electricity generated by 3-phase AC generator generally has extremely high voltage (23000V) and high current (~10000A).

Electricity is fed into step-up transformer that increases voltage to 330000V and results in decreased current.

- This minimises power loss during transmission.

After electricity transmitted over large distance, voltage is stepped down by transformers in sub-stations for safety reasons near cities. Current thus increases.

Eventually voltages stepped down to 240V by telegraph pole transformers. Voltage is stepped down gradually after sub-station by local step-down transformers.

When electricity is transmitted through wires, energy is lost mainly in the form of heat.

This is mainly due to resistance of wires. (Note: The thicker the wire, the smaller the resistance).

The power loss is given by . (Note: power is rate of energy loss).

Resistance of wire depends on length and so long distance transmission lines will inevitably have high resistance.

Thus the only way to minimise loss of energy is by changing the current.

Transformers can increase the voltage of electricity and decrease its current as energy is conserved.

This effectively decreases the current without changing power being transmitted.

Since power loss is proportional to the square of the current, it follows that decreasing the current will minimise energy loss.

Thus, transformers make electricity transmission more efficient.

## Applications of the Motor Effect

### The DC Motor

**Electric motor** – a device that converts electrical energy into mechanical energy usually through rotation.

A DC motor has the following typical structure:

The Motor effect causes a force to be generated on the sides ab and cd of the coil in the above image.

Notice how the forces resulting from the motor effect acting on the sides of the coil form a turning moment with the pivot of the rotational axis.

The torque produced allows the coil to rotate on the axis.

As the coil spins, a small force forms on sides BC and AD which stretches the coil but since the coil is rigid this is not noticeable.

The torque in a motor can be quantified:

- N is the number of coils of wire.
- B is magnetic field strength (T).
- I is current (A).
- A is surface area of coil
- θ is the angle between the plane of the coil and the magnetic field

Torque is a maximum when coil plane is flat.

Torque slowly decreases as the coil turns because the angle between the coil and magnetic field increases.

Torque is 0 when coil plane is vertical.

At this position, there is no turning effect so the DC motor can no longer spin continuously and coil eventually stops.

Describing changing torque in DC motor:

Initially τ is maximum and LHS/RHS will turn upwards (Using RHP rule).

Τ declines as motor moves through 90o and coil reaches vertical position. Due to its momentum coil will not stop moving and presence of split ring commutator will ensure that it will keep turning in the same direction.

As coil turns, there will also be back EMF as the coil experiences a magnetic flux change. Back EMF will increase as coil speeds up, and it will oppose supply voltage.

This continues until back EMF balances out supply voltage and net current in coil is 0. There is no torque at this point and coil rotates at constant speed.

### Main features of a DC Motor

When drawing a “real life” DC motor must have:

- Radial magnets
- Iron Core
- Indicate multiple coils of wire

Split-ring commutator

A split-ring commutator is used to maintain one-directional torque in a DC motor.

The coil is still able to spin at the vertical position due to inertia but will eventually stop.

This is because as soon as the coil leaves the vertical position, the torque formed will attempt to push the coil back to the vertical position.

Thus coil oscillates around vertical until it comes to a rest.

A split-ring commutator consists of a two halves of a metal (usually copper) cylinder insulated from each other

The rotation of the split-ring commutator reverses the direction of the current at vertical positions every half cycle by changing the contact of the two halves of the commutator with carbon brushes of different polarity.

Allows torque and force to change direction every half revolution so DC motor can continue to spin.

Carbon brushes are used because it is solid lubricant and a conductor which reduces friction and sparks. This elongates the life of the motor.

- Also provides sliding contact between circuit and commutator.

The above position of the commutator occurs when maximum torque is experienced.

When the insulator of the commutator is in contact with the carbon brushes, the current through the coils is 0 but inertia will keep it moving so this is not a problem.

Magnetic field

Magnetic fields are required for sustained motor effect to rotate the motor.

Provided by permanent magnets or electromagnets.

This is also known as the stator as the magnet usually remains stationary in the motor.

Permanent magnet

- Ferromagnetic magnets that retain magnetic property at all times.

Electromagnet

- Consists of a soft iron core with a current-carrying wire wrapped around it to form
**solenoid**. - Able to provide stronger magnetic fields but requires electric input.
- Magnet strength can be adjusted.

Radial magnets

- If bar magnets are used in a motor, the torque varies because the component of the force responsible for the torque decreases as the angle between the coil plane and the magnetic field increases.
- Hence the motor will operate at varying speeds as it rotates.
- A radial magnetic field may be used to remedy this problem.
- This ensures that the coil plane is perpendicular to the magnetic field for the majority of the time so a constant pair of forces acts on the coil.
- Results in almost constant torque.
- Also used to generate constant EMF as flux change is constant

Note: the magnetic field is not a uniform magnetic field. - Multiple coil sets and magnets

Alternatively, can have multiple sets of coils and magnets to produce a near constant torque.

Armature

- Coil of wire placed inside magnetic field.
- Wound around a soft iron core to enhance motor effect.
- Soft iron core enhances magnetic field and increases inertia of motor to maintain coil rotation.
- Rotor refers to the soft iron core and the wires if it is the armature that spins.
- Armature + soft iron core is called the stator if it is not moving and the magnets are rotating.

## Simple AC and DC generators and AC induction motors

### DC Generator

**Electric generator** – A device that converts mechanical energy to electrical energy using the principle of electromagnetic induction.

A DC Generator is basically a DC motor in reverse.

Most of the components of a generator and motor are largely the same.

Both have armatures and magnets.

Difference in function:

- A generator converts mechanical energy into electrical energy via electromagnetic induction.
- A motor converts electrical energy into mechanical energy via motor effect.

### AC Generator

Exactly the same as the DC Generator except with a slip-ring commutator

The purpose of the slip ring commutator is to provide constant contact between the internal and external circuit so the induced EMF will be replicated without entangling the circuits.

### AC Induction motors

Functional principle:

- As the magnet rotates, there is a flux change in the aluminium disc.
- This induces an EMF in the disc and consequently eddy currents such that they oppose the induction via Lenz’s Law.
- Since the magnet cannot be stopped, it follows that to minimise flux change, the aluminium disc rotates with the magnet. i.e. it rotates in the same direction as the magnet.

Similarly, in an AC induction motor, there are a series of stator coils around the motor.

Squirrel cage chases magnetic field to minimise relative motion between magnetic field and cage.

AC current is fed into the coils to create magnetic fields but they are fed in such a way that they are slightly out of phase with each other.

Consequently, each coil will be magnetised at different times which creates a rotating magnetic field as each pair of stator coils are magnetised in turn.

Motor effect (not really true).

- This rotating magnetic field will result in a flux change in the squirrel cage.
- This flux change induces EMF and current in the squirrel cage in the form of induced currents.
- The currents will interact with the external magnetic field such that it will experience a magnetic force and torque.
- This results in the squirrel cage rotating in the same direction as the magnetic field.

No current is fed into the rotor; it is the induced current of the rotor that interacts with the magnetic field.

Minimal back EMF and eddy currents in squirrel cage as little relative motion between rotor and magnetic field (squirrel cage chases field).

Advantages:

- Very reliable, long lasting and compact as no commutator required.
- Self – start, little noise

Problems:

- AC current is weaker than direct current for a given voltage so it can only power small appliances.
- Has low starting torque as rotor has inertia preventing it reaching high speed rotation in a small amount of time.
- Low power, constant torque
- Speed decreases with load.
- Fixed speed as frequency of AC current needs to be changed.

Uses: fans, fridges, washing machine.

## Conservation of energy

### In DC Motors

Lenz’s law is basically a restatement of the conservation of energy.

Work cannot be done unless you put in additional energy into the system.

In the context of DC motors, the motor eventually achieves constant rotational velocity due to the existence of back emf.

If you place a load on the motor, you want it to do work on the load. Since the rotational speed of the motor will slow down, the flux change will decrease and the back emf acting against the input emf will decrease. This will ‘free up’ more available emf to increase the motor effect so that the motor can do work on this additional load. However, this will also mean that a larger current will now flow through the coils, thus causing the coil to heat up as there isn’t as much back emf to oppose the input emf as before.

### In magnetic breaking

**Eddy currents** – Loops of current that flow on the plane of solid conductors as a result of induced EMF.

Eddy currents exist via Lenz’s Law to oppose induction.

Only occurs at the area of the conductor where there is a flux change.

Although positive charges don’t move, we are dealing with conventional current in this case.

Notice that eddy currents only occur on the edge of the magnetic field, this is where there is a flux change and not the middle of the metal plate.

The eddy currents are induced such that there is an induced force that opposes the force of the metal plate as a result of the interaction between two magnetic fields.

In the presence of the magnetic field, the positive charges move to the top of the plate.

When there is no magnetic field, the positive charges move back to the bottom of the plate which explains the downward motion of positive charges as motor effect no longer applies.

Apply right hand palm rule only to movement of positive charges within the magnetic field and not outside of it.

Note: NO EMF or current is induced if conductor moves at constant velocity through a uniform magnetic field. Exception is if it moves outside of magnetic field.

Strong magnets are lowered next to the train’s wheels to produce a magnetic field to induce eddy currents.

The eddy currents within the wheels oppose the motion of the rotating wheel.

However, the braking effect reduces as the wheel slows because the flux change decreases so the size of the eddy currents decreases.

Thus braking force decreases as the rotation of the wheel decreases.

Kinetic energy is converted to heat energy. (Wheels become very hot)

Hence if the train wishes to stop completely, then it must apply mechanical brakes.

Advantages

- No mechanical contact, no mechanical friction and wearing. Thus minimal maintenance required.
- It is smooth. Braking force is proportional to velocity so braking force decreases with velocity.

Disadvantages

- Lots of heat generated in wheels.