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Charged Particles: Conductors, Electric & Magnetic Fields

 » Electric Fields

• A force experienced by a charge due to an electric field:
• 
• A source of charge (Q), is the charge produced by the electric field.
• A test charge (q), is a charge used to measure the source charge’s electric field.
• The test charge experiences a force (F) inside the source charge’s electric field.
• The force can either be attractive or repulsive; it depends on the charges of Q & q.

                   

 » Electric Fields Between ll Plates

 

• The plates produce a uniform electric field throughout the plate.
• Field lines always travel perpendicular.
• Density of field lines represent field strength

             -  High amount of lines indicates a strong electric field.    

                         

• A charge placed anywhere in a uniform field will experience a constant electric
  field (F=Eq). Therefore, acceleration is also constant (F=ma).

 » Work in Electric Fields

•   Work is done when a charged object is moved in an electric field.

     

• Furthermore, F = Eq is substituted into W = Fd to yield.

• If the charge moves against the electric field, work is done onto the field by the
  charge.
• If the charge moves along the electric field, work is done by the field by the
  charge.
• Equipotential lines consist of points where they all have the same electric potential energy.

                                             

 » Comparing Gravitational Fields & Electric Fields

          

                             

 » Projectile Motion Equations

   

 » The Effect of a Charged Particle in a Magnetic Field

• It was observed that when a charge (q) was moving at a velocity (v) in the presence of a magnetic field (B), it would experience a force (F):

           

 » Right Hand Palm Rule

 -  Use right hand:

• Thumb points in the direction of the conventional current.

                 ‣  Proton’s velocity

• Fingers point to the direction of the magnetic field.
• Palm faces the direction of force.

                                             

 » The Effect of a Charged Particle in a Magnetic Field

• If the velocity if not perpendicular to the magnetic field. The result is a helical motion
• Break the oblique velocity into its horizontal & vertical components.
• The horizontal component, which is parallel to the field, does not experience a magnetic force.
• Only the vertical component experiences the force
• Combining both horizontal & vertical components result in a spiral/helical motion.

                              

 » The Effect of a Charged Particle in a Magnetic Field

• If a charge were to enter a magnetic field perpendicular, it will undergo circular motion & experience centripetal motion.
• Therefore, the magnetic force on the charge is equivalent to its centripetal force.

   

 

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Motor Effect

 » Introduction

• When a current-carrying conductor is placed in a magnetic field, it experiences a force.
• A moving charge with a constant velocity produces a magnetic field.
• Electrons move as a stream in conductor with uniform velocity & produces a magnetic field surrounding the conductor.
• It converts EPE to KE.
• This magnetic field interacts with an external magnetic field, which causes a constant field in the same direction acting on the conductor.

 

• The magnetic field produced by a current carrying conductor surrounds the wire in a circular fashion.

                     

 » Carrying Conductors - Force Between Parallel Current

• There will be a force experienced by two wires if line two conductors in a parallel fashion.
• This force is a consequence of the magnetic fields, produced by the individual wires, interacting with one another.   

• Recall Newton’s Third Law of Motion

            ‣  “Every action has an equal & opposite force”

                • The force experienced by wire 1 is equal in magnitude to the force experiences by wire 2.

                     • However, they act in the opposite direction.

 » Defining the Ampere

    

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Electromagnetic Induction

 » Introduction

• Michael Faraday discovered that an electric current can be induced in a conductor if there was a change in the magnetic field acting on that conductor.
• This is called electromagnetic induction.

 » Magnetic Flux

• A measure of the total magnetic field that passes through a particular area.
• Measured in Webes (Wb) k= kgm²s^-2A^-1 

  

• Faraday demonstrated that a change in magnetic field would result in an induced electromotive force (EMF) that in turn produces inducing current.
• This would be done by moving a magnet near a conductor & observe the needle of a galvanometer deflect in a certain direction.

                                            

                 ‣  A deflection in a galvanometer indicates a producing current.

• He observed that the magnitude of the current produced was dependent on the speed at which the magnet was moving.
• A current can also be induced by moving a conductor.

                       

• Recall when a charge moves in a magentic field, it experiences a force equal to 𝑞𝑣𝐵 𝑠𝑖𝑛𝜃.
• Using the RHP, the positive charge goes out of the page while the negative charge goes into the page.

 » Faraday’s Law

• “The induced EMF, in a closed circuit, is directly proportional to the rate of change of magnetic flux.”

     

 » Lenz’s Law

• “An induced EMF always gives rise to a current whose magnetic field will oppose the original change in flux”
• This law is an extension of the Law of Conservation of Energy, which states that energy cannot be created nor destroyed.
• Lenz’s Law accounts for the negative sign in Faraday’s law. The current induced in the coil opposes any change in the magnetic flux by flowing opposite to the current which caused such changes.
• Thus, this law is used to explain the direction of the induced EMF.

          

• Case Studies

    ‣ Case 1:

• Consider Faraday’s experiment, where a magnet moves into a
  conducting coil.
• We know that an induced current is generated in the coil.
• This indicates that the initial kinetic energy has been converted to electrical energy.
• If this was not the case & the induced current somehow achieved to speed the magnet up, more kinetic energy would be produced.
• This violates Law of Conservation of Energy.

     ‣ Case 2:

 

• The magnetic field that arises from the induced EMF MUST oppose the external flux change. Otherwise, energy is created form nowhere, violating the Law of Conservation of Energy.
• Therefore, the current induced flows clockwise. 

 » Applying Lenz’s Law to a Coil

1. Determine if the flux through the coil is increasing or decreasing.
2. By Lenz’s Law, the induced current’s magnetic field must oppose this change in flux
3. Fingers point in direction which opposes change in flux.

                i.  If the magnetic field directs into the page, it is decreasing.

                

 » Eddy Current

• It is a special type of current that is induced when a metal plate experiences a change in magnetic flux.
• Lenz’s Law governs the direction of eddy currents’ flow in a loop.

                                 

 » Determining the Direction of Eddy Currents

• Bad Method

1. Fingers point in the direction of the magnetic field.
2. Use the right-hand palm rule, such that the direction of the force (palm) faces opposite to the direction of movement of the conductor.
3. Identify the direction in which the conducting plate is moving
4. Draw a loop at the location/boundary where the change in magnetic flux is occurring.    ‣ Perform this step inside the boundary containing the magnetic field.
5. Thumb points to the direction of the induced current.
6. The current loops around the boundary.

• Good Method

    -  Direction of current can be found by using the right-hand coil rule

1. Thumb points in the direction that opposes the initial change in flux.
2. Finger wrap around the eddy current in the direction of conventional current.

 » Transformers

• They allow generated AC voltage to either by increased or decreased before it is used.
• They function by mutual induction where a changing current in one coil causes an induced EMF in the area of another coil.
• A transformer has two coils (primary & secondary coils) of conducting wires wound on a laminated iron core.

                                      

         - Iron core:

• Material with high permeability to concentrate & guide the magnetic field lines inside the core.
• AC current is fed into the primary coil which induces a current in the secondary coil.
• AC current switches current direction periodically, which results in a changing magnetic field.
• The secondary coil experiences a change in magnetic flux; therefore, AC current is induced.

  

 » From Faradays’ Law:

• By the law of conservation of energy, energy in the primary coil is conserved when transmitted to the second coil through electromagnetic induction.

           

• Combining these equations, we obtain the full transformer equation:

                                            

 » Types of Transformers

          

 

 » Applications of Transformers

 

 » Power Loss in Transmission Lines

• The resistance in the transmission lines causes an inefficient transfer of energy
  when the electrical energy transmitted is eventually transformed into heat
  energy.
• The power lost can be calculated through;
• To reduce the power loss, current must be reduced before entering the transmission lines.
• This is done by set-up transformers.

    

 » Applications of Transformers

• Not all appliance run on the same voltage. Therefore, transformers exist in household appliances to increase/decrease the supplied voltage for suitable use, allowing them to be conveniently connected to the same power supply.
• Substations near power-plants use step-up transformers to reduce the amount of power loss
• Substations near households/consumers use step-down transformers to reduce voltages to safe & practical levels
• Allows remote communities to access grid electricity.

 » Limitations of Transformers

• Resistive Heat Production

• The iron core experiences changes in flux. Therefore, eddy currents are induced.
• Eddy currents in the iron core generates a significant amount of heat energy via resistance. The iron core heats up.
• This is an inefficient transfer of energy. It also poses a fire hazard.

• Solutions

⁛  Lamination:

- Iron core is laminated by insulation sheets, introducing electrical discontinuity in the core.

- This reduces the eddy currents that are formed inside the core. Therefore, decreasing the heat producing and increasing the efficiency of the transformers.

⁛  Other:

- A ferrite core (made from iron oxides) can substitute the iron core.

•  Ferrites are great magnetic flux transmitters; but poor electrical conductors, so the magnitude of eddy currents are significantly reduced. 

-  Overheating can cause the isolation to foil, leading to larger currents to flow & extreme heat will follow.

-  To prevent overheating, heat can be dissipated by using:

• Water & oil as coolants.

• Fans to increase air circulation through & around transformer.

• Heat sinks to disperse heat elsewhere.

• Incomplete Flux Linkage

• Flux Linkage is the total magnetic flux passing through the turns of a coil (the flux ‘links’ the turns).
• Each turn links the flux identically.

• This is applicable in an ideal scenario. In reality, it is extremely rare for the flux to link all the turns. However, there is a flux linkage.
• This results in incomplete flux transmission between the primary & secondary coil, hence inefficiency is inevitable.

 » Electromagnetic Braking

• Electromagnetic braking applies the EM induction principle.
• When the metal wheels pass through a magnetic field, eddy currents are
  produced in the wheel. This is due to the change in magnetic flux (Faraday’s
  Law) that the wheel experiences
• The eddy currents are induced in a specific way to counteract the motion of the wheel. Hence, slowing down its rotation (Lenz’s Law).
• EM braking is used by modern trains and rollercoasters.

 

• Advantages

• Since there is no direct contact between components, friction is
  removed. This reduces the need for maintenance & replacement.

⁙  Little to no noise

⁙  Smooth brake effect

• Disadvantages:

• EM braking is incapable of holding the transport system after coming to rest. In this case, mechanical braking is required.

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Applications of Motor Effect

 » DC Motors

• A DC motor is a device which converts electrical energy into mechanical energy
• DC current is fed through a coil in a magnetic field, which produces a rotation motion due to the motor effect.

• Using a simple design and wires attached to a power supply results in two problems:

1. The direction of force acting on each side of the coil reverses every halfrotation meaning that the coil can’t completely rotate & gets stuck at the vertical position.
2. Even if the coil completes its rotations, eventually the wires attached would get tangled.

• We can solve both issues by using a split-ring commutator.

 

 » Components of a DC Motor:

           

        

• Armature/Coil (Rotor):

  ⁛  Armature:

• The frame around which the coil of wire is wound. It
  has an axis on which it can freely rotate.

  ⁛ Coil:

• One or more turns of wire wound around the
  armature. Current-carrying coils experience forces
  which act in certain directions due to the motor
  effect. This force is a rotational force, known as
  torque.
• ↑ turns = ↑ force = ↑ torque

• Stator:

• The stationary permanent/electromagnet that provides the
  external, radial magnetic field around the coils.
• Radial magnetic field ensures the sides of the coil are always travelling perpendicular to the magnetic field to produce the possible maximum torque throughout its rotation.

• Split-Ring Commutator:

• A device with two metal semi-circular rings that reverses the
  direction of the current flowing in each coil at every 180^o,allowing the rotor to continuously rotate in the same direction.

•  Brushes:

 ⁛  Conducting contacts (generally graphite or carbon) that connect
        the external circuit to the split-ring commutator.

• Carbon & graphite are preferential as they are both
  good conductors of electricity & good lubricants,
  thereby reducing the friction between commutator. 

 » Operation of a DC Motor

 

            

1. Side WX experiences a force upwards because it is connected to the + terminal. Side YZ                          experiences a force downwards because it is connected to the – terminal.

• The combination of these two forces initiate the clockwise rotation
  of the coil.

      

2. As the coil rotates, it will reach a perpendicular position where the terminals are fully                         disconnected & there is no force.

• However, the momentum of the coil keeps the coil rotating
  clockwise & help reconnect the commutators with the brushes.

  

• Notice that any further rotation without a commutator, will cause the sides to produce a force that causes the coil to rotate anti-clockwise back to its perpendicular position

                 

• A commutator prevents this. Just after the coils’ perpendicular
  position, the split-ring commutator changes the direction of the
  current through the coil.

       

• The forces now act to allow the coil to continoulsy rotate clockwise.

3. Now side WX experiences a force downwards because it is connected to the terminal. On the              other hand, side YZ experiences a force upwards because it is now connected to the + terminal.

• This process repeats as the motor rotates.

        

 » Torque

• Torque is the required force to cause an object to rotate.

• The net torque is the sum of all acting torques.

 » Torque in a DC Motor

• Consider the torque produced within a DC motor:

     

• Now, what happens if the coil is at an angle?

• The turning force that the coil experiences in an electric motor is referred to as
  the torque & is caused by the forces acting on the sides.

        

» Back EMF

• Back EMF is the induced EMF produced in the coil of a motor due to its rotation
  in a magnetic field.
• The rotation supplies the change in flux, thus a current is induced (Faraday’s
  Law).
• As a consequence of Lenz’s Law, the induced EMF opposes the change that
  causes it & therefore acts in the opposite direction to the EMF creating it.
• Therefore, back EMF works against the input voltage from the power supply.
• Back EMF reduces the net EMF:

                                     

• Net current also decreases, as voltage & current share a directly proportional
  relationship (V 𝖺 I).

» Significance of Back EMF

-  Magnitude of back EMF is directly proportional to speed of rotation.

• i.e. a faster rotating motor induces a larger back EMF.

-  When there is no load:

• Back EMF is initially zero when the coil is stationary & increases to a maximum as the coil reaches its maximum rate of rotation.
• A maximum rate of rotation, the coil rotates at a constant angular
  speed.
• So, the next force acting on the coil is ZERO!

-  When a load is introduced:

• ↑ loads = ↓ speed of rotation = ↓ back EMF = ↑ current
• ↑ loads (slower rotations) = ↑ currents in the coil.
• ↑ current = ↑ torque (sacrificing heat production).

 ⁛  Back EMF keeps dropping until a high enough current & torque is reached to meet the load experiment.

• When the motor comes to a sudden halt (drill gets stuck), back EMF will be completely removed.
• This leads to extremely high currents that could burn out the motor.
• When a DC motor starts, there is little back EMF or rotation. This
  means the coil experiences the full initial current.
• In order to reduce this, a variable resistor is placed in series with the armature to provide a starting resistance.
• When the motor speeds up, the back EMF increases (acts like
  resistance itself).
• The resistor switches out at higher speeds because the back EMF is sufficient to lower the current in the coil.
• This resistor can switch on any time thereafter to protect the coil
  against high currents, thereafter, preventing a burn out (e.g. stuck
  drill).

» Generators

• A generator is a device which converts mechanical energy into electrical energy
  by applying the principle of electromagnetic induction.
• Its anatomy is extremely similar to a DC motor. The key difference is the lack of
  a power supply (power is now generated).
• A coil of wire is forced to rotate about an axis in a magnetic field.
• This causes the coil to experience a change in magnetic flux, inducing an EMF
  (Faraday’s Law). This then transferred to an external circuit.
• There are two types of generators: AC & DC.

- Both need an external source to mechanically turn the coil. E.g. water or steam turbine.

  

‣ AC Generator:

• Slip Rings:

• Two cylindrical metal conductors that rotate freely with the
  armature.
• They provide constant electrical contact between the rotating
  armature & external circuit.
• The armature’s rotation naturally produced an AC voltage which
  is transmitted to the slip rings, giving an AC output.

‣ DC Generator:

• Split Rings:

• The flux-time graph of a DC generator is identical to an AC’s.
• However, its EMF-time graph is always in one direction.
• The split-ring commutator reverses the direction of the natural AC voltage every half-cycle.
• This rectifies the output voltage to become unidirectional (DC).

                           

                  

‣ AC vs DC

  

» AC Induction Motors

• AC induction motors use AC current as opposed to DC current from DC motors.
• These motors are different since they use the principle of electromagnetic
  induction to rotate the rotor, instead of the motor effect in traditional motors.
• The rotor isn’t supplied current from a supply. It is induced.

» Structure of an AC induction Motor

• AC has a stator and rotor, just like DC.
• The rotor is an assembly of parallel conductors & end rings. They produce a
  similar shape to ‘squirrel cages’.
• The stator is the stationary electrical component, which is made up of pairs of electromagnets connected to an AC power supply.
• The coils are wound in a way that when a current flows through the coils, one coil would be the north pole & its pair a south pole.

                                                      

» Operation of an AC Induction Motor

• Initially, one pair of electromagnets is fed AC current to produce a magnetic field, while the remaining two pairs do not.
• Due to the nature of the AC power supply, this pair is then turned off & the adjacent pair is turned on.
• This process continues, ultimately producing rotating magnetic field.
• Phase AC Motor:

               

• Due to the rotating magnetic field, an electric current is induced in the rotor (Faraday’s Law). This induced electric current produces its own magnetic field.
• Due to Lenz’s Law, this magnetic field is induced in such a way that opposes the change that causes it, effectively causing the rotor to spin the same direction as the rotating magnetic field.

• A magnetic field rotating clockwise equivalent to the rotor rotating
  anti-clockwise (same relative motion). Lenz’s Law acts to counteract the change that causes it, resulting in a clockwise rotation of the rotor.

• The rotor continually ‘chases’ the rotating magnetic field, always slower &
  never catches up.
• The difference in rotational speed between the magnetic field & rotor is known
  as the slip speed. If there is slip speed, there is relative motion between the
  stator and rotor.
• Relative motion is required for torque generation in AC induction motors.

» AC Induction Motor

• Advantages:

• Cost effective (absence ofsplit-ring commutators & brushes)
• Reduced maintenance/reduced wear & tear (absence of split-ring commutator & brushes)

• Disadvantages:

• Poor starting torque
• Used only in fixed-speed applications

» Definitions:

 

» Formulas