Chapter 9 Magnetic Fields Flashcards
A magnetic field is a field of force that is created either by
- Moving electric charge
- Permanent magnets
- Permanent magnets are materials that produce a magnetic field
- A stationary charge will not produce a magnetic field
- A magnetic field is sometimes referred to as a B–field
- A magnetic field is created around a current carrying wire due to the movement of electrons
- Although magnetic fields are invisible, they can be observed by the force that pulls on magnetic materials, such as iron or the movement of a needle in a plotting compass
Magnetic fields are represented by
- magnetic field lines
- These can be shown using iron filings or plotting compasses
- Field lines are best represented on bar magnets, which consist of a north pole on one end and south pole on the other
- The magnetic field is produced on a bar magnet by the movement of electrons within the atoms of the magnet
- This is a result of the electrons circulating around the atoms, representing a tiny current and hence setting up a magnetic field
- The direction of a magnetic field on a bar magnet is always from north to south
Magnetic field lines are directed from the north pole to the south pole
When two bar magnets are pushed together, they either attract or repel each other:
- Two like poles (north and north or south and south) repel each other
- Two opposite poles (north and south) attract each other
The key aspects of drawing magnetic field lines:
- The lines come out from the north poles and into the south poles
- The direction of the field line shows the direction of the force that a free magnetic north pole would experience at that point
- The field lines are stronger the closer the lines are together
- The field lines are weaker the further apart the lines are
- Magnetic field lines never cross since the magnetic field is unique at any point
- Magnetic field lines are continuous
A uniform magnetic field is where the magnetic field strength is the
- same at all points
- This is represented by equally spaced parallel lines, just like electric fields
- The direction of the magnetic field into or out of the page in 3D is represented by the following symbols:
- Dots (sometimes with a circle around them) represent the magnetic field directed out of the plane of the page
- Crosses represent the magnetic field directed into the plane of the page
The magnetic field into or out of the page is represented by circles with dots or crosses
A current-carrying conductor produces its own
-
magnetic field
- When interacting with an external magnetic field, it will experience a force
- A current-carrying conductor will only experience a force if the current through it is perpendicular to the direction of the magnetic field lines
- A simple situation would be a copper rod placed within a uniform magnetic field
- When current is passed through the copper rod, it experiences a force which makes it move
A copper rod moves within a magnetic field when current is passed through it
The strength of a magnetic field is known as the
-
magnetic flux density, B
- This is also known as the magnetic field strength
- It is measured in units of Tesla (T)
- The force F on a conductor carrying current I at right angles to a magnetic field with flux density B is defined by the equation
F = BIL sinθ
- Where:
- F = force on a current carrying conductor in a B field (N)
- B = magnetic flux density of external B field (T)
- I = current in the conductor (A)
- L = length of the conductor (m)
- θ = angle between the conductor and external B field (degrees)
- This equation shows that the greater the current or the magnetic field strength, the greater the force on the conductor
Magnitude of the force on a current carrying conductor depends on the angle of the conductor to the external B field
The maximum force occurs when
- sin θ = 1
- This means θ = 90o and the conductor is perpendicular to the B field
- This equation for the magnetic force now becomes:
F = BIL
- The minimum force (0) is when sin θ = 0
- This means θ = 0o and the conductor is parallel to the B field
- It is important to note that a current-carrying conductor will experience no force if the current in the conductor is parallel to the field
The direction of the force on a charge moving in a magnetic field is determined by the
- direction of the magnetic field and the current
- Recall that the direction of the current is the direction of conventional current flow (positive to negative)
- When the force, magnetic field and current are all mutually perpendicular to each other, the directions of each can be interpreted by Fleming’s left-hand rule:
- On the left hand, with the thumb pointed upwards, first finger forwards and second finger to the right ie. all three are perpendicular to each other
- The thumb points in the direction of motion of the rod (or the direction of the force) (F)
- The first finger points in the direction of the external magnetic field (B)
- The second finger points in the direction of conventional current flow (I)
Fleming’s left hand rule
Using Fleming’s left-hand rule: with papers
B = into the page
F = vertically downwards
I = from right to left
The magnetic flux density B is defined as
The force acting per unit current per unit length on a current-carrying conductor placed perpendicular to the magnetic field
- Rearranging the equation for magnetic force on a wire, the magnetic flux density is defined by the equation:
- Note: this equation is only relevant when the B field is perpendicular to the current
- Magnetic flux density is measured in units of tesla, which is defined as:
A straight conductor carrying a current of 1A normal to a magnetic field of flux density of 1 T with force per unit length of the conductor of 1 N m-1
- To put this into perspective, the Earth’s magnetic flux density is around 0.032 mT and an ordinary fridge magnet is around 5 mT
- The magnetic force on an isolating moving charge, such an electron, is given by the equation:
F = BQv sinθ
- Where:
- F = force on the charge (N)
- B = magnetic flux density (T)
- Q = charge of the particle (C)
- v = speed of the charge (m s-1)
- θ = angle between charge’s velocity and magnetic field (degrees)
The force on an isolated moving charge is perpendicular to its motion and the magnetic field B
- Equivalent to the force on a wire, if the magnetic field B is perpendicular to the direction of the charge’s velocity, the equation simplifies to:
F = BQv
- According to Fleming’s left hand rule:
- When an electron enters a magnetic field from the left, and if the magnetic field is directed into the page, then the force on it will be directed upwards
- The equation shows:
- If the direction of the electron changes, the magnitude of the force will change too
- The force due to the magnetic field is always perpendicular to the velocity of the electron
- Note: this is equivalent to circular motion
Fleming’s left-hand rule can be used again to find the direction of the
force, magnetic field and velocity
- The key difference is that the second finger representing current I (direction of positive charge) is now the direction of velocity v of the positive charge
The Hall voltage is a product of the
- Hall effect
- Hall voltage is defined as:
The potential difference produced across an electrical conductor when an external magnetic field is applied perpendicular to the current through the conductor
When an external magnetic field is applied
- perpendicular to the direction of current through a conductor, the electrons experience a magnetic force
- This makes them drift to one side of the conductor, where they all gather and becomes more negatively charged
- This leaves the opposite side deficient of electrons, or positively charged
- There is now a potential difference across the conductor
- This is called the Hall Voltage, VH
The positive and negative charges drift to opposite ends of the conductor producing a hall voltage when a magnetic field is applied
- An equation for the Hall voltage VH is derived from
the electric and magnetic forces on the charges
The electric and magnetic forces on the electrons are equal and opposite
The voltage arises from the
- electrons accumulating on one side of the conductor slice
- As a result, an electric field is set up between the two opposite sides
- The two sides can be treated like oppositely charged parallel plates, where the electric field strength E is equal to:
- Where:
- VH = Hall voltage (V)
- d = width of the conductor slice (m)
A single electron has a drift velocity of v within the conductor. The magnetic field is into the plane of the page, therefore the electron has a magnetic force FB to the right:
FB = Bqv
- This is equal to the electric force FE to the left:
FE = qE
qE = Bqv
- Substituting E and cancelling the charge q
VH/d = Bv
- Recall that current I is related to the drift velocity v by the equation:
I = nAvq
- Where:
- A = cross-sectional area of the conductor (m2)
- n = number density of electrons (m-3)
- Rearranging this for v and substituting it into the equation gives:
VH/d = B(I/nAq)
- The cross-sectional area A of the slice is the product of the width d and thickness t:
A = dt
- Substituting A and rearranging for the Hall voltage VH leads to the equation:
VH/d = B(I/n(dt)q)
VH/d = B(I/ntq)
- Where:
- B = magnetic flux density (T)
- q = charge of the electron (C)
- I = current (A)
- n = number density of electrons (m-3)
- t = thickness of the conductor (m)
- This equation shows that the smaller the electron density n of a material, the larger the magnitude of the Hall voltage
- This is why a semiconducting material is often used for a Hall probe
-
Note: if the electrons were placed by positive charge carriers, the negative and positive charges would still deflect in opposite directions
- This means there would be no change in the polarity (direction) of the Hall voltage
A Hall probe can be used to measure the
- magnetic flux density between two magnets based on the Hall effect
- It consists of a cylinder with a flat surface at the end
A Hall probe consists of a flat surface and is held so the magnetic field lines are perpendicular to it
A Hall probe consists of a flat surface and is held so the magnetic field lines are perpendicular to it
- the flat surface of the probe must be directed between the magnets so the magnetic field lines pass completely perpendicular to this surface
- The probe is connected to a voltmeter to measure the Hall voltage
- If the probe is not held in the correct orientation (perpendicular to the field lines), the voltmeter reading will be reduced
- Since the Hall voltage is directly proportional to the magnetic flux density, the flux density of the magnets can be obtained
- A Hall probe is sensitive enough to measure even the Earth’s magnetic flux density
The Hall voltage depends on
angle between the magnetic field and the plane of the probe
The Hall voltage reaches a maximum when the
field is perpendicular to the probe
The Hall voltage is zero when the field is
- parallel to the probe
A charged particle in uniform magnetic field which is perpendicular to its direction of motion travels in a
- circular path
- This is because the magnetic force FB will always be perpendicular to its velocity v
- FB will always be directed towards the centre of the path
- The magnetic force FB provides the centripetal force on the particle
- Recall the equation for centripetal force:
F= Mv2/r
- Where:
- m = mass of the particle (kg)
- v = linear velocity of the particle (m s-1)
- r = radius of the orbit (m)
- Equating this to the force on a moving charged particle gives the equation:
Mv2/r = Bqv
- Rearranging for the radius r obtains the equation for the radius of the orbit of a charged particle in a perpendicular magnetic field:
r = mv/Bq
- This equation shows that:
- Faster moving particles with speed v move in larger circles (larger r): r ∝ v
- Particles with greater mass m move in larger circles: r ∝ m
- Particles with greater charge q move in smaller circles: r ∝ 1 / q
- Particles moving in a strong magnetic field B move in smaller circles: r ∝ 1 / B
- A velocity selector is:
A device consisting of perpendicular electric and magnetic fields where charged particles with a specific velocity can be filtered
Velocity selectors are used in devices, such as
mass spectrometers, in order to produce a beam of charged particles all travelling at the same velocity
The construction of a velocity selector consists of two
horizontal oppositely charged plates situated in a vacuum chamber
- The plates provide a uniform electric field with strength E between them
There is also a uniform magnetic field with flux density B applied perpendicular to the electric field
- If a beam of charged particles enter between the plates, they may all have the same charge but travel at different speeds v
- The electric force does not depend on the velocity: FE = EQ
- However, the magnetic force does depend on the velocity: FB = BQv
- The magnetic force will be greater for particles which are travelling faster
- To select particles travelling at exactly the desired the speed v, the electric and magnetic force must therefore be equal, but in opposite directions
FE = FB
The particles travelling at the desired speed v will travel through undeflected due to the equal and opposite electric and magnetic forces on them
The resultant force on the particles at speed v will be
- zero, so they will remain undeflected and pass straight through between the plates
- By equating the electric and magnetic force equations:
EQ = BQv
- The charge Q will cancel out on both sides to give the selected velocity v equation:
V= E/B
- Therefore, the speed v in which a particle will remain undeflected is found by the ratio of the electric and magnetic field strength
- If a particle has a speed greater or less than v, the magnetic force will deflect it and collide with one of the charged plates
- This would remove the particles in the beam that are not exactly at speed v
the gravitational force on the charged particles will be negligible compared to the
electric and magnetic forces and therefore can be ignored in these calculations
Magnetic field patterns are not only observed around bar magnets, magnetic fields are formed wherever current is flowing, such as in:
- Long straight wires
- Long solenoids
- Flat circular coils
Field Lines in a Current-Carrying Wire
- Magnetic field lines in a current carrying wire are circular rings, centered on the wire
- The field lines are strongest near the wire and become further part away from the wire
- Reversing the current reverses the direction of the field
The direction of the magnetic field is determined by
Maxwell’s right hand screw rule
- This is determined by pointing the right-hand thumb in the direction of the current in the wire and curling the fingers onto the palm
- The direction of the curled fingers represents the direction of the magnetic field around the wire
- For example, if the current is travelling vertically upwards, the magnetic field lines will be directed anticlockwise, as seen from directly above the wire
- Note: the direction of the current is taken to be the conventional current ie. from positive to negative, not the direction of electron flow
Field Lines in a Solenoid
- As seen from a current carrying wire, an electric current produces a magnetic field
- An electromagnetic makes use of this by using a coil of wire called a solenoid which concentrates the magnetic field
- One ends becomes a north pole and the other the south pole
, the magnetic field lines around a solenoid are very similar to a bar magnet how?
- The field lines emerge from the north pole
- The field lines return to the south pole
- The field lines emerge from the north pole
- Which is the north or south pole depends on the direction of the current
- This is found by the right hand grip rule
- This involves gripping the electromagnet so the fingers represent the direction of the current flow of the wire
- The thumb points in the direction of the field lines inside the coil, or in other words, point towards the electromagnet’s north pole
Field Lines in a Flat Circular Coil
- A flat circular coil is equal to one of the coils of a solenoid
- The field lines will emerge through one side of the circle (north pole) and leave the other (south pole)
- As before, the direction of the north and south pole depend on the direction of the current
- This can be determined by using the right hand thumb rule
- It easier to find the direction of the magnetic field on the straight part of the circular coil to determine which direction the field lines are passing through
Factors Affecting Magnetic Field Strength
- The strength of the magnetic field of a solenoid can be increased by:
- Adding a core made from a ferrous (iron-rich) material eg. an iron rod
- Adding more turns in the coil
- When current flows through the solenoid with an iron core, it becomes magnetised, creating an even stronger field
- The addition of an iron core can strengthen the magnetic field up to a several hundred times more
- When more turns are added in the coil, this concentrates the magnetic field lines, causing the magnetic field strength to increase
Newton’s Third Law states
- When two bodies interact, the force on one body is equal but opposite in direction to the force on the other body
- Therefore, the forces on the wires act in equal but opposite directions
A current carrying conductor, such as a wire, produces a
- magnetic field around it
- The direction of the field depends on the direction of the current through the wire
- This is determined by the right hand thumb rule
Parallel current-carrying conductors will therefore either
attract or repel each other
- If the currents are in the same direction in both conductors, the magnetic field lines between the conductors cancel out – the conductors will attract each other
- If the currents are in the opposite direction in both conductors, the magnetic field lines between the conductors push each other apart – the conductors will repel each othe
Both wires will attract if their currents are in the same direction and repel if in opposite directions
- When the conductors attract, the direction of the magnetic forces will be towards each other
- When the conductors repel, the direction of the magnetic forces will be away from each other
- The magnitude of each force depend on the amount of current and length of the wire
Electromagnetic induction is when an
- e.m.f is induced in a closed circuit conductor due to it moving through a magnetic field
- This happens when a conductor cuts through magnetic field lines
- The amount of e.m.f induced is determined by the magnetic flux
The amount of magnetic flux varies as the coil rotates within the field
- The flux is the total magnetic field that passes through a given area
- It is a maximum when the magnetic field lines are perpendicular to the area
- It is at a minimum when the magnetic field lines are parallel to the area
- The magnetic flux is defined as:
The product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density
- In other words, magnetic flux is the number of magnetic field lines through a given area
Magnetic flux is defined by the symbol
- Φ (greek letter ‘phi’)
- It is measured in units of Webers (Wb)
- Magnetic flux can be calculated using the equation:
Φ = BA
- Where:
- Φ = magnetic flux (Wb)
- B = magnetic flux density (T)
- A = cross-sectional area (m2)
- When the magnet field lines are not completely perpendicular to the area A, then the component of magnetic flux density B perpendicular to the area is taken
- The equation then becomes:
Φ = BA cos(θ)
- Where:
- θ = angle between magnetic field lines and the line perpendicular to the plane of the area (often called the normal line) (degrees)
- This means the magnetic flux is:
- Maximum and Minimum at?
- Maximum = BA when cos(θ) =1 therefore θ = 0o. The magnetic field lines are perpendicular to the plane of the area
- Minimum = 0 when cos(θ) = 0 therefore θ = 90o. The magnetic fields lines are parallel to the plane of the area
An e.m.f is induced in a circuit when the
magnetic flux linkage changes with respect to time
- This means an e.m.f is induced when there is:
- A changing magnetic flux density B
- A changing cross-sectional area A
- A change in angle θ
The magnetic flux linkage is a quantity commonly used for
Magnetic flux linkage is defined as: and is calculated using the equation:
The product of the magnetic flux and the number of turns
- It is calculated using the equation:
ΦN = BAN
- Where:
- Φ = magnetic flux (Wb)
- N = number of turns of the coil
- B = magnetic flux density (T)
- A = cross-sectional area (m2)
- The flux linkage ΦN has the units of Weber turns (Wb turns)
- As with magnetic flux, if the field lines are not completely perpendicular to the plane of the area they are passing through
- Therefore, the component of the flux density which is perpendicular is equal to:
ΦN = BAN cos(θ)
Magnetic flux linkage is defined as: and is calculated using the equation:
The product of the magnetic flux and the number of turns
- It is calculated using the equation:
ΦN = BAN
- Where:
- Φ = magnetic flux (Wb)
- N = number of turns of the coil
- B = magnetic flux density (T)
- A = cross-sectional area (m2)
- The flux linkage ΦN has the units of Weber turns (Wb turns)
- As with magnetic flux, if the field lines are not completely perpendicular to the plane of the area they are passing through
- Therefore, the component of the flux density which is perpendicular is equal to:
ΦN = BAN cos(θ)
Electromagnetic induction is a
- phenomenon which occurs when an e.m.f is induced when a conductor moves through a magnetic field
- When the conductor cuts through the magnetic field lines:
- This causes a change in magnetic flux
- Which causes work to be done
- This work is then transformed into electrical energy
- Therefore, if attached to a complete circuit, a current will be induced
electromagnetic induction and is defined as:
The process in which an e.m.f is induced in a closed circuit due to changes in magnetic flux
- This can occur either when:
- A conductor cuts through a magnetic field
- The direction of a magnetic field through a coil changes
Electromagnetic induction is used in:
- Electrical generators which convert mechanical energy to electrical energy
- Transformers which are used in electrical power transmission
- Electrical generators which convert mechanical energy to electrical energy
- This phenomenon can easily be demonstrated with a magnet and a coil, or a wire and two magnets
Moving a magnet through a coil
Factors that will increase the induced e.m.f are:
- Moving the magnet faster through the coil
- Adding more turns to the coil
- Increasing the strength of the bar magnet
An e.m.f is induced only when the bar magnet is moving through the coil
A wire is moved between two magnets connected to a voltmeter to induce an e.m.f
Experiment 1: Moving a magnet through a coil expected results
- When the bar magnet is not moving, the voltmeter shows a zero reading
- When the bar magnet is held still inside, or outside, the coil, the rate of change of flux is zero, so, there is no e.m.f induced
- When the bar magnet begins to move inside the coil, there is a reading on the voltmeter
- As the bar magnet moves, its magnetic field lines ‘cut through’ the coil, generating a change in magnetic flux
- This induces an e.m.f within the coil, shown momentarily by the reading on the voltmeter
- When the bar magnet is taken back out of the coil, an e.m.f is induced in the opposite direction
- As the magnet changes direction, the direction of the current changes
- The voltmeter will momentarily show a reading with the opposite sign
- Increasing the speed of the magnet induces an e.m.f with a higher magnitude
- As the speed of the magnet increases, the rate of change of flux increases
- The direction of the electric current, and e.m.f, induced in the conductor is such that it opposes the change that produces it
Experiment 2: Moving a wire through a magnetic field expected results
- When the wire is not moving, the voltmeter shows a zero reading
- When the wire is held still inside, or outside, the magnets, the rate of change of flux is zero, so, there is no e.m.f induced
- As the wire is moved through between the magnets, an e.m.f is induced within the wire, shown momentarily by the reading on the voltmeter
- As the wire moves, it ‘cuts through’ the magnetic field lines of the magnet, generating a change in magnetic flux
- When the wire is taken back out of the magnet, an e.m.f is induced in the opposite direction
- As the wire changes direction, the direction of the current changes
- The voltmeter will momentarily show a reading with the opposite sign
- As before, the direction of the electric current, and e.m.f, induced in the conductor is such that it opposes the change that produces it
When Moving a wire through a magnetic field Factors that will increase the induced e.m.f are:
- Increasing the length of the wire
- Moving the wire between the magnets faster
- Increasing the strength of the magnets
Faraday’s law tells us the
- magnitude of the induced e.m.f in electromagnetic induction and is defined as:
The magnitude of the induced e.m.f is directly proportional to the rate of change in magnetic flux linkage
Faraday’s law equation
- Where:
- ε = induced e.m.f (V)
- N = number of turns of coil
- Δɸ = change in magnetic flux (Wb)
- Δt = time interval (s)
Lenz’s Law gives the
- direction of the induced e.m.f as defined by Faraday’s law:
The induced e.m.f acts in such a direction to produce effects which oppose the change causing it
- Lenz’s law combined with Faraday’s law is:
- This equation shows:
- When a bar magnet goes through a coil, an e.m.f is induced within the coil due to a change in magnetic flux
- A current is also induced which means the coil now has its own magnetic field
- The coil’s magnetic field acts in the opposite direction to the magnetic field of the bar magnet
If a direct current (d.c) power supply is replaced with an alternating current (a.c) supply, the e.m.f induced will also be
alternating with the same frequency as the supply
Showing that F= BIL and F = BQV are consistent with one another
-We know that I = Q/T -Sub for I in F = BIL - F= BQL/T -but L/t = velocity -F= BQV AND F= BEW FOR AN ELECTRON
The equation F= BEV on a charged particle
Tells us perspective on all the electrons within a conductor ( wire )
The charge to mass ratio of an electron
r= m(electron)v/Be = e/m(electron) = V/B
Speed of electrons
We use the cathode voltage to measure b and r The cathode and anode voltage is the Vca is the pd between them which cause each electron to accelerate as it moves from the cathode to the anode W= eVca W= 1/2Mev^2 r= MeV/ Be (1) 1/2mev^2 = eVca (2) Vca = 1/2B^2R^2/Me (3) Vca= 1/2 x B^2er^2/Me
