7.5 magnetic fields

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electromagnetic induction
wire ‘cuts’ magnetic field inducing electromotive force
the faster the wire ‘cuts’, the greater the emf
lenz’s law
when a magnet is pushed into a solenoid, the direction of the induced emf/current is always as to oppose the change that causes that emf/current
movement against repulsion
south pole induced - clockwise
movement against attraction
north pole induced - anticlockwise
magnetic flux
the product of magnetic flux density and the area perpendicular to the field
area - amount of field ‘cut’
measured in weber (Wb)
Magnetic field
Region where a force is exerted on magnetic material
Represented by field lines
Flux lines
From the north to the south pole of a magnet
Closer together the lines, the stronger the field
Magnetic field is induced around a wire when a current flows through it
Field lines are concentric circles centered on the wire
The direction of a magnetic field is determined using the right-hand rule
Thumb = direction of current
Fingers = direction of field
Field is doughnut-shaped around a coil
Solenoid forms a field like a bar magnet
Force on a current-carrying wire
A current-carrying wire placed into an external magnetic field will result in the field around the wire and the magnetic field adding together
Resultant field where the field is stronger, causing a pushing force on the wire
Direction of force is always perpendicular to the current direction and the magnetic field
Flemming’s left-hand rule
Force, field, current
If current is parallel to the field lines, there is no component of the magnetic field perpendicular to the current
Force is 0N
Flux density
Force on a current-carrying wire at a right angle to an external magnetic field is proportional to the magnetic flux density
Magnetic flux density = the force on one meter of wire carrying a current of one amp at right angles to the magnetic field
Strength of the magnetic field
When current is 90° to the magnetic field, the size of the force is proportional to the current/length of wire/flux density
$F=$ $BIl$
Flux density is a vector with direction and magnitude
Measured in teslas
$T=$ $\frac{Wb}{m^2}$
Number of flux lines per unit area
Core practical 10: Force on a wire varies with flux density, current and length of wire
Square hoop of metal wire positioned so the top of the hoop passes through the magnetic field perpendicular to it
When current flows the length of the wire in the field, it will experience a downwards force
Power supply connected to variable resistor to alter the current
Zero the balance when there is no current through the wire
Mass reading is due to electromagnetic force only
Record mass and current using a variable resistor to change the current and record the new mass
...practical: magnetic flux density
Repeat with a large range of currents, recording three mass readings for each and find the mean
Convert mass into force using $F=$ $mg$ and plot a graph of force against current
Gradient is $B\times l$
Divide gradient by length to find B
Can vary the length of wire perpendicular to the field by using different-sized hoops, or vary the magnetic field instead, using different strengths of the magnet
Force on charged particle in magnetic field
Electric current in a wire is the flow of negatively charged electrons
Force on current-carrying wire in magnetic field is perpendicular to the currency
$F=$ $BIl$
Electric current = the flow of charge per unit time
$I=$ $\frac{Q}{t}$
Charged particle moving with velocity
$v=$ $\frac{l}{t}\ \ \ \rightarrow\ \ \ l=$ $vt$
Force acting on single charged particle moving through magnetic field where velocity is perpendicular to magnetic field
$F=$ $BIl=$ $\frac{BQ}{t}vt\ \ \ \rightarrow\ \ \ F=$ $BQv$
Movement of charged particles in a magnetic field
Flemmings left-hand rule
Force on a moving charge in a magnetic field is always perpendicular to its direction of travel
Condition for circular motion
Conventional current = movement of positively charged particles
If particles are negatively charged, flip hand so second finger (current) is facing opposite direction - or use right hand
...Movement of charged particles in magnetic field
Force due to magnetic field experienced by particle travelling through magnetic field is independent of particle’s mass
Centripetal acceleration depends on mass
$a=$ $\frac{v^2}{r}$
Force on particle in circular orbit $F=$ $\frac{mv^2}{r}$
Radius of circular path found by combining force on charged particle in magnetic field and circular orbit
$\frac{mv^2}{r}=$ $BQv$
$r=$ $\frac{mv}{BQ}$
Radius increases if mass / velocity increases
Radius decreases if the strength of magnetic field or charge increases
Cyclotrons
Circular deflection is used in particle accelerators
Used in medicine
To produce radioactive tracers
High-energy beams of radiation used in radiotherapy
Made up of two hollow semicircular electrodes with uniform magnetic field applied perpendicular to the plane of the electrodes
Has an alternating potential difference between the electrodes
Particle acceleration in a cyclotron
Charged particles fired into one electrode
Magnetic field makes them follow a circular path before leaving the electrode
Applied pd between electrodes accelerates particles across the gap
Increased speed results in a circular path with a larger radius
Potential difference is reversed, accelerating the particle again and the process repeats as the particle spirals outwards
Speed increases before the particle exits the cyclotron
Magnetic flux
$\phi=$ $BA$
Magnetic flux density is perpendicular to the area
Magnetic flux linkage
More coils in a wire mean a larger emf is induced
Used for solenoids
Made of N turns of wire
Flux linkage = product of the magnetic flux and the number of turns of the coil
$\phi N=$ $BAN$
$\phi$ = magnetic flux (Wb)
N = number of turns of the coil
B = magnetic flux density (T)
A = cross-sectional area (m2)
Flux linkage ( $\phi N$ ) measured in weber turns
Number of turns cutting the flux
Flux linkage in a rotating coil
When the magnetic field lines aren't completely perpendicular to the area, then the perpendicular component of magnetic flux density is taken
$\phi=$ $BA\cos\left(\theta\right)$
Magnetic flux maximum is BA
when cos( $\theta$ )=1
The angle is 0°
The magnetic field lines are perpendicular to the plane of the area
Magnetic flux minimum is 0
when cos( $\theta$ )=0
The angle is 90°
The magnetic field lines are parallel to the plane of the area
Magnetic flux through a rectangular coil decreases as the angle between the field lines and plane decrease
Field lines might not be completely perpendicular to the plane of the area they pass through
$\phi N=$ $BAN\cos\left(\theta\right)$
Induced emf
emf is induced in a circuit when the magnetic flux linkage changes with time
induced when there is:
A changing magnetic flux density
A changing cross-sectional area
A change in angle
Core practical 11
Investigating the effect of angle to the flux lines on effective magnetic flux linkage
A stretched metal spring acts as a solenoid and is connected to an alternating power supply
The flux through the search coil is constantly changing
The search coil should have a known area and a set number of loops of fine wire
Connected to an oscilloscope to record the induced emf in the coil
Set up the oscilloscope so it only shows the amplitude of the emf as a vertical line
Position the search coil so it is halfway along the solenoid
Orientate the search coil so it is parallel to the solenoid and its area is perpendicular to the field
Record the induced emf in the search coil from the amplitude of the oscilloscope trace
Rotate the search coil so its angle to the solenoid changes by 10°
Record induced emf and repeat until the search coil has been rotated by 90°
...core practical 11
As the coil rotates, the induced emf decreases
As the coil is cutting fewer flux lines as the component of the magnetic field perpendicular to the area of the coil gets lower
The total magnetic flux passing through the search coil is lower
The magnetic flux linkage experienced by the coil is lower
Plot a graph of induced emf against the angle
Induced emf should be maximum at 0° and zero at 90°
Inducing electromotive forces
If there is a relative motion between the conducting rod and a magnetic field, the electrons in the rod will experience a force
Causes electrons to accumulate at one end of the rod
Electromagnetic induction = induces an electromotive force across the ends of the rod
Can induce an emf in a flat coil or solenoid
By moving the coil towards or away from the poles of a magnet
By moving a magnet towards or away from the coil
Emf is caused by the changing of the magnetic field that passes through the coil
If the coil is part of a complete circuit, an induced current will flow through it
Size of emf
The size of the emf is dependent on the magnetic flux passing through the coil and the number of turns in the coil that cut the flux
When a coil is moved in a magnetic field
Flux linkage = product of magnetic flux and the number of turns that cut the flux
The rate of change in flux is the strength of electromotive force in volts
A change in flux-linkage of one weber per second will induce an electromotive force of 1 volt in a loop of wire
Faraday’s law
The induced emf is directly proportional to the rate of the change of flux linkage
$\epsilon=$ $\frac{flux\ linkage\ change}{time\ taken}=$ $\frac{N\Delta\Phi}{\Delta t}$
The size of the emf is shown by the gradient of a graph of flux linkage against time
The area under the graph of the magnitude of the emf against time gives the flux linkage change
Lenz’s law
The induced emf is always in such a direction as to oppose the change that caused it
$\epsilon=$ $\frac{-flux\ linkage\ change}{time\ taken}=$ $\frac{-N\Delta\phi}{\Delta t}$
The minus shows the direction of the induced emf
An induced emf opposing the change that causes it agrees with the principle of the conservation of energy
The energy used to pull a conductor through a magnetic field against the resistance caused by magnetic attraction is what produces the induced current
Lenz’s law can be used to find the direction of an induced emf and current in a conductor travelling at right angles to a magnetic field
Using Fleming’s left-hand rule
Lenz’s law says that the induced emf will produce a force that opposes the motion of the conductor
Resistance
Thumb is the direction of the force of resistance
The opposite direction to the motion of the conductor
The first finger is the direction of the field
The second finger is the direction of the induced emf
If the conductor is connected as part of a circuit, a current will be induced in the same direction as the induced emf
Alternator
Generators, dynamos convert kinetic energy into electrical energy
Induce an electric current by rotating a coil in a magnetic field
It has slip rings and brushes to connect the coil to an external circuit
The output voltage and current change direction with every half rotation of the coil
Produces alternating current
Flux linkage and induced voltage
90° out of phase
$N\phi=$ $BAN\cos\left(\theta\right)$ is the amount of flux cut by the coil
Flux linkage
As the coil rotates, the angle changes, so the flux linkage varies sinusoidally between +BAN and -BAN
The angular speed determines the rate of change of the angle
$\theta=$ $\omega t$
flux linkage = $N\phi=$ $BAN\cos\omega t$
The induced emf depends on the rate of change of flux linkage, so it varies sinusoidally
Due to Faraday’s law
$\epsilon=$ $BAN\omega\sin\omega t$
Emf induced in a coil rotating uniformly in a magnetic field
Alternating current
Alternating current = changes direction with time
Same with voltage
Voltage across a resistance oscillates in a regular pattern
Can use an oscilloscope to display the voltage of an alternating or direct current
The vertical height of the trace shows the input voltage at that point
An alternating current source gives a regularly repeating sinusoidal waveform
A direct current source is a constant voltage, so it shows a horizontal line
If the time base is turned off, ac voltage is a straight line, and dc voltage is a dot
Voltage values
Peak voltage = maximum value of voltage an ac supply can have
Has a lower power output than a dc supply of that peak voltage
rms voltage = the root of the mean voltage squared
Use the root mean square voltage (rms) to compare ac and dc supply
For a sine wave, the rms voltage and current can be found by dividing the peak voltage by root 2
$V_{rms}=$ $\frac{V_0}{\sqrt{2}}$
$I_{rms}=$ $\frac{I_0}{\sqrt{2}}$
The average power for an ac supply is the product of the rms voltage and current
$P_{average}=$ $V_{rms}\times I_{rms}$
The time period is the distance between successive peaks along the time axis and can be used to calculate frequency
$frequency=$ $\frac{1}{time\ period}\ \ \ \ \ f=$ $\frac{1}{T}$
Calculating mains electricity
The mains electricity supply is approximately 230V
It is an alternating supply, so 230V is the rms value
$V_0=$ $V_{rms}\times\sqrt{2}$
$230\times\sqrt{2}=$ $325.26$
$V_{peak\ to\ peak}=$ $2\ \times325.26=$ $650.53V$
Transformers
Transformers = devices that use electromagnetic induction to change the size of the voltage for an alternating current
An alternating current flowing in the primary coil produces magnetic flux
The changing magnetic field is passed through the iron core to the secondary coil, where it induces an alternating voltage of the same frequency as the input voltage
From Faraday's law, the induced emf in the two coils can be calculated
Primary coil: $V_P=$ $N_P\ \frac{\Delta\Phi}{\Delta t}$
Secondary coil: $V_S=$ $N_S\ \frac{\Delta\Phi}{\Delta t}$
The transformer equation shows their relationship in an ideal transformer
$\frac{N_S}{N_P}=$ $\frac{V_S}{V_P}$
Step-up transformers increase the voltage by having more turns on the secondary coil than the primary
Step-down transformers reduce the voltage by having fewer turns on the secondary coil
Transformer efficieny
If a transformer is 100% efficient, the input power would equal the output power
Power is the product of current and voltage
For an ideal transformer $I_PV_P=$ $I_SV_S$
$\frac{N_S}{N_P}=$ $\frac{V_S}{V_P}=$ $\frac{I_P}{I_s}$
In practice, there are losses of power from the transformer
Mostly due to eddy currents in the transformer’s iron core
Eddy currents = looping currents induced by the changing magnetic flux in the core
They create a magnetic field that acts against the field that induced them, reducing field strength
Dissipate energy by generating heat
Laminating the core with layers of insulation reduces the effect of eddy currents
Heat is generated by resistance, so thick copper wire is used as it has a low resistance
Efficiency = the ratio of power out to power in
efficiency= $\frac{I_SV_S}{I_PV_P}$
The national grid
Electricity from power stations is sent round the country in the national grid at the lowest possible current
The power losses due to the resistance of the cables $P=$ $I^2R$
A low current means a high voltage
P=VI
Transformers step up the voltage to around 400 000V for transmission and reduce it again to 230V for domestic use