Topic 11: Electromagnetic Induction (HL) Flashcards
Define: magnetic flux density
Unit?
The magnetic field strength.
The force acting on a current, divided by the current and the length of the current element.
B = (F/IL)
Units: Tesla (T) or kgA-1s-2
Derive the magnetic field force equation in terms of charge and velocity
F = B I L sinθ
as I = q/t
F = (BqL/t) sinθ
as L = vt
F = (Bqvt/t) sinθ
F = Bqv sinθ
Define: magnetic flux
Units?
The product of an area, A, and the component of the magnetic field strength perpendicular to that area
Units: Weber, Wb or kgm2A-1s-2
What is the equation for magnetic flux?
Φ = BA cosθ
How can you determine the radius of a coil using the magnetic flux equation?
Φ = BA cosθ
as A = πr2
Φ = Bπr2 cosθ
rearrrange for r
As a coil turns in a magnetic field, how does the magnetic flux change?
It follows a cosine wave
What does e.m.f. mean?
Electromotive force. The work done per unit charge when energy is transferred from other forms into electric energy.
When a conductor of length l, moves up through a magnetic field going into the page, at a constant velocity, v. What happens?
A current flows to the left, and subsequently an e.m.f. is induced.
What factors determine the magnitude of the induced e.m.f for a conductor in a magnetic field?
(3)
- The velocity of the conductor
- The magnetic flux density
- The length of the conductor
Derive the equation for e.m.f induced in a conductor moving through a magnetic field
As work done = Fd
Work done = BILΔx
E.m.f. = work done/charge = (BILΔx)/q
as q = It
E.m.f. = BILΔx/It
= BLΔx/t
= BLv
One side of a rectangular coil of n loops is in a magnetic field, what is the formula for the e.m.f. induced
e.m.f = n (BLv)
How is Fleming’s left hand rule applied to:
a) a conductor with a current flowing through it going into a magnetic field
b) a conductor in a magnetic field moving at a constant velocity
a) Index point in magnetic field direction, middle finger points in current, the thumb is the force on the conductor.
b) Index points in magnetic field direction, middle finger point in the direction of the conductor’s velocity, the thumb is the direction of the current induced.
What factors will determine the magnitude of induced e.m.f for a coil in a magnetic field?
(4)
- Magnetic flux density
- Rate at which flux density changes
- Area of the coil
- Number of turns in a coil
For a single turn coil, what is the equation for the e.m.f. induced in terms of differences?
e.m.f = ΔΦ/Δt
or
e.m.f. = Δ(BA)/Δt
What e.m.f. would a 1Wbs-1 change of flux induce?
1V across the conductor
What is magnetic flux linkage?
Equation?
A magnetic flux passing through the coil is said to link the coil.
Magnetic flux linkage is the product of the number of turns in a coil and the flux passing through the coil.
Flux linkage = nΦ
What is faraday’s law?
What is its equation?
The magnitude induced e.m.f in a circuit is equal to the rate of change of magnetic flux linkage.
e. m.f = -n (ΔΦ/Δt)
e. m.f = -n ((ΔBAcosθ)/Δt)
What does the graph for magnetic flux linkage look like for a coil turning? (starting at perpendicular to the magnetic field)
What does the graph for the e.m.f. induced look like?
nΦ would follow a cosine graph
e.m.f would follow a sine graph
What is Lenz’s law?
Where is it seen mathematically?
The direction of an induced e.m.f will be such that it will oppose the change the produced it.
This is shown as the negative sign on the faraday’s law equation.
A north end of a magnetic is moving at constant velocity into a coil, in what direction does the induced current flow?
When it is leaving?
Entering: rightwards
Leaving: leftwards
When a coil is fixed and a magnet rotates in the iron core, what will be induced?
An alternating e.m.f. would be induced, and hence an AC current as there is continuously changing magnetic flux in the coil.
Why will a coil placed between two magnets continue to spin?
There would be a force acting up on one end and a force acting down on the other.
What is a back e.m.f.?
When a coil spins in a generation, its magnetic flux linkage will be continuously changing (as it is at a different angle each time), inducing an e.m.f.
From Lenz’s law, we know this e.m.f. would act in the opposite direction to the current that drives the motor.
This is called back e.m.f.
What happens to an e.m.f. - time graph when the rate of rotation of an AC generator is doubled?
- Double amplitude
- Half the time period
In an AC system, what is the equation for the power dissipated at any instant?
P = I2R
For an AC system what does the power - time graph look like for:
a) power dissipated at the instant
b) mean power
for time period, T.
a) Two peaks within the time period, T.
b) A horizontal line at half the value of Pmax
What are the symbols for:
a) Maximum power
b) Maximum current
c) Maximum e.m.f.
a) Pmax
b) I0
c) V0
What is the equation for:
a) Maximum power dissipated
b) Mean power dissipated
a) Pmax = I02R
b) Pmean = 1/2 I02R
How are the graphs of AC current and power dissipated related?
Power is always positive and at double the frequency of current
What does mean AC power show?
The equivalent power output of a DC system
What does the root mean square current show?
The value of direct current that would dissipate the same power in the resistor as an AC current with max current (I0)
What are the 6 equations for mean power?
Pmean = 1/2 I0V0
Pmean = IrmsVrms
Pmean = Irms2R
Pmean = 1/2 I02R
Pmean = Vrms2/R
Pmean = 1/2 (V02/R)
Steps in explaining how a transformer works
(6)
- An alternating current is supplied to the primary coil
- The AC current in the primary coil produces a magnetic field that links around a core made of a soft magnetic material, usually soft iron.
- As the current in the primary coil is alternating, the magnetic field in the core alternates at the same frequency. The field reverses twice for every current cycle.
- The shape of the core also means that the magnetic flux links with the secondary coil.
- Since the secondary coil has a charging flux inside it, an induced emf appears across its terminals. If connected to an external load, a current will flow.
- This means that energy has transferred from the primary to the secondary circuit through the core.
What is a step down transformer?
What is the relationship between the number of turns on each coil?
Step down transformers reduce the voltage supplied (current increases)
NS < NP
What is a step up transformer?
What is the relationship between the number of turns on each coil?
Step up transformers increase the secondary voltage (decreasing current).
NS > NP
What does magnetically soft mean?
Gains and loses magnetism very quickly.
Not a permanent magnet.
Why is iron used as the magnetic core?
What problems can this cause?
Iron is a good electrical conductor.
However, this means that the changing flux can induce eddy currents.
What can be done to make an iron core more efficient?
The core can be split into slabs and laminated.
Increasing the electrical resistance, but having minimal impact on the magnetic properties.
This reduces eddy currents and heat/energy loss.
What properties should the wires used in a transformer have?
- Should be enamelled or insulated
- The wire material should have a low resistivity
What does the national grid use for power transfer and why?
They use step-up transformers for high voltage transmission, to reduce energy loss through the heating effect.
Define: capacitor
A device used for storing and releasing charge. It consists of two charged plates separated by an insulator called a dielectric.
Explain how a capacitor is charged
(3)
- When the capacitor is in a circuit and an e.m.f. is supplied, the electrons are displaced from one of the plates to another.
- The e.m.f. drives the electrons through the circuit until the potential difference between the plates is equal to that of the e.m.f.
- At that point the capacitor is fully charged.
Define: capacitance
Units?
The charge stored on one of the plates, per unit p.d. across the plates
Units: Farad (F) or A2s4kg-1m-2
What is the equation for capacitance?
C = Q/V
where C is the capacitance in Farads
Q is charge
V is potential difference
How can you determine capacitance from a charge - pd graph?
as Q = C V
y = m x
the capacitance would be the gradient
What are the three equations for energy stored by capacitors?
How are they derived?
E = 1/2 QV [derived from work done]
E = 1/2 CV2 [derived by substituting Q=CV into the first one]
E = 1/2 (Q2/C) [derived by substituting V=Q/C into the second equation]
Where Q is charge, C is capacitance, V is p.d.
What is the dielectric equation?
How can you make capacitance the subject?
Q/A = ε0 V/d
where Q is charge, A is area, ε0 is the permittivity of free space, V is pd, d is distance between the plates.
Q/V = ε0 A/d
C = ε0 A/d
where C is capacitance
How can you increase a capacitor’s capacitance?
- Increase the area of the plates (rolled up)
- Decrease the distance between the plates
- Add a dielectric of higher ε value
What are suitable materials for dielectrics?
Why?
Insulators which become polarized when an electric field is set up across them.
This is because the molecules are more negative on one side and more positive on the other. When a field is applied, they rotate to counter the field.
What is the equation for electric field strength with a dielectric and what effect does this has?
Etotal = Ecapacitor - Edielectric
As Etotal = V/d
d is fixed so V must decrease
and Q=CV, V decreasing would mean that C increases.
What rule do capacitors in parallel follow?
Ctotal = C1 + C2 + C3 …
[Same rule resistors in series]
NOTE: the capacitor and resistance rules are SWITCHED
Prove the capacitors in parallel rule
For capacitors in series, the potential difference between all of them is the same as the potential difference between each individual one.
From the conservation of charge, we know that Qtotal = Q1 + Q2 etc.
In using Q=CV
CV = C1V + C2V + C3V etc.
C = C1 + C2 + C3 etc.
What rule do capacitors in series follow?
1/Ctotal = 1/C1 + 1/C2 + 1/C3
[Same as the rule for resistors in parallel]
NOTE: the capacitor and resistance rules are SWITCHED
Prove the capacitors in series rule
Kirchhoff’s second law tells us that the total pd over the capacitors is equal to the sum of the pd over each individual one.
V = V1 + V2
Conservation of charge tells us that the charge on each resistor is the same
Q/C = Q/C1 + Q/C2 + Q/C3
1/Ctotal = 1/C1 + 1/C2 + 1/C3 etc.
What factors will affect how long it takes for a capacitor to discharge?
(3)
- Capacitance of the capacitor
- Resistance of the resistor in the circuit
- (pd across the capacitor)
Derive the equation for the change in charge with respect to capacitance and resistance
Vc = IR
as I = ΔQ/Δt
ΔQ/Δt = Vc/R
as Vc = Q/C
ΔQ/Δt = Q/RC
ΔQ = - Q/RC Δt
[the negative sign is added because the charge stored has decreased]
Define: time constant
Units?
The time constant is the taken for the current, charge, and potential difference to reach 1/e of what they were at t=0. (Around 37% of the original value).
It is given the τ (tau)
Units: usually seconds (s) but can be any unit of time
What relationship does charge over time follow when a capacitor is discharging?
A negatively exponential relationship
Derive the *accurate* version of the capacitor discharge equation
ΔQ/Q = - 1/τ Δt
1/Q dQ = -1/τ dt
[integrate]
lnQ - lnQ0 = -t/τ
[take exponentials and solve for Q]
Q = Q0e-t/τ
What is the fomula for current and pd when a capacitor is discharging?
It follows the charge formula (Q = Q0e-t/τ)
I = I0 e-t/τ
V = V0 e-t/τ
How can you present the capacitor discharge equation as a linear relationship?
How could you find the time constant?
lnQ = -1/τ t + lnQ0
y = m x + c
The gradient would be -1/τ so the time constant = - 1/gradient
