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IB Physics HL/Notes/D.4 Induction

IB Physics HLD.4 InductionNotes

Predict Magnetic flux

Flux questions often turn on the angle definition. Draw the surface normal first, then place θ between the normal and B. A single loop has flux Φ; a coil has flux linkage NΦ, which is the quantity used in Faraday’s law.

Magnetic flux Φ measures the magnetic field passing through an area: Φ = BA cosθ.
θ is the angle between the magnetic field and the normal to the surface, not the plane of the surface.
Magnetic flux is measured in webers, Wb, where 1 Wb = 1 T m^2.
Flux is maximum when B is perpendicular to the surface, so θ = 0° to the normal.
Flux is zero when B is parallel to the surface, so θ = 90° to the normal.
For a coil of N turns, flux linkage is NΦ.

Assemble the magnetic-flux model.

Formula
Target formula Φ = BA cosθ
Φ
magnetic flux
Wb
B
magnetic flux density
T
A
area through which field passes
m^2
θ
angle between B and the area normal
degrees or radians
N
number of turns in a coil
1Measure the angle to the normal of the surface.θ = angle between B and normal
2Use the magnetic-flux expression.Φ = BA cosθ
3For a coil, multiply by the number of turns.flux linkage = NΦ
4State the flux unit.1 Wb = 1 T m^2

Define magnetic flux and flux linkage for a coil.

Using the angle to the coil plane instead of the normal.

Define magnetic flux and flux linkage for a coil.

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Predict Faraday’s law

Faraday’s law is a rate-of-change law. A large flux with no change induces no emf; a changing flux linkage induces emf. The sign is not just algebra: it encodes the opposition described by Lenz’s law.

Faraday’s law states that induced emf equals the negative rate of change of magnetic flux linkage: ε = -NΔΦ/Δt.
The magnitude is |ε| = N|ΔΦ|/Δt for a uniform change over time interval Δt.
Flux can change if B changes, A changes, or the angle θ changes.
A larger rate of change of flux linkage produces a larger induced emf.
The negative sign represents Lenz’s law: the induced emf opposes the change that caused it.

Assemble Faraday’s law.

Formula
Target formula ε = -NΔΦ/Δt
ε
induced emf
V
N
number of turns
ΔΦ
change in magnetic flux per turn
Wb
Δt
time interval for the change
s
Φ
magnetic flux BA cosθ
Wb
1Start from flux through one turn.Φ = BA cosθ
2Use flux linkage for N turns.NΦ
3Induced emf magnitude is rate of change of linkage.|ε| = N|ΔΦ|/Δt
4Add Lenz’s law sign.ε = -NΔΦ/Δt

State Faraday’s law and give two ways magnetic flux through a coil can change.

Saying emf depends on flux size rather than rate of change of flux linkage.

State Faraday’s law and give two ways magnetic flux through a coil can change.

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Analyze Motional emf

Motional emf can be explained two ways: moving charges feel magnetic force, or the moving conductor changes the magnetic flux through a circuit. Both point to ε = BvL for the perpendicular straight-conductor case.

A straight conductor of length L moving with speed v perpendicular to a uniform magnetic field B has induced emf ε = BvL.
Free charges in the conductor experience magnetic force F = qvB, causing charge separation along the conductor.
The charge separation creates a potential difference between the ends of the conductor.
The formula assumes the conductor, velocity, and magnetic field are mutually perpendicular in the required way.
The same result follows from Faraday’s law because the conductor sweeps area LvΔt in time Δt.

Assemble the motional-emf model.

Formula
Target formula ε = BvL
ε
induced emf
V
B
magnetic flux density
T
v
speed of the conductor
m s^-1
L
length of conductor in the field
m
q
mobile charge in the conductor
C
1Moving charges feel magnetic force.F = qvB
2Charge separation creates an emf.potential difference between conductor ends
3For perpendicular motion, use motional emf.ε = BvL
4Link to swept area.ΔA = LvΔt -> ΔΦ/Δt = BLv

A straight conductor moves at right angles through a uniform magnetic field. Calculate the induced emf and explain its origin.

Using Faraday’s law without recognizing the required perpendicular condition.

A straight conductor moves at right angles through a uniform magnetic field. Calculate the induced emf and explain its origin.

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Predict Lenz’s law

Lenz’s law is about opposing the change, not necessarily opposing the magnetic field itself. Decide whether the flux is increasing or decreasing, then choose the induced field that resists that change.

Lenz’s law states that the direction of induced emf or current is such that its magnetic effect opposes the change in magnetic flux that caused it.
If flux through a coil is increasing in one direction, the induced current produces a field in the opposite direction.
If flux is decreasing, the induced current produces a field that tries to maintain it.
Lenz’s law is a consequence of conservation of energy.
If induction reinforced the original change, energy would be produced without external work.

Use Lenz’s law to choose the induced field direction.

Decision
Flux into the coil is increasing.
Flux into the coil is decreasing.
Flux out of the coil is increasing.

Explain Lenz’s law using conservation of energy.

Saying induced current opposes the magnetic field rather than the change in flux.

Explain Lenz’s law using conservation of energy.

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Analyze Rotating-coil emf

An AC generator is a rotating-coil Faraday-law machine. The coil does not produce maximum emf when flux is maximum; it produces maximum emf when flux is changing fastest.

For a coil of N turns and area A rotating in a uniform magnetic field B, flux linkage varies as NΦ = NBA cosωt.
Faraday’s law gives induced emf ε = NBAω sinωt, up to sign convention.
The peak emf is ε0 = NBAω.
Emf is zero when flux linkage is maximum or minimum because the rate of change is momentarily zero.
Emf is maximum when the coil is parallel to the field because the rate of change of flux linkage is greatest.
The output is alternating because the sign of the rate of change reverses every half-turn.

Assemble the AC generator equations.

Formula
Target formula ε = NBAω sinωt
ε
instantaneous induced emf
V
N
number of turns
B
magnetic flux density
T
A
coil area
m^2
ω
angular frequency of rotation
rad s^-1
t
time
s
1Write flux linkage for a rotating coil.NΦ = NBA cosωt
2Differentiate flux linkage using Faraday’s law.ε = -d(NΦ)/dt
3State the sinusoidal emf form.ε = NBAω sinωt
4Identify peak emf.ε0 = NBAω

A rectangular coil rotates in a uniform magnetic field. Explain why the induced emf is alternating and when it is maximum.

Equating maximum flux with maximum emf.

A rectangular coil rotates in a uniform magnetic field. Explain why the induced emf is alternating and when it is maximum.

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Analyze Rotation frequency and emf

A faster generator changes flux linkage more quickly. That raises the rate of change of flux linkage, so the peak emf increases. Because the coil is rotating faster, the AC waveform also has a shorter period.

For a rotating coil, peak emf is ε0 = NBAω.
Angular frequency and rotation frequency are related by ω = 2πf.
Increasing the frequency of rotation increases peak emf in direct proportion.
Increasing frequency also decreases the period of the alternating emf waveform.
Increasing N, B, or A also increases the peak emf, but changing f specifically changes how quickly the waveform repeats.

Predict how generator frequency changes emf output.

Formula
Target formula ε0 = NBA(2πf)
ε0
peak induced emf
V
N
number of turns
B
magnetic flux density
T
A
coil area
m^2
ω
angular frequency
rad s^-1
f
rotation frequency
Hz
1Start from the rotating-coil peak emf.ε0 = NBAω
2Replace angular frequency.ω = 2πf
3Write peak emf in terms of frequency.ε0 = NBA(2πf)
4Relate graph period to frequency.T = 1/f

Explain the effect of increasing the frequency of rotation of an AC generator.

Saying frequency changes only the number of cycles per second and not the peak emf.

Explain the effect of increasing the frequency of rotation of an AC generator.

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Retrieve the D.4 Induction Model

Review

D.4 is held together by one question: how fast is magnetic flux linkage changing? Faraday’s law gives size, Lenz’s law gives direction, and generator examples are rotating versions of the same idea.

Magnetic flux is Φ = BA cosθ, with θ measured between B and the area normal; flux linkage is NΦ.
Faraday’s law gives induced emf from changing flux linkage: ε = -NΔΦ/Δt.
Lenz’s law gives the direction: the induced magnetic effect opposes the change in flux that caused it.
For a straight conductor moving perpendicularly through a uniform magnetic field, ε = BvL.
For a rotating coil, NΦ = NBA cosωt and ε = NBAω sinωt, with peak emf ε0 = NBAω.
Increasing rotation frequency increases ω, increases peak emf, and decreases the AC period.

Match each D.4 cue to the induction idea it retrieves.

Match
Reasons
0/8

Summarize the electromagnetic induction model for an HL response.

Writing formulas without identifying the changing flux linkage or Lenz-law direction.

Summarize the electromagnetic induction model for an HL response.

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