DP IB Physics: SL

D. Fields

D.4 Induction

DP IB Physics: SL D. Fields D.4 Induction Understandings Standard level and higher level: 8 hours
a)	Magnetic flux [math]\Phi[/math] as given by [math]\Phi = B A \cos \theta[/math]
b)	That a time-changing magnetic flux induces an emf ε as given by Faraday’s law of induction [math]\varepsilon = -N \frac{\Delta \Phi}{\Delta t}[/math]
c)	That a uniform magnetic field induces an emf in a straight conductor moving perpendicularly to it as given by [math]ε = BvL[/math]
d)	That the direction of induced emf is determined by Lenz’s law and is a consequence of energy conservation
e)	That a uniform magnetic field induces a sinusoidal varying emf in a coil rotating within it
f)	The effect on induced emf caused by changing the frequency of rotation.

a) Magnetic Flux (Φ)
[math]\Phi = B A \cos \theta[/math]
Where:
– Φ = magnetic flux (measured in webers, Wb)
– B = magnetic field strength (tesla, T)
– A = area of the loop or surface (m²)
– θ = angle between the magnetic field direction and the normal (perpendicular) to the surface
Magnetic flux measures how much magnetic field passes through a given surface.
Think of magnetic field lines as invisible arrows; flux tells you how many arrows go through a surface.
Figure 1 Magnetic flux
⇒ Understanding the Angle θ:
[math]θ = 0^0[/math]: Field is perpendicular to the surface → maximum flux
[math]Φ = B ⋅ A[/math]
[math]θ = 90^0[/math]: Field is parallel to the surface → zero flux
[math]Φ = 0[/math]
So, flux is strongest when the field “cuts” directly through the surface, and weakest when it slides past it.
Figure 2 Magnetic field lines and magnetic flux
b) Faraday’s Law of Induction
[math]\varepsilon = -N \frac{\Delta \Phi}{\Delta t}[/math]
Where:
– ε = induced emf (volts, V)
– N = number of turns in the coil
– ΔΦ = change in magnetic flux (Wb)
– Δt = time over which the change happens (s)
– The negative sign indicates the direction of the induced emf (explained below)
Figure 3 Faraday’s Law of induction
⇒ Electromagnetic Induction:
When magnetic flux through a closed loop change, an emf
(voltage) is induced in the loop.
This is called electromagnetic induction.
This is the basic principle behind electric generators, transformers, and induction coils.
Figure 4 Electromagnetic Induction
How Flux Can Change:
There are 3 ways to induce an emf:
1. Change the magnetic field strength B
2. Change the area A through which field lines pass
3. Change the angle θ between the field and the surface
Figure 5 Flux can change
Any of these changes cause ΔΦ, which then creates ε.
Negative Sign — Lenz’s Law:
[math]\varepsilon = -N \frac{\Delta \Phi}{\Delta t}[/math]
The negative means the induced emf opposes the change in flux.
This is called Lenz’s Law.
It’s a consequence of the law of conservation of energy.
The system resists change — for example, if flux increases, the induced emf tries to reduce it.
Example:
Imagine a loop of wire in a magnetic field, and the magnetic field strength increases with time. Then:
– Magnetic flux Φ increases
– The loop experiences a change in flux ΔΦ
– An emf is induced in the loop, producing a current
– The induced current creates its own magnetic field that opposes the increasing original field (Lenz’s Law)
Real-World Applications:
Electric Generators: Rotate a coil in a magnetic field to induce current
Transformers: Use changing magnetic flux in coils to transfer energy between circuits
Electric Guitar Pickups: Vibrating strings disturb magnetic flux, generating electrical signals
Induction Cooktops: Changing fields induce currents in metal pots to heat them
c) Induced EMF in a Moving Conductor
When a straight conductor moves in a uniform magnetic field, an emf is induced across it.
[math]ε = BvL[/math]
Where:
– ε = induced emf (volts, V)
– B = magnetic field strength (tesla, T)
– v = velocity of the conductor (m/s)
– L = length of the conductor inside the magnetic field (m)
Figure 6 Induced EMF in a moving conductor
How It Works:
Imagine a straight metal rod moving perpendicularly through a uniform magnetic field (e.g. rod moving horizontally while magnetic field points vertically into the page).
The free electrons inside the metal experience a magnetic force due to their motion in the magnetic field.
This force is given by the Lorentz force:
[math]F = qvB[/math]
This force pushes electrons to one side of the rod, building up positive and negative charges at opposite ends.
As a result, an electric potential difference (emf) is induced across the conductor.
This is electromagnetic induction due to motion, also called motional emf.
⇒ Conditions for Maximum EMF:
– Conductor must move perpendicular to the magnetic field lines.
– If it moves parallel to the field, no emf is induced.
General form:
[math]ε = BvLsinθ[/math]
Where θ is the angle between the direction of motion and the field.
d) Direction of the Induced EMF – Lenz’s Law
Lenz’s Law Statement:
The direction of the induced emf is such that the current it causes opposes the change in magnetic flux that produced it.
This law is represented by the negative sign in Faraday’s law:
[math]\varepsilon = -\frac{d\Phi}{dt}[/math]
Figure 7 Lenz’s Law
Energy Conservation
Lenz’s law ensures that:
You can’t create energy from nothing
If an induced current helped the change (instead of opposing it), it would create a feedback loop that violates energy conservation.
So:
If flux increases, induced current flows in a direction to oppose the increase.
If flux decreases, current tries to increase it again.
⇒ In the Moving Conductor Example:
– Suppose a rod moves to the right in a magnetic field going into the page.
– Electrons are pushed to one end → emf is generated.
– If the circuit is closed, current flows.
– This induced current creates its own magnetic field that opposes the motion, exerting a magnetic drag force on the rod.
Thus, mechanical work must be done to keep the rod moving — this energy is converted into electrical energy. No energy is lost or created, only transformed.
Real-World Applications:
– Railgun: Uses magnetic fields and induced current to launch projectiles.
– Electric generators: Rotating coils through magnetic fields to induce emf.
– Moving wire in magnetic field: Classic demonstration of motional emf.
– Braking in trains: Magnetic braking uses induced currents to oppose motion (eddy currents + Lenz’s Law).
(e) Sinusoidal EMF Induced in a Rotating Coil
Faraday’s Law of Electromagnetic Induction:
[math]\varepsilon = -\frac{d\Phi}{dt}[/math]
Where:
ε = induced emf (volts)
Φ = magnetic flux =
θ = angle between magnetic field and coil’s normal (perpendicular)
Figure 8 Sinusoidal EMF induced in a rotating coil
How a Rotating Coil Works:
A coil with N turns rotates in a uniform magnetic field.
Let’s define:
– B = magnetic field strength
– A = area of the coil
– ω = angular velocity (in radians per second)
– [math]θ = ωt[/math] = angle between coil’s normal and magnetic field at time
So, the magnetic flux at time t is:
[math]Φ(t) = BAcos(ωt)[/math]
Now, applying Faraday’s Law:
[math]\varepsilon = -N \frac{d\Phi}{dt} \\
\varepsilon = -N \frac{d}{dt} \left(B A \cos(\omega t)\right) \\
\varepsilon = -N B A \omega \sin(\omega t)[/math]
Final Expression:
[math]\varepsilon(t) = \varepsilon_{\text{max}} \sin(\omega t)[/math]
Where:
– [math]\varepsilon_{\text{max}} = NBA\omega[/math]
– The emf varies sinusoidally with time as the coil rotates.
This is the principle of AC generators: a rotating coil in a magnetic field produces an alternating (sinusoidal) emf.
⇒ Characteristics of the Induced EMF:
AC (Alternating Current): The emf continuously changes direction.
Zero Crossing: emf becomes zero twice every full rotation.
Frequency of Oscillation: Related to how fast the coil spins.
f) Effect of Changing the Frequency of Rotation
The angular velocity [math]ω = 2πf[/math], where f is the frequency (rotations per second).
So, from the emf expression:
[math]\varepsilon_{\text{max}} = NBA\omega \\
\varepsilon_{\text{max}} = NBA(2 \pi f)[/math]
Therefore:
If frequency increases:
– Angular velocity ω increases
– Maximum emf [math]ε_{max[}/math] increases
– The emf oscillates faster (higher frequency)
– The rate of change of magnetic flux is greater → more emf
If frequency decreases:
– emf decreases in amplitude
– emf oscillates more slowly
Figure 9 Changing the frequency of rotational in magnetic field