Physics | Induction (HL)

👋 Welcome to Induction (HL): Mastering Change in Fields!

Hello future physicists! You've already explored the fascinating world of stationary electric fields (D.2) and the forces generated by magnetic fields (D.3). Now, we are diving into one of the most important concepts in modern physics: Electromagnetic Induction (Topic D.4).

Simply put, induction explains how we can generate electricity (an Electric Field) using magnetism (a Magnetic Field). This is the principle behind every power station, generator, and even wireless charging pads! It bridges the gap between electricity and magnetism, proving they are two sides of the same coin—Electromagnetism.

Don't worry if this seems tricky at first. We will break it down into three simple ideas: What is changing? How much EMF is produced? Which direction does the current flow? Let's get started!

1. Magnetic Flux (\(\Phi\)) and Flux Linkage (\(N\Phi\))

Before we can talk about inducing a voltage (or Electromotive Force, EMF), we need a way to quantify how much magnetic field is passing through an area. This is the concept of Magnetic Flux.

1.1 Defining Magnetic Flux (\(\Phi\))

Magnetic flux (\(\Phi\)) is essentially the amount of magnetic field "flow" passing perpendicularly through a given surface area.

• Symbol: \(\Phi\) (the Greek letter "Phi")
• Units: Weber (Wb), which is equivalent to \(\text{Tesla} \cdot \text{metre}^2\) (\(\text{T}\cdot\text{m}^2\)).

The magnetic flux is calculated using the formula:

\(\Phi = BA \cos \theta\)

Where:

• \(B\) is the magnetic field strength (or magnetic flux density) in Tesla (T).
• \(A\) is the area of the loop or coil through which the field passes (\(\text{m}^2\)).
• \(\theta\) is the angle between the magnetic field vector (\(B\)) and the normal (perpendicular line) to the area (\(A\)).

1.2 Magnetic Flux Linkage (\(N\Phi\))

In physics, we rarely deal with just one loop of wire. Most devices (like solenoids or generators) use coils with many turns.

The Magnetic Flux Linkage is the total flux passing through *all* the turns of the coil.

Flux Linkage = \(N \Phi\)

Where \(N\) is the number of turns in the coil. If the flux through one turn is \(\Phi\), the total flux linked is simply \(N\) times that amount.

---

Key Takeaway 1:

The key concept is that EMF is induced only when the magnetic flux linkage (\(N\Phi\)) is changing. This change can be caused by changing \(B\), changing \(A\), or changing the angle \(\theta\).

2. Faraday's Law of Induction (HL Core)

This is the heart of electromagnetic induction. Faraday's Law quantifies the relationship between the changing magnetic field and the induced EMF.

2.1 The Principle

Faraday's Law states that the magnitude of the induced EMF (\(\varepsilon\)) in a circuit is directly proportional to the rate of change of magnetic flux linkage (\(N\Phi\)) through the circuit.

In simpler terms: The faster you change the magnetic environment around a wire, the bigger the voltage you generate!

2.2 Mathematical Formulation (HL)

The induced EMF (\(\varepsilon\)) is given by:

\(\varepsilon = -N \frac{\Delta \Phi}{\Delta t}\) (Average induced EMF)

For instantaneous EMF (required for HL differentiation work):

\(\varepsilon = -N \frac{d\Phi}{dt}\) (Instantaneous induced EMF)

Where:

• \(\varepsilon\) is the induced EMF (voltage, V).
• \(\frac{d\Phi}{dt}\) is the rate of change of magnetic flux (Wb s\({}^{-1}\)).
• \(N\) is the number of turns.
• The negative sign is crucial and is explained by Lenz’s Law (coming next!).

---

Key Takeaway 2:

Induced EMF is proportional to the number of turns and how quickly the magnetic flux linkage changes. Maximum change = maximum induced voltage.

3. Lenz's Law and Conservation of Energy

Faraday’s Law tells us the size of the induced EMF; Lenz’s Law tells us the direction of the induced current. It explains the purpose of that negative sign in Faraday's equation.

3.1 The Principle of Opposition

Lenz's Law states that the direction of the induced current (and hence the induced EMF) is always such that it opposes the change in magnetic flux that produced it.

Analogy: Imagine your coil is a very grumpy child. Whatever you try to give it (increasing flux) or take away from it (decreasing flux), it tries to push back or hold on.

3.2 Step-by-Step Application of Lenz's Law

1. Determine the initial change: Is the external magnetic flux through the coil increasing or decreasing? (e.g., A magnet is moving *towards* the coil, so flux is increasing).
2. Determine the required opposition: The induced current must create its own magnetic field (the induced field, \(B_{\text{ind}}\)) that opposes this change.
   • If flux is increasing (getting stronger), \(B_{\text{ind}}\) must point opposite the external field.
   • If flux is decreasing (getting weaker), \(B_{\text{ind}}\) must point in the same direction as the external field to try and maintain it.
3. Use the Right-Hand Grip Rule: Once you know the direction of the induced field (\(B_{\text{ind}}\)), use the Right-Hand Grip Rule (fingers curl in direction of current, thumb points in direction of B-field) to find the direction of the induced current.

3.3 Lenz's Law and Energy Conservation

Lenz's Law is a direct consequence of the Law of Conservation of Energy.

If the induced current aided the change (rather than opposing it), the system would accelerate itself, generating more and more energy without external work input. This is impossible.

Therefore: To induce current, you must do mechanical work against the opposing magnetic force. The electrical energy generated comes directly from the work you put in (e.g., pushing the magnet).

---

Key Takeaway 3:

Lenz's Law ensures that the induced current fights the action that caused it, upholding the conservation of energy. If you push, the coil pushes back.

4. Motional EMF (Induced EMF in a Moving Conductor)

A special case of Faraday's law occurs when a straight conducting rod moves through a uniform magnetic field. This is known as Motional EMF.

4.1 The Mechanism

When a conductor of length \(L\) moves with velocity \(v\) perpendicular to a magnetic field \(B\):

1. The magnetic force \(F = qvB\) acts on the free electrons inside the conductor.
2. This force pushes the electrons to one end of the rod (say, the bottom), leaving the other end positive (the top).
3. This charge separation creates an electric field (\(E\)) inside the conductor.
4. Equilibrium is reached when the magnetic force (\(F_B\)) balances the electric force (\(F_E\)).

The induced EMF (\(\varepsilon\)) acts like a battery, with the voltage across the ends of the rod given by:

\(\varepsilon = B L v\)

This formula is valid only when B, L, and v are all mutually perpendicular.

4.2 Relating Motional EMF to Flux

Consider a sliding rod forming a closed circuit on a U-shaped rail. As the rod moves a distance \(\Delta x\) in time \(\Delta t\), the area of the circuit increases by \(\Delta A = L \Delta x\).

The change in flux is \(\Delta \Phi = B \Delta A = B L \Delta x\).

Applying Faraday's Law (ignoring the negative sign for magnitude):

\(\varepsilon = \frac{\Delta \Phi}{\Delta t} = \frac{B L \Delta x}{\Delta t}\)

Since \(\frac{\Delta x}{\Delta t}\) is the velocity \(v\):

\(\varepsilon = B L v\)

---

Key Takeaway 4:

Motional EMF is a special case of induction where movement of a conductor through a field creates a change in area, generating a voltage proportional to the field strength, length, and speed.

5. Applications: Alternating Current (AC) Generators

The AC generator (or dynamo) is the most important practical application of electromagnetic induction, converting mechanical energy into electrical energy.

5.1 How an AC Generator Works

An AC generator consists of a coil (armature) rotating at a constant angular velocity (\(\omega\)) within a uniform magnetic field (\(B\)).

1. Rotation causes flux change: As the coil rotates, the angle \(\theta\) between the B-field and the area normal constantly changes.
2. Flux follows cosine: The magnetic flux linkage (\(N\Phi\)) through the coil varies sinusoidally, described by \(N\Phi = NBA \cos(\omega t)\).
3. EMF follows sine: According to Faraday's Law, the induced EMF is the negative derivative of the flux linkage with respect to time:

   \(\varepsilon = -N \frac{d}{dt} (BA \cos(\omega t))\)
   \(\varepsilon = NBA \omega \sin(\omega t)\)

This result shows that the EMF generated is sinusoidal (a sine wave), creating Alternating Current (AC).

5.2 Maximum EMF (\(\varepsilon_0\))

The maximum (peak) value of the induced EMF occurs when \(\sin(\omega t) = 1\):

\(\varepsilon_0 = NBA \omega\)

This confirms that the generator produces maximum voltage when the coil is passing through the orientation where the flux change is fastest (i.e., when the flux itself is zero, but the coil is slicing perpendicularly across the field lines).

Memory Aid: To get more voltage, increase N-B-A-W (Number of turns, B-field, Area, Angular speed).

---

Key Takeaway 5:

AC generators work by continuously changing the angle (\(\theta\)) between the coil area and the magnetic field, producing a sinusoidal (AC) output voltage proportional to the rotation speed.

6. Transformers (Stepping Voltage Up and Down)

Transformers are essential devices in power distribution. They rely entirely on induction to change (step up or step down) AC voltages.

6.1 Construction and Principle

An ideal transformer consists of two coils, the primary coil (\(N_p\)) and the secondary coil (\(N_s\)), wrapped around a common soft iron core.

1. AC Input: An alternating voltage (\(V_p\)) applied to the primary coil generates an alternating current.
2. Changing Flux: This AC current creates a continually changing magnetic flux (\(\Phi\)).
3. Flux Linkage: The soft iron core guides nearly all of this changing flux to link with the secondary coil.
4. Induction: The changing flux induces an alternating EMF (\(V_s\)) in the secondary coil, according to Faraday's Law.

Crucial Point: Transformers only work with Alternating Current (AC). A steady DC current creates constant flux (\(\frac{d\Phi}{dt}=0\)), so no EMF is induced in the secondary coil.

6.2 The Transformer Equation (Ideal Transformer)

For an ideal transformer (where no energy is lost and 100% of flux links both coils), the ratio of voltages equals the ratio of turns:

\(\frac{V_p}{V_s} = \frac{N_p}{N_s}\)

• Step-Up Transformer: \(N_s > N_p\), so \(V_s > V_p\). (More turns on the output coil.)
• Step-Down Transformer: \(N_s < N_p\), so \(V_s < V_p\). (Fewer turns on the output coil.)

6.3 Power and Current Conservation

Assuming an ideal transformer, input power equals output power: \(P_{\text{input}} = P_{\text{output}}\).

\(V_p I_p = V_s I_s\)

This leads to the relationship between current and turns:

\(\frac{I_s}{I_p} = \frac{N_p}{N_s}\)

This means if you step up the voltage (say, by a factor of 10), you must step down the current by the same factor (1/10th). This is vital for power transmission, as low current minimizes heat loss (\(P_{\text{loss}} = I^2 R\)) over long distances.

6.4 Real Transformer Losses (HL Considerations)

Real transformers are highly efficient (often 99% or more) but do have some losses:

• Resistance Loss (\(I^2 R\)): Energy lost as heat in the copper wires of the coils. Minimized by using thick wires (low R).
• Eddy Currents: Circulating currents induced in the iron core itself. These are minimized by laminating the core (making it from thin, insulated sheets) to increase resistance in the plane of the induced currents.
• Hysteresis Loss: Energy lost continually magnetizing and demagnetizing the iron core. Minimized by using a soft magnetic material.
• Flux Leakage: Not all the magnetic flux generated by the primary coil links the secondary coil. Minimized by wrapping the coils closely together.

---

Key Takeaway 6:

Transformers use mutual induction (AC input creates changing flux, which links the secondary coil) to efficiently change AC voltage. The voltage ratio is dictated by the turn ratio, while power must be conserved (\(V_p I_p = V_s I_s\)).

🎯 Quick Review: Induction Checklist

Congratulations! Induction is a complex topic, but by understanding that change is the requirement, you have unlocked the secret to how nearly all electrical power is generated and distributed globally. Keep practicing those directional rules with Lenz's Law—it's the trickiest part! You've got this!