# D.5 Electromagnetic induction (AHL)

> IB Physics Higher Level · IB Physics HL 2025+
> Source: https://www.owlsprep.com/study/ib-physics-hl-u4-d-5-electromagnetic-induction/

This sub-topic explores induced electromotive force (emf) generated by changing magnetic fields, covering Faraday's law, Lenz's law, motional emf, self-inductance and transformers for IB Physics HL AHL.

**Prerequisites:** [Magnetic flux and magnetic fields](https://www.owlsprep.com/study/ib-physics-hl-u4-d-4-magnetic-fields/); [Magnetic force on moving charges](https://www.owlsprep.com/study/ib-physics-hl-u4-d-2-magnetic-forces/)

## Learning objectives

- State and apply Faraday's law of electromagnetic induction
- Use Lenz's law to determine the direction of induced current
- Calculate motional emf for moving conductors
- Solve problems involving self-inductance and ideal transformers
- Explain energy losses in real transformers

## 1. Faraday's Law of Induction

**Faraday's Law of Induction** — The magnitude of the induced emf equals the negative rate of change of magnetic flux linkage through a coil. The negative sign indicates direction per Lenz's law.

*Notation:* \varepsilon = -\frac{d(N\Phi)}{dt}

*Example:* Any change to the magnetic environment through the coil generates an emf.

Induced emf only arises from a change in flux, not a constant flux. Changes can come from changing magnetic field strength, changing coil area, changing coil orientation, or moving the coil into/out of a field.

**Worked example:** A 50-turn coil of area 0.1 m² is placed perpendicular to a uniform magnetic field that increases linearly from 0.2 T to 1.2 T in 2.0 s. Calculate the magnitude of the induced emf.

1. Step 1: Calculate the change in magnetic flux linkage
2. $$\Delta(N\Phi) = N A \Delta B = 50 \times 0.1 \times (1.2 - 0.2) = 5.0 \text{ Wb}$$
3. Step 2: Find the magnitude of induced emf using Faraday's law
4. $$|\varepsilon| = \frac{\Delta(N\Phi)}{\Delta t} = \frac{5.0}{2.0} = 2.5 \text{ V}$$

> **tip**
>
> Always take the absolute value when asked for the magnitude of induced emf; the negative sign only indicates direction.

## 2. Lenz's Law and Induced Current Direction

**Lenz's Law** — The direction of the induced current is such that the magnetic field it produces opposes the change in flux that created it.

Lenz's law is a consequence of conservation of energy: if the induced current assisted the change instead of opposing it, we would generate free energy, which violates physical laws. A consistent step-by-step process avoids direction errors.

**Worked example:** The north pole of a bar magnet is moved towards a stationary circular coil. Find the direction of the induced current when viewed from the magnet's side.

1. Step 1: Original magnetic field points through the coil away from the magnet, and flux is increasing as the magnet approaches.
2. Step 2: By Lenz's law, the induced magnetic field must oppose the increase, so it points towards the incoming magnet (opposite the original field direction).
3. Step 3: Use right-hand grip rule: thumb points towards the magnet (direction of induced field), so fingers curl counter-clockwise when viewed from the magnet side.

> **IODA Mnemonic for Lenz's Law**
>
> Increase Oppose, Decrease Agree: If flux increases, induced field opposes original field; if flux decreases, induced field agrees with original field.

## 3. Motional Emf in Moving Conductors

A conducting rod moving perpendicular to a uniform magnetic field has an induced emf across its ends, called motional emf. This arises from magnetic force separating free charges in the conductor, and can be derived directly from Faraday's law.

**Derivation:** Derive motional emf for a perpendicular moving conductor

*Starting from:* Faraday's law: $|\varepsilon| = \frac{d\Phi}{dt}$

1. A rod of length $l$ slides along rails at speed $v$, forming a loop of area $A = l x$ in a perpendicular field $B$
2. $$\Phi = B A = B l x$$
3. Differentiate with respect to time, where $\frac{dx}{dt} = v$
4. $$\frac{d\Phi}{dt} = B l \frac{dx}{dt} = B l v$$

*Conclusion:* For perpendicular motion: $\varepsilon = B l v$

**Worked example:** A 0.5 m long metal rod moves at 4 m/s perpendicular to a 0.2 T magnetic field into the page. If velocity is to the right, calculate the induced emf and find which end is positive.

1. Step 1: Calculate emf magnitude
2. $$\varepsilon = B l v = 0.2 \times 0.5 \times 4 = 0.4 \text{ V}$$
3. Step 2: For negative electrons, force is $F = q v \times B$. $v$ right, $B$ in, so $v \times B$ is upwards. Electrons are pulled down, leaving the top end positive.

## 4. Applications: Inductance and Transformers

**Self-Inductance** — A changing current in a coil induces a back emf that opposes the change in current. Inductance $L$ is measured in henries (H).

*Notation:* \varepsilon = -L \frac{dI}{dt}, \quad L = \frac{N\Phi}{I}

Transformers use mutual inductance between two coils wrapped around a common core to step up or step down AC voltage. For an ideal 100% efficient transformer, the relationship between voltage, current and turns is:

$$\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}$$

**Worked example:** A 100% efficient transformer steps down 240 V AC to 12 V for a 60 W light bulb. The primary coil has 4000 turns. Find the number of secondary turns and primary current.

1. Step 1: Use the turns-voltage ratio
2. $$N_2 = N_1 \frac{V_2}{V_1} = 4000 \times \frac{12}{240} = 200$$
3. Step 2: Power is conserved for ideal transformers, so input power equals output power
4. $$P_1 = V_1 I_1 = 60 \text{ W} \implies I_1 = \frac{60}{240} = 0.25 \text{ A}$$

> **info**
>
> Transformers only work with alternating current (AC). Constant DC input produces no changing flux, so there is zero induced emf in the secondary coil.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using total flux instead of change in flux to calculate induced emf
  - Why it fails: Induced emf only arises from a change in flux. A large constant flux produces zero emf
  - Correct: Always calculate the rate of change of flux linkage, not the total flux value
- **Wrong:** Opposing the original magnetic field instead of the change in flux in Lenz's law
  - Why it fails: Induced current opposes the change, not the field itself. If flux is decreasing, induced field adds to the original field
  - Correct: Use the IODA mnemonic: check if flux is increasing or decreasing first
- **Wrong:** Applying the transformer equation to DC input
  - Why it fails: Transformers require changing flux to induce emf, which does not occur for constant DC
  - Correct: Remember transformers only operate with AC; they produce no output for steady DC
- **Wrong:** Reporting the negative sign from Faraday's law as the magnitude of emf
  - Why it fails: The negative sign only indicates direction, not the size of the induced potential difference
  - Correct: Always take the absolute value when asked for the magnitude of emf

## Cheatsheet

| Concept | Formula/Rule | Key Notes |
| --- | --- | --- |
| Faraday's Law | $\varepsilon = - \frac{d(N\Phi)}{dt}$ | Magnitude = $\|d(N\Phi)/dt\|$ |
| Motional Emf (perpendicular) | $\varepsilon = Blv$ | For moving conducting rods |
| Lenz's Law | IODA: Increase Oppose, Decrease Agree | Induced current opposes flux change |
| Self Inductance | $\varepsilon = -L \frac{dI}{dt}$ | Back emf opposes current change |
| Ideal Transformer | $\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}$ | Only works for AC, 100% efficiency |

## What's next

Electromagnetic induction is the fundamental principle behind almost all grid electricity generation, from coal and gas power stations to wind turbines, and it underpins modern AC power distribution. Mastery of Faraday's and Lenz's laws is required for nearly all further electromagnetic topics in IB Physics, and this sub-topic regularly appears in both Paper 1 multiple choice and Paper 2 extended response questions. Building on these concepts, you will next explore the behaviour of alternating current circuits and the role of induction in Maxwell's description of electromagnetic waves.

- [Theme E: Nuclear and quantum physics](https://www.owlsprep.com/study/ib-physics-hl-u5-overview/)

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