# Faraday's law and Lenz's law

> CIE A-Level Physics · 9702 A-Level Physics 2022-2024
> Source: https://www.owlsprep.com/study/cie-9702-u24-faraday-s-law-and-lenz/

This sub-topic covers the two core laws of electromagnetic induction: Faraday's law for the magnitude of induced emf, and Lenz's law for the direction of induced current. We apply both to common CIE exam problems.

**Prerequisites:** [Magnetic flux and flux linkage](https://www.owlsprep.com/study/cie-9702-u24-magnetic-flux-flux-linkage/)

## Learning objectives

- State Faraday's law and Lenz's law in exam-correct form
- Calculate the magnitude of induced emf from changing flux linkage
- Determine the direction of induced current using Lenz's law
- Explain the link between Lenz's law and conservation of energy

## Faraday's Law of Induction

**Faraday's Law** — The magnitude of the induced electromotive force (emf) is equal to the rate of change of magnetic flux linkage through a circuit.

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

*Example:* A falling magnet approaching a coil produces an increasing magnitude of induced emf.

For most CIE problems involving uniform changes in flux, the equation simplifies to calculate the magnitude of induced emf:

$$\varepsilon = N \frac{|\Delta \Phi|}{\Delta t}$$

**Worked example:** A 50-turn coil with cross-sectional area $2.0 \times 10^{-4} \text{ m}^2$ is placed perpendicular to a uniform magnetic field that increases from 0.10 T to 0.35 T in 0.20 s. Calculate the magnitude of the induced emf.

1. First, calculate the change in magnetic flux per turn: $\Delta \Phi = \Delta B \cdot A \cdot \cos\theta$. $\theta = 0^\circ$ so $\cos\theta = 1$:
2. $$\Delta \Phi = (0.35 - 0.10) \times 2.0 \times 10^{-4} = 5.0 \times 10^{-5} \text{ Wb}$$
3. Substitute into Faraday's law, multiplying by the number of turns $N$:
4. $$\varepsilon = N \frac{\Delta \Phi}{\Delta t} = 50 \times \frac{5.0 \times 10^{-5}}{0.20} = 0.0125 \text{ V} = 12.5 \text{ mV}$$

> **tip**
>
> Always include the number of turns $N$ in your calculation — this is the most frequently missed mark in Faraday's law questions.

## Lenz's Law and Conservation of Energy

**Lenz's Law** — The direction of the induced current is such that it creates a magnetic effect that opposes the change in magnetic flux that produced the current.

Lenz's law is not just an arbitrary direction rule: it is a direct consequence of the law of conservation of energy. If the induced current aided the change that produced it, you would gain mechanical and electrical energy without input, violating energy conservation.

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

1. 1. Identify the change in flux: As the north pole approaches, flux pointing towards the coil through the magnet increases.
2. 2. Apply Lenz's law: Induced current must create a magnetic field that opposes this increase. So the induced magnetic field points away from the magnet (towards the incoming north pole to repel it).
3. 3. Use the right-hand grip rule: Thumb points in the direction of the induced magnetic field, fingers curl to give current direction: counter-clockwise when viewed from the magnet side.

> **mnemonic**
>
> Approach repels, recede attracts: An incoming pole gets an identical induced pole to repel it, a receding pole gets an opposite induced pole to attract it.

## Combined Applications of Both Laws

Most exam questions require you to use Faraday's law for magnitude and Lenz's law for direction. We demonstrate this for a moving conducting rod on rails, a common CIE problem:

**Worked example:** A 0.5 m long conducting rod moves at $2.0 \text{ m s}^{-1}$ right along parallel rails, perpendicular to a 0.4 T magnetic field pointing into the page. Find the magnitude and direction of induced current.

1. 1. Calculate the rate of change of flux. Flux $\Phi = BA = Blx$, where $x$ is the length of the circuit. $\frac{d\Phi}{dt} = Bl \frac{dx}{dt} = Blv$:
2. $$\frac{d\Phi}{dt} = 0.4 \times 0.5 \times 2.0 = 0.4 \text{ Wb s}^{-1}$$
3. 2. Faraday's law: $N=1$ for a single loop, so induced emf $\varepsilon = 0.4 \text{ V}$.
4. 3. Lenz's law: Flux into the page is increasing as the rod moves right. Induced current must create flux out of the page to oppose the change. By right-hand rule, current flows counter-clockwise around the loop, so up through the moving rod.

**Check your understanding**

Test your understanding of Lenz's direction rules

1. A south pole is pulled away from a coil. What is the polarity of the coil side facing the magnet?

   - North pole
   - South pole
   - No polarity

   *Answer:* North pole

   *Why:* Correct: The receding south pole must be attracted by the induced pole, so the facing side is north.

2. Flux pointing right through a coil is decreasing. What direction is the induced magnetic field?

   - Left
   - Right
   - Zero

   *Answer:* Right

   *Why:* Correct: Induced field opposes the change. A decrease in right flux means the change is loss of right flux, so induced field adds right flux to oppose the change.

## Common pitfalls

- **Wrong:** Forgetting to multiply by the number of turns $N$ when calculating induced emf.
  - Why it fails: Faraday's law uses flux linkage (total for all turns), not flux per turn. Missing N loses 1-2 marks in most calculation questions.
  - Correct: Always multiply the change in flux per turn by the number of turns to get the change in flux linkage before calculating emf.
- **Wrong:** Stating Lenz's law as 'induced current opposes the magnetic field' instead of the change in flux.
  - Why it fails: This is a common misstatement that loses full marks in definition or explanation questions.
  - Correct: Always reference the change in flux: 'induced current opposes the change in magnetic flux that produced it'.
- **Wrong:** Getting direction wrong for decreasing flux, defaulting to opposing the original field.
  - Why it fails: Students forget that we oppose the change, not the field. If flux is decreasing, the change is a reduction of flux, not the flux itself.
  - Correct: First ask: is flux increasing or decreasing? If decreasing, induced field is in the same direction as the original field.
- **Wrong:** Using degrees instead of radians for sinusoidal rotating coil flux problems.
  - Why it fails: Differentiation of sine/cosine functions only gives correct results when angles are in radians, leading to wrong emf values.
  - Correct: Always switch your calculator to radians mode for rotating coil induced emf calculations.

## Cheatsheet

| Concept | Equation/Rule | Key Note |
| --- | --- | --- |
| Faraday's Law (magnitude) | $\varepsilon = N \frac{\|\Delta \Phi\|}{\Delta t}$ | Uses flux linkage, not flux per turn |
| Lenz's Law Direction | Induced current opposes change in flux | Opposes change, not the flux itself |
| Approaching magnet | Induced pole = incoming pole | Repels the approaching magnet |
| Receding magnet | Induced pole = opposite to receding pole | Attracts the receding magnet |
| Moving rod emf | $\varepsilon = Blv$ | Valid when $v$, $B$, $l$ are mutually perpendicular |

## What's next

Faraday's and Lenz's laws are the foundation of all electromagnetic induction, the principle behind modern grid electricity generation, transformers, and inductive electronic components. Mastering these laws is critical for all subsequent topics in electromagnetism for CIE A-Level. You will apply these rules to extended response questions on generators, transformers, and eddy current braking, which make up a large portion of the marks for this unit. Build on your understanding with the following related sub-topics to continue your exam preparation.

- [Induced e.m.f.](https://www.owlsprep.com/study/cie-9702-u24-induced-e-m-f/)
- [Alternating currents](https://www.owlsprep.com/study/cie-9702-u25-overview/)

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