Faraday's law Equation, Derivation & Rearrangements

Core idea

Overview

Faraday's law of induction quantifies how a temporal change in magnetic flux through a conducting loop generates an electromotive force. It establishes the fundamental link between electricity and magnetism, stating that the magnitude of induced voltage is proportional to the rate of flux change.

When to use: Use this equation when a magnetic field moves relative to a conductor or when the strength of a magnetic field changes within a fixed loop. It is essential for analyzing systems where the area of a coil or its orientation relative to magnetic field lines is being modified.

Why it matters: This principle is the bedrock of modern power grids, enabling the operation of electrical generators that convert mechanical energy into electricity. It also powers everyday technology like wireless chargers, induction cooktops, and transformers.

Symbols

Variables

\Delta\Phi = Change in Flux, t = Time, \varepsilon = Induced EMF

ΔΦ

Change in Flux

W b

t

Time

s

ε

Induced EMF

V

Walkthrough

Derivation

Formula: Faraday's Law of Electromagnetic Induction

States that induced emf is proportional to the rate of change of magnetic flux linkage.

The coil has N turns experiencing the same flux.

1

State the Law:

Emf depends on change in flux $Δ$ $Φ$ over time. N $Φ$ is flux linkage.

ε = - N \frac{ΔΦ}{Δ t}

Note: The minus sign is Lenz’s law: induced emf opposes the change that produces it.

Result

ε = - N \frac{ΔΦ}{Δ t}

Source: AQA A-Level Physics — Magnetic Fields

Free formulas

Rearrangements

Solve for $ε$

Make emf the subject

ε = \frac{ΔΦ}{Δ t}

emf is already the subject of the formula.

Difficulty: 1/5

Solve for $ΔΦ$

Make DeltaPhi the subject

ΔΦ = ε Δ t

Start from Faraday's law. To make DeltaPhi the subject, clear Delta t, then simplify to isolate DeltaPhi.

Difficulty: 2/5

Solve for $t$

Make t the subject

Δ t = \frac{ΔΦ}{ε}

Start from Faraday's law. To make t the subject, clear Delta t, then divide by varepsilon.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin because the induced emf is directly proportional to the change in flux. For a student, this means that a larger change in flux over the same time interval results in a proportionally larger induced emf, while a smaller change in flux results in a smaller emf. The most important feature is that the linear relationship means doubling the change in flux will exactly double the induced emf.

Graph type: linear

Why it behaves this way

Intuition

Imagine a loop of wire; as the number of magnetic field lines passing through this loop changes over time, an electric 'push' (voltage) is generated around the loop, trying to create a current that opposes this change.

ε

Induced electromotive force (emf)

This is the voltage or 'electrical push' generated in a conductor or circuit due to the changing magnetic flux. It drives an induced current if the circuit is closed.

ΔΦ

Change in magnetic flux

This represents how much the total amount of magnetic field lines passing through a given area (like a coil) changes. A larger change in flux means a stronger induced emf.

Δ t

Change in time

This is the duration over which the magnetic flux changes. A shorter time interval (faster change) for the same change in flux results in a larger induced emf.

Signs and relationships

Negative sign (in ε = -dΦ/dt): The negative sign, often included in the full expression of Faraday's Law (ε = -dΦ/dt), represents Lenz's Law. It indicates that the direction of the induced electromotive force (and thus the induced current)

Free study cues

Insight

Canonical usage

This equation is used to calculate the magnitude of induced electromotive force (emf) in volts, given a change in magnetic flux in webers over a time interval in seconds.

Common confusion

A common mistake is confusing magnetic flux (Weber, Wb) with magnetic field strength (Tesla, T). Another is using non-SI units for time (e.g., minutes instead of seconds)

Unit systems

$ε$ V · Represents the induced electromotive force (voltage).

$ΔΦ$ Wb · Represents the change in magnetic flux through a surface.

$Δ t$ s · Represents the time interval over which the magnetic flux changes.

One free problem

Practice Problem

A sensor detects a change in magnetic flux of 0.8 Webers over 4.0 seconds. What is the average induced electromotive force in the circuit?

Change in Flux0.8 Wb

Time4 s

Solve for: $e m f$

Hint: Divide the total change in flux by the time interval to find the EMF.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating emf in a moving coil generator.

Study smarter

Tips

Ensure the change in flux (Phi) accounts for both final and initial states.
Time (t) must be in seconds for the result to be in Volts.
Remember that flux represents the magnetic field lines passing through a given area.

Avoid these traps

Common Mistakes

Forgetting to divide by time.
Using total flux instead of change in flux.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

States that induced emf is proportional to the rate of change of magnetic flux linkage.

Use this equation when a magnetic field moves relative to a conductor or when the strength of a magnetic field changes within a fixed loop. It is essential for analyzing systems where the area of a coil or its orientation relative to magnetic field lines is being modified.

This principle is the bedrock of modern power grids, enabling the operation of electrical generators that convert mechanical energy into electricity. It also powers everyday technology like wireless chargers, induction cooktops, and transformers.

Forgetting to divide by time. Using total flux instead of change in flux.

Estimating emf in a moving coil generator.

Ensure the change in flux (Phi) accounts for both final and initial states. Time (t) must be in seconds for the result to be in Volts. Remember that flux represents the magnetic field lines passing through a given area.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics
Wikipedia: Faraday's law of induction
IUPAC Gold Book: Electromotive force
IUPAC Gold Book: Magnetic flux
NIST CODATA
Wikipedia: Volt
Wikipedia: Weber
Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.