Faraday's Law of Electromagnetic Induction Equation, Derivation & Rearrangements

Core idea

Overview

Faraday's Law of Electromagnetic Induction describes how a changing magnetic field through a coil of wire generates an electromotive force (EMF), which can drive an electric current. The law states that the magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linkage. The negative sign, often referred to as Lenz's Law, indicates that the direction of the induced EMF opposes the change in magnetic flux that produced it, a manifestation of energy conservation.

When to use: Apply this law when a conductor or coil is exposed to a time-varying magnetic field, or when a conductor moves through a magnetic field, to determine the resulting induced voltage. It is crucial for understanding generators, transformers, and other electromagnetic devices. Ensure consistent units for magnetic flux (Webers) and time (seconds).

Why it matters: Faraday's Law is fundamental to modern electrical engineering and technology, underpinning the operation of almost all electrical power generation. It explains how generators convert mechanical energy into electrical energy, how transformers efficiently change AC voltages, and is essential for designing induction motors, RFID systems, and many other devices that rely on electromagnetic induction.

Symbols

Variables

N = Number of Turns, \Delta\Phi = Change in Magnetic Flux, \Delta t = Change in Time, \mathcal{E} = Induced EMF

N

Number of Turns

t u r n s

ΔΦ

Change in Magnetic Flux

W b

Δ t

Change in Time

s

E

Induced EMF

V

Walkthrough

Derivation

Formula: Faraday's Law of Electromagnetic Induction

Faraday's Law states that the induced electromotive force (EMF) in a closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

The magnetic flux is changing over time.
The coil is a closed circuit or has defined ends for EMF measurement.

1

Define Magnetic Flux Linkage:

Magnetic flux Φ_B through a surface is the integral of the magnetic field B over the area dA. For a coil with N turns, the flux linkage is NΦ_B.

Φ_{B} = \int B \cdot d A

2

Faraday's Experimental Observation:

Faraday observed that the induced EMF 𝓔 is proportional to the rate of change of magnetic flux linkage NΦ_B.

E \propto - \frac{Δ ( N Φ _{B} )}{Δ t}

3

Introducing the Constant of Proportionality:

In SI units, the constant of proportionality k is 1, leading to the direct relationship. The negative sign is due to Lenz's Law.

E = - k \frac{Δ ( N Φ _{B} )}{Δ t}

4

Final Form of Faraday's Law:

This is the standard form of Faraday's Law, where ΔΦ represents the change in magnetic flux over a time interval Δt.

E = - N \frac{ΔΦ}{Δ t}

Note: For instantaneous EMF, the derivative form 𝓔 = -N * (dΦ/dt) is used.

Result

E = - N \frac{ΔΦ}{Δ t}

Source: AQA A-Level Physics — Electromagnetism (7408/7407)

Free formulas

Rearrangements

Solve for $N$

Faraday's Law: Make N the subject

N = - \frac{E Δ t}{ΔΦ}

To make N (Number of Turns) the subject of Faraday's Law, multiply both sides by Δt, then divide by -ΔΦ.

Difficulty: 2/5

Solve for $ΔΦ$

Faraday's Law: Make ΔΦ the subject

ΔΦ = - \frac{E Δ t}{N}

To make ΔΦ (Change in Magnetic Flux) the subject, multiply both sides by Δt, then divide by -N.

Difficulty: 2/5

Solve for $d e l t a T im e$

Faraday's Law: Make Δt the subject

Δ t = - N \frac{ΔΦ}{E}

To make Δt (Change in Time) the subject, first swap 𝓔 and (ΔΦ/Δt) terms, then isolate Δt.

Difficulty: 2/5

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

Visual intuition

Graph

The graph forms a hyperbola because the change in time appears in the denominator. A small change in time results in a large induced EMF, while a large change in time causes the induced EMF to approach zero, illustrating that rapid flux changes generate more voltage. The most important feature is that the curve never reaches zero, meaning that as long as there is a change in magnetic flux, some induced EMF will always exist regardless of how long the time interval becomes.

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine a coil of wire as a net. When the number of magnetic field lines passing through this net changes, an electrical 'push' (EMF) is generated in the wire, trying to create a magnetic field that opposes that change.

E

Induced electromotive force (EMF)

The 'voltage' or electrical 'push' generated across the coil's terminals, capable of driving a current.

N

Number of turns in the coil

Each loop of wire contributes to the total induced EMF. More turns mean the magnetic effect is multiplied, leading to a larger total EMF.

ΔΦ

Change in magnetic flux

How much the total amount of magnetic field lines passing through the coil's area has increased or decreased. A larger change means a stronger effect.

Δ t

Change in time

The duration over which the magnetic flux changes. A shorter time for the same flux change means a faster rate of change, leading to a larger induced EMF.

\frac{ΔΦ}{Δ t}

Rate of change of magnetic flux

How quickly the magnetic field 'threading' through the coil is changing. A faster change in magnetic flux generates a larger induced EMF.

Signs and relationships

-: The negative sign represents Lenz's Law. It indicates that the direction of the induced electromotive force (and any resulting current)

Free study cues

Insight

Canonical usage

In the International System of Units (SI), electromotive force (EMF) is expressed in Volts, magnetic flux in Webers, and time in seconds, with the number of turns being dimensionless.

Common confusion

A frequent error is to confuse magnetic field strength (Tesla) with magnetic flux (Weber) or to use inconsistent units for time (e.g., minutes instead of seconds) when calculating the rate of change.

Unit systems

$E$ V · Represents the induced electromotive force (EMF), measured in Volts (V).

$N$ dimensionless · Represents the number of turns in the coil. It is a pure, dimensionless number.

$ΔΦ$ Wb · Represents the change in magnetic flux, measured in Webers (Wb). One Weber is equivalent to one Tesla-meter squared (T·m2).

$Δ t$ s · Represents the change in time over which the magnetic flux changes, measured in seconds (s).

One free problem

Practice Problem

A coil with 150 turns experiences a change in magnetic flux from 0.02 Wb to 0.08 Wb over a period of 0.5 seconds. Calculate the magnitude of the induced electromotive force (EMF) across the coil.

Number of Turns150 turns

Change in Magnetic Flux0.06 Wb

Change in Time0.5 s

Solve for: $r es u l t$

Hint: Calculate the change in magnetic flux first.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Operation of an electrical generator producing electricity.

Study smarter

Tips

Remember the negative sign (Lenz's Law) indicates the direction of the induced EMF opposes the change in flux.
Magnetic flux linkage (NΦ) is the product of the number of turns and the magnetic flux through each turn.
Ensure ΔΦ and Δt are measured over the same interval.
The rate of change of flux (ΔΦ/Δt) is key; a constant flux produces no induced EMF.

Avoid these traps

Common Mistakes

Forgetting the negative sign or misinterpreting its meaning (Lenz's Law).
Confusing magnetic flux (Φ) with magnetic flux density (B).
Incorrectly using units, especially for time or magnetic flux.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Faraday's Law states that the induced electromotive force (EMF) in a closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Apply this law when a conductor or coil is exposed to a time-varying magnetic field, or when a conductor moves through a magnetic field, to determine the resulting induced voltage. It is crucial for understanding generators, transformers, and other electromagnetic devices. Ensure consistent units for magnetic flux (Webers) and time (seconds).

Faraday's Law is fundamental to modern electrical engineering and technology, underpinning the operation of almost all electrical power generation. It explains how generators convert mechanical energy into electrical energy, how transformers efficiently change AC voltages, and is essential for designing induction motors, RFID systems, and many other devices that rely on electromagnetic induction.

Forgetting the negative sign or misinterpreting its meaning (Lenz's Law). Confusing magnetic flux (Φ) with magnetic flux density (B). Incorrectly using units, especially for time or magnetic flux.

Operation of an electrical generator producing electricity.

Remember the negative sign (Lenz's Law) indicates the direction of the induced EMF opposes the change in flux. Magnetic flux linkage (NΦ) is the product of the number of turns and the magnetic flux through each turn. Ensure ΔΦ and Δt are measured over the same interval. The rate of change of flux (ΔΦ/Δt) is key; a constant flux produces no induced EMF.

References

Sources

Halliday, Resnick, Walker. Fundamentals of Physics.
Griffiths, David J. Introduction to Electrodynamics.
Wikipedia: Faraday's law of induction
Halliday, Resnick, and Walker, Fundamentals of Physics
Griffiths, David J. Introduction to Electrodynamics
NIST Special Publication 330, The International System of Units (SI)
Halliday, Resnick, Walker Fundamentals of Physics
Griffiths Introduction to Electrodynamics

Faraday's Law of Electromagnetic Induction

Overview

Variables

Derivation

Define Magnetic Flux Linkage:

Faraday's Experimental Observation:

Introducing the Constant of Proportionality:

Final Form of Faraday's Law:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Faraday's law

Motor effect

Transformer Turns Ratio

Transformer Efficiency

Frequently Asked Questions

Sources