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Faraday's Law — Electromagnetic Induction

A changing magnetic flux induces EMF: ε = -N dΦ/dt. Flux Φ = B×A×cos(θ). Lenz's law: induced current opposes the change. Generators, transformers, and induction motors rely on this principle.

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EMF = -N dΦ/dt; negative sign from Lenz's law Φ = B×A×cos(θ) for flux through loop Rotating coil: ε_peak = N×B×A×ω Transformers use mutual induction between coils

Key quantities
ε = -N dΦ/dt
Faraday's Law
Key relation
Φ = B×A×cos θ
Flux
Key relation
opposes change
Lenz's Law
Key relation
N×B×A×ω
Peak EMF
Key relation

Ready to run the numbers?

Why: Faraday's law underlies generators, transformers, induction heating, and wireless charging. Changing flux induces voltage; Lenz's law ensures energy conservation.

How: Enter magnetic field, area, turns, and rate of change (or frequency for AC). The calculator computes induced EMF, flux, and current for coils and rotating generators.

EMF = -N dΦ/dt; negative sign from Lenz's lawΦ = B×A×cos(θ) for flux through loop
Sources:HyperPhysics ElectromagnetismNIST Physical Constants

Run the calculator when you are ready.

Solve Faraday's Law EquationCalculate induced EMF and current

Input Parameters

Number of turns must be a positive number

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

🧲

Faraday discovered electromagnetic induction in 1831.

— HyperPhysics

⚡

Lenz's law: induced current opposes the flux change.

— NIST

🔄

AC generators use rotating coils in magnetic fields.

— MIT

📐

Transformer EMF ratio = turns ratio N₁/N₂.

— HyperPhysics

What is Faraday's Law?

🔬 Electromagnetic Induction

Faraday's Law of electromagnetic induction is one of the fundamental principles of electromagnetism, discovered by Michael Faraday in 1831. It describes how a changing magnetic field induces an electromotive force (EMF) in a conductor, leading to the generation of electric current.

This phenomenon is the basis for generators, transformers, electric motors, and many other electrical devices that power our modern world.

⚡ Key Concepts

  • Magnetic Flux (Φ): The total magnetic field passing through a surface, measured in Weber (Wb)
  • Induced EMF (ε): The voltage generated by changing magnetic flux, measured in Volts (V)
  • Lenz's Law: The induced current flows in a direction that opposes the change causing it
  • Flux Linkage: The product of number of turns and magnetic flux (NΦ)

📐 Mathematical Foundation

Faraday's Law states that the induced EMF in a coil is proportional to the rate of change of magnetic flux:

ε=−NdΦdt\varepsilon = -N \frac{d\Phi}{dt}

Where N is the number of turns, and the negative sign represents Lenz's law.

How Does Faraday's Law Work?

Faraday's Law operates through several mechanisms that can induce EMF in a conductor:

1. Changing Magnetic Field

When the magnetic field strength changes while the coil area and orientation remain constant, an EMF is induced. This is common in transformers where alternating current creates a changing magnetic field.

ΔΦ=(B2−B1)×A×cos⁡(θ)\Delta \Phi = (B_2 - B_1) \times A \times \cos(\theta)

2. Changing Area

When a conductor moves through a magnetic field, changing the area exposed to the field, EMF is induced. This occurs in generators where coils rotate through magnetic fields.

ΔΦ=B×(A2−A1)×cos⁡(θ)\Delta \Phi = B \times (A_2 - A_1) \times \cos(\theta)

3. Rotating Coil

When a coil rotates in a uniform magnetic field, the angle between the field and the coil's normal changes, creating a sinusoidal EMF. This is the principle behind AC generators.

ε(t)=N×B×A×ω×sin⁡(ωt)\varepsilon(t) = N \times B \times A \times \omega \times \sin(\omega t)

4. Lenz's Law Direction

Lenz's law ensures that the induced current creates a magnetic field that opposes the change in flux. If flux is increasing, the induced current flows to decrease it; if flux is decreasing, the current flows to increase it.

When to Use Faraday's Law Calculator

⚡ Generator Design

Calculate EMF output for AC/DC generators, determine optimal coil configurations, and predict power generation capacity.

🔌 Transformer Analysis

Design step-up or step-down transformers, calculate voltage transformation ratios, and analyze transformer efficiency.

🚂 Moving Conductor Systems

Analyze EMF generation in linear motors, maglev trains, and systems with conductors moving through magnetic fields.

🛑 Electromagnetic Braking

Calculate eddy current braking forces, design electromagnetic brakes for vehicles, and analyze energy dissipation.

🔄 Motor Design

Analyze back EMF in electric motors, design motor windings, and calculate motor performance characteristics.

📡 Wireless Power Transfer

Design inductive charging systems, calculate coupling efficiency, and analyze wireless energy transmission.

Formulas Explained

Faraday's Law

ε=−NdΦdt\varepsilon = -N \frac{d\Phi}{dt}

The induced EMF equals the negative of the number of turns times the rate of change of magnetic flux. The negative sign indicates Lenz's law.

Magnetic Flux

Φ=B×A×cos⁡(θ)\Phi = B \times A \times \cos(\theta)

Magnetic flux depends on field strength (B), area (A), and the angle (θ) between the field and the surface normal.

Rotating Coil EMF

εpeak=N×B×A×ω\varepsilon_{peak} = N \times B \times A \times \omega

For a coil rotating with angular velocity ω in a uniform field B, the peak EMF is proportional to all these factors.

Induced Current

I=εRI = \frac{\varepsilon}{R}

The induced current equals the induced EMF divided by the circuit resistance, following Ohm's law.

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