Faraday's Law Formula

The Formula

\mathcal{E} = -N\frac{d\Phi_B}{dt} where N is number of turns and \Phi_B = BA\cos\theta is magnetic flux.

When to use: The faster you change the magnetic field through a loop, the bigger the voltage you get. Faraday's law tells you exactly how much.

Quick Example

A coil with 100 turns in a field that drops from 0.5 T to 0 in 0.1 s: if the coil area is 0.01 m^2, the induced EMF is 5 V.

Notation

\mathcal{E} is the induced EMF in volts, N is the number of turns, \Phi_B is the magnetic flux in webers (Wb), \vec{B} is the magnetic field in tesla, and d\vec{A} is the differential area element.

What This Formula Means

The induced EMF in a circuit equals the negative rate of change of magnetic flux through the circuit.

The faster you change the magnetic field through a loop, the bigger the voltage you get. Faraday's law tells you exactly how much.

Formal View

Faraday's law states that the induced electromotive force in a closed loop equals the negative time derivative of magnetic flux: \mathcal{E} = -N\frac{d\Phi_B}{dt}, where \Phi_B = \int \vec{B} \cdot d\vec{A}.

Worked Examples

Example 1

medium
A coil with 100 turns experiences a change in magnetic flux from 0.05 \text{ Wb} to 0.02 \text{ Wb} in 0.1 \text{ s}. What is the induced EMF?

Solution

  1. 1
    Faraday's law: \mathcal{E} = -N\frac{\Delta \Phi}{\Delta t}.
  2. 2
    Change in flux: \Delta \Phi = 0.02 - 0.05 = -0.03 \text{ Wb}.
  3. 3
    \mathcal{E} = -100 \times \frac{-0.03}{0.1} = -100 \times (-0.3) = 30 \text{ V}

Answer

\mathcal{E} = 30 \text{ V}
Faraday's law states that a changing magnetic flux through a coil induces an EMF. More turns and faster flux changes produce larger EMFs. This is the basis of electric generators.

Example 2

hard
A single loop of area 0.04 \text{ m}^2 is in a magnetic field that increases uniformly from 0 to 0.6 \text{ T} in 0.3 \text{ s}. What is the induced EMF?

Example 3

medium
A coil of 50 turns has a magnetic flux that changes from 0.02 \text{ Wb} to 0.08 \text{ Wb} in 0.1 \text{ s}. Find the induced EMF.

Common Mistakes

  • Forgetting to multiply by the number of turns N — a coil with 100 turns produces 100 times more EMF than a single loop.
  • Ignoring the angle \theta between the magnetic field and the area vector when computing flux (\Phi_B = BA\cos\theta, not just BA).
  • Dropping the negative sign and then getting the direction of induced current wrong — the sign encodes Lenz's law.

Why This Formula Matters

Faraday's law is the quantitative foundation for designing generators, transformers, and induction-based sensors.

Frequently Asked Questions

What is the Faraday's Law formula?

The induced EMF in a circuit equals the negative rate of change of magnetic flux through the circuit.

How do you use the Faraday's Law formula?

The faster you change the magnetic field through a loop, the bigger the voltage you get. Faraday's law tells you exactly how much.

What do the symbols mean in the Faraday's Law formula?

\mathcal{E} is the induced EMF in volts, N is the number of turns, \Phi_B is the magnetic flux in webers (Wb), \vec{B} is the magnetic field in tesla, and d\vec{A} is the differential area element.

Why is the Faraday's Law formula important in Physics?

Faraday's law is the quantitative foundation for designing generators, transformers, and induction-based sensors.

What do students get wrong about Faraday's Law?

The negative sign means the induced EMF opposes the change (Lenz's law) — it's not just a math convention.

What should I learn before the Faraday's Law formula?

Before studying the Faraday's Law formula, you should understand: electromagnetic induction.

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