Faraday's Law of Induction Equation, Derivation & Rearrangements

Core idea

Overview

Interprets the induced electric field around a fixed loop from changing magnetic flux. It is content-only because the displayed integral or circulation law is a decision rule before choosing a specific geometry.

When to use: Use this to decide which source, path, or geometry controls the magnetic or induced-electric field.

Why it matters: It explains where the simpler magnetic-field formulas in this batch come from.

Walkthrough

Derivation

Derivation of Faraday Law for a Stationary Path

Interprets the induced electric field around a fixed loop from changing magnetic flux.

The path and surface orientation are chosen consistently.
The electromagnetic fields are described by classical electromagnetism.

1

Read the law

Identify the field circulation, source, and sign convention.

\oint E \cdot d l = - \frac{d Φ _{B}}{d t}

2

Match geometry

Only then can the law be reduced to a scalar formula.

choose path/surface/current element

Result

choose path/surface/current element

Source: Moebs, Ling, and Sanny, University Physics Volume 2, OpenStax, 2016, chapter 13, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $law$

Use the law as a geometry decision rule

draw geometry and choose orientation

Choose the correct path, surface, or current element before reducing the relation.

Difficulty: 3/5

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

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

A changing magnetic flux induces a circulating electric field, and the minus sign records Lenz-law opposition.

\oint

closed-loop integral

The circulation of the electric field around the loop.

E

electric field

The field induced around the loop.

d l

path element

A tiny segment along the loop.

Φ_{B}

magnetic flux

The magnetic field threading the loop.

t

time

The variable over which the flux changes.

Signs and relationships

-: The induced field opposes the change in flux.

Free study cues

Insight

Canonical usage

This equation relates the line integral of the electric field around a closed loop to the rate of change of magnetic flux through the surface bounded by that loop.

Common confusion

Students may confuse the rate of change of magnetic flux with magnetic flux itself, or misapply the negative sign, which indicates the direction of the induced electric field according to Lenz's Law.

Dimension note

While the equation itself involves quantities with physical units, the relationship it describes is fundamental to electromagnetism and is applied across various unit systems.

Unit systems

$E$ V/m · Represents the electric field vector.

dlm · Represents an infinitesimal element of length along the path of integration.

$P h i_{B}$ Wb · Represents the magnetic flux, which is the integral of the magnetic field over an area.

$t$ s · Represents time.

One free problem

Practice Problem

According to Faraday's Law, what must happen to the magnetic flux through a loop to induce an electric field along that loop?

Solve for: $\oint E \cdot d l$

Hint: Look at the derivative term in the formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In These laws are used to derive fields of wires, loops, solenoids, and induction setups, Faraday's Law of Induction is used to calculate \oint\vec{E}\cdot d\vec{l} from the measured values. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Draw the loop or current element first.
Use the sign/orientation convention consistently.
Choose a simpler derived formula only after matching the geometry.

Avoid these traps

Common Mistakes

Treating an integral law as a one-line scalar calculator.
Ignoring path orientation or enclosed current/flux.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Interprets the induced electric field around a fixed loop from changing magnetic flux.

Use this to decide which source, path, or geometry controls the magnetic or induced-electric field.

It explains where the simpler magnetic-field formulas in this batch come from.

Treating an integral law as a one-line scalar calculator. Ignoring path orientation or enclosed current/flux.