KAHIBARO
Table of Contents
Changing magnetic flux and induced emf
Faraday's law describes how a changing magnetic flux produces an electromotive force, usually called an induced emf, in a closed loop. This law is one of the key links between electricity and magnetism. A magnetic field that changes with time can create an electric effect, even when no battery is present.
The central idea is not simply that a magnetic field exists, but that the magnetic flux through a loop changes. If the flux stays constant, no emf is induced. If the flux changes, an emf appears.
Faraday's law for a single loop is
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
where $\mathcal{E}$ is the induced emf and $\Phi_B$ is the magnetic flux through the loop.
The minus sign is very important. It tells us the induced effect opposes the change in flux. The physical meaning of that opposition is treated more fully in Lenz's law, but the sign already appears in Faraday's law itself.
Magnetic flux in this context
To use Faraday's law, we need the magnetic flux through a surface bounded by the loop. For a uniform magnetic field through a flat surface,
$$
\Phi_B = BA\cos\theta
$$
where $B$ is the magnetic field magnitude, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the surface normal.
This means the flux can change in several different ways. The magnetic field strength can change, the loop area can change, or the angle can change.
| Quantity | How flux changes |
|---|---|
| $B$ | Stronger or weaker magnetic field |
| $A$ | Larger or smaller loop area |
| $\theta$ | Rotation of the loop relative to the field |
So a changing flux does not require only a changing field. Any process that changes $BA\cos\theta$ can induce an emf.
Meaning of induced emf
An emf is not necessarily a force in the ordinary mechanical sense. In circuit language, it is the energy supplied per unit charge around a loop. When Faraday's law says an emf is induced, it means charges in the conductor are driven around the loop by the electric field associated with the changing magnetic environment.
If the loop is a conducting wire, the induced emf can produce a current. If the loop is broken, there may still be an induced emf, but no steady current can circulate because the path is incomplete.
For a loop with resistance $R$, the induced current magnitude is often found from Ohm's law as
$$
I = \frac{|\mathcal{E}|}{R}
$$
when the loop behaves like an ordinary resistor.
A simple example
Imagine a circular wire loop sitting in a region where the magnetic field points upward through the loop. If the field strength increases with time, then the magnetic flux through the loop increases. Faraday's law says an emf is induced.
If the field changes at a constant rate and the loop area and orientation stay fixed, then
$$
\mathcal{E} = -A\frac{dB}{dt}\cos\theta
$$
For the special case where the field is perpendicular to the loop, $\theta = 0$, so $\cos\theta = 1$, and
$$
\mathcal{E} = -A\frac{dB}{dt}
$$
The larger the loop area, the larger the induced emf for the same rate of field change.
Coils with many turns
Many practical devices use coils rather than single loops. If a coil has $N$ identical turns and each turn experiences the same changing flux, then the induced emf adds from turn to turn.
For a coil of $N$ turns,
$$
\mathcal{E} = -N\frac{d\Phi_B}{dt}
$$
This is why coils are widely used in generators, transformers, and sensors. More turns usually means a larger induced emf.
Ways to produce induction
Faraday's law can be applied in several common situations.
If the magnetic field changes with time while the loop stays fixed, the flux changes and an emf is induced.
If the loop moves into or out of a magnetic field region, the part of the area threaded by the field changes, so the flux changes.
If the loop rotates in a magnetic field, the angle $\theta$ changes, so the flux changes.
These are all different physical setups, but mathematically they are all expressions of the same law.
Time rate of change matters
Faraday's law depends on how fast the flux changes, not just how much it changes overall. A large change in flux over a long time can produce a small emf, while a small change in flux over a very short time can produce a large emf.
Suppose the flux changes from $\Phi_{B1}$ to $\Phi_{B2}$ in a time interval $\Delta t$. Then the average induced emf is
$$
\mathcal{E}_{\text{avg}} = -\frac{\Delta \Phi_B}{\Delta t}
$$
where
$$
\Delta \Phi_B = \Phi_{B2} - \Phi_{B1}
$$
For exact instantaneous behavior, we use the derivative form with $d\Phi_B/dt$.
Integral form of Faraday's law
Faraday's law is one of Maxwell's equations. In general form, it relates the electric field around a closed path to the rate of change of magnetic flux through the surface bounded by that path.
The integral form of Faraday's law is
$$
\oint \vec{E}\cdot d\vec{\ell} = -\frac{d}{dt}\int \vec{B}\cdot d\vec{A}
$$
The left side is the circulation of the electric field around the loop. The right side is the negative time rate of change of the magnetic flux through the surface.
This equation shows something profound. A changing magnetic field can create an electric field whose field lines form closed loops. This differs from the electric field produced by stationary charges, whose field lines begin and end on charges.
Induced electric fields
Faraday's law is not only about wires. Even in empty space, a changing magnetic field creates an electric field. If a conductor is placed there, charges respond to that field and may move, producing current.
This means electromagnetic induction is a field phenomenon first, and a circuit phenomenon second. The wire helps reveal the effect, but the changing magnetic field creates the electric field whether or not the wire is present.
Sign and direction
The sign in Faraday's law tells us that nature resists changes in magnetic flux. If the flux through a loop increases in one direction, the induced current, if allowed to flow, creates its own magnetic field that tends to oppose that increase. If the flux decreases, the induced current tends to support the original flux direction.
The detailed method for finding the direction of induced current is the subject of Lenz's law, so here it is enough to remember that the minus sign is not optional. It is a fundamental part of the law.
Faraday's law without the minus sign is incomplete. The correct law is
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
Example with a rotating loop
Consider a loop of area $A$ rotating with angular speed $\omega$ in a uniform magnetic field $B$. If the angle between the field and the surface normal is
$$
\theta = \omega t
$$
then the flux is
$$
\Phi_B = BA\cos(\omega t)
$$
For a coil with $N$ turns, the induced emf is
$$
\mathcal{E} = -N\frac{d}{dt}\big(BA\cos(\omega t)\big)
$$
so
$$
\mathcal{E} = NBA\omega \sin(\omega t)
$$
This is a time-varying emf. It changes continuously as the loop rotates. This idea is the basis of alternating current generation.
Visualizing flux change
A loop can experience induction in several geometrically different ways.
In this drawing, the upward magnetic field through the loop is increasing. Because the flux changes, an emf is induced around the loop.
Common special cases
For a flat loop in a uniform magnetic field, some useful forms are:
| Situation | Flux | Induced emf |
|---|---|---|
| Changing field, fixed loop | $\Phi_B = BA\cos\theta$ | $\mathcal{E} = -A\cos\theta \, \dfrac{dB}{dt}$ |
| Changing area, fixed $B$ and $\theta$ | $\Phi_B = BA\cos\theta$ | $\mathcal{E} = -B\cos\theta \, \dfrac{dA}{dt}$ |
| Rotating loop, fixed $B$ and $A$ | $\Phi_B = BA\cos\theta$ | $\mathcal{E} = -BA \, \dfrac{d(\cos\theta)}{dt}$ |
| $N$ turns | $N\Phi_B$ effectively | $\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$ |
These formulas are all direct consequences of the same principle.
Physical importance
Faraday's law explains how electrical energy can be generated from mechanical motion and how changing currents and fields are connected. It is the operating principle behind electric generators, transformers, induction cooktops, and many sensing devices.
It also reveals a deep symmetry in electromagnetism. Just as electric charges can produce electric fields, changing magnetic fields can produce electric fields. This idea is essential for understanding electromagnetic waves and the full structure of Maxwell's equations.
Key point to remember
The essential statement of Faraday's law is this:
A changing magnetic flux through a closed loop induces an emf around that loop.
Mathematically,
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
and in full field form,
$$
\oint \vec{E}\cdot d\vec{\ell} = -\frac{d}{dt}\int \vec{B}\cdot d\vec{A}
$$
Faraday's law is therefore the rule that turns changing magnetism into electricity.