KAHIBARO
Table of Contents
Changing magnetic flux and induced emf
Faraday's law tells us how a changing magnetic environment creates an electromotive force, usually called an emf. This is one of the central ideas of electromagnetism. It explains how generators work, why a moving magnet can light a bulb in a coil, and why changing currents can produce voltages in nearby circuits.
The key idea is simple. If the magnetic flux through a loop changes with time, an emf is induced in that loop. Magnetic flux measures how much magnetic field passes through a surface.
For one loop, Faraday's law is
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
If a coil has $N$ turns, then
$$
\mathcal{E} = -N \frac{d\Phi_B}{dt}
$$
Here, $\mathcal{E}$ is the induced emf and $\Phi_B$ is the magnetic flux through one turn.
Faraday's law states that the induced emf equals the negative rate of change of magnetic flux:
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}, \qquad \mathcal{E} = -N\frac{d\Phi_B}{dt} \text{ for } N \text{ turns}
$$
No change in magnetic flux means no induced emf.
Magnetic flux in Faraday's law
To use Faraday's law, we need the magnetic flux through a surface. For a uniform magnetic field crossing a flat loop,
$$
\Phi_B = BA \cos\theta
$$
where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the surface.
This means the flux can change in three basic ways. The magnetic field $B$ can change, the area $A$ can change, or the orientation $\theta$ can change.
A loop does not care only about whether a magnetic field exists. It responds to whether the flux through it is changing.
What produces an induced emf
There are several common situations where Faraday's law applies. In each case, the induced emf comes from changing flux.
| Situation | What changes | Result |
|---|---|---|
| Magnet moved toward or away from a coil | $B$ through the coil changes | Induced emf appears |
| Coil moved into or out of a magnetic field | Flux changes because the part of the loop in field changes | Induced emf appears |
| Loop rotates in a magnetic field | Angle $\theta$ changes | Induced emf appears |
| Magnetic field itself changes with time | $B$ changes directly | Induced emf appears |
| Coil area changes | $A$ changes | Induced emf appears |
Meaning of the negative sign
The negative sign in Faraday's law is very important. It shows that the induced emf acts in a direction that opposes the change in magnetic flux. This direction rule is more fully discussed in Lenz's law, but even here we can see the meaning.
If the magnetic flux through a loop increases, the induced emf drives current in such a way that the magnetic field created by that current tries to reduce the increase.
If the magnetic flux decreases, the induced emf drives current in such a way that the loop tries to maintain the original flux.
The negative sign in Faraday's law does not mean the emf is always negative as a number. It expresses direction, the induced effect opposes the change in flux.
Integral form of Faraday's law
Faraday's law can also be written in a more general form:
$$
\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}
$$
This says that a changing magnetic flux produces an electric field whose line integral around a closed loop is not zero. This is different from the electrostatic field created by stationary charges. In electrostatics, the line integral around a closed path is zero. In electromagnetic induction, a changing magnetic field creates a circulating electric field.
This form is especially useful because it shows that induction is not only about wires. A changing magnetic field can create an electric field in space itself.
A simple example with a changing field
Suppose a circular loop of area $A = 0.020 \, \text{m}^2$ is in a magnetic field perpendicular to the loop. The field increases from $0.10 \, \text{T}$ to $0.40 \, \text{T}$ in $0.50 \, \text{s}$.
Since the field is perpendicular, $\cos\theta = 1$, so
$$
\Phi_B = BA
$$
The change in flux is
$$
\Delta \Phi_B = A \Delta B = (0.020)(0.40 - 0.10) = 0.0060 \, \text{Wb}
$$
The magnitude of the induced emf is
$$
|\mathcal{E}| = \frac{\Delta \Phi_B}{\Delta t} = \frac{0.0060}{0.50} = 0.012 \, \text{V}
$$
So the induced emf is $0.012 \, \text{V}$.
If the loop had 100 turns, then
$$
|\mathcal{E}| = 100 \times 0.012 = 1.2 \, \text{V}
$$
This shows why coils with many turns are useful in devices.
Rotating loop and sinusoidal emf
A very important application of Faraday's law is a loop rotating in a uniform magnetic field. If the angle changes with time as
$$
\theta = \omega t
$$
then the flux is
$$
\Phi_B = BA \cos(\omega t)
$$
For a coil with $N$ turns, the induced emf becomes
$$
\mathcal{E} = -N\frac{d}{dt}\bigl(BA\cos(\omega t)\bigr)
$$
so
$$
\mathcal{E} = NBA\omega \sin(\omega t)
$$
This is an alternating emf. It changes sign periodically. This is the basic idea behind electrical generators.
For a rotating coil in a uniform magnetic field,
$$
\Phi_B = NBA\cos(\omega t)
$$
and
$$
\mathcal{E} = NBA\omega \sin(\omega t)
$$
The maximum emf is
$$
\mathcal{E}_{\max} = NBA\omega
$$
Visual picture of changing flux
A loop in a magnetic field can experience more or less flux depending on its orientation.
In this drawing, the crosses represent a magnetic field going into the page. If the loop rotates, the angle between the field and the loop's normal changes, so the flux changes, and an emf is induced.
Conducting rod moving in a magnetic field
Another useful case is a straight conducting rod moving through a magnetic field. This is a special example of Faraday's law because the motion changes the area of the loop and therefore changes the flux.
If a rod of length $\ell$ moves with speed $v$ perpendicular to a magnetic field $B$, then the induced emf is
$$
\mathcal{E} = B\ell v
$$
This result can be understood as a flux change. If the rod sweeps out area at a rate
$$
\frac{dA}{dt} = \ell v
$$
then
$$
\frac{d\Phi_B}{dt} = B\frac{dA}{dt} = B\ell v
$$
and therefore
$$
|\mathcal{E}| = B\ell v
$$
This is one of the most direct examples of induction.
Practical meaning of emf
An induced emf is not exactly the same thing as current. The emf is the cause that can drive current if the loop is a closed conducting path. If the loop is open, an emf can still exist, but no steady current can flow around the loop.
If the loop has resistance $R$ and forms a closed circuit, then the induced current magnitude is often found from Ohm's law:
$$
I = \frac{|\mathcal{E}|}{R}
$$
The chapter on circuits explains this in more detail. Here the main point is that Faraday's law gives the induced emf first.
Summary relationships
The most important forms of Faraday's law are collected here.
| Case | Formula |
|---|---|
| General law for one loop | $\mathcal{E} = -\dfrac{d\Phi_B}{dt}$ |
| Coil with $N$ turns | $\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$ |
| Uniform field through flat loop | $\Phi_B = BA\cos\theta$ |
| Rotating coil | $\mathcal{E} = NBA\omega \sin(\omega t)$ |
| Maximum rotating coil emf | $\mathcal{E}_{\max} = NBA\omega$ |
| Sliding rod | $\mathcal{E} = B\ell v$ |
To apply Faraday's law, always check what is changing in the flux:
$$
\Phi_B = BA\cos\theta
$$
Ask whether $B$, $A$, or $\theta$ changes with time. Any of these can produce an induced emf.
Physical significance
Faraday's law shows a deep connection between electricity and magnetism. A changing magnetic field can create an electric effect, even without direct contact and even in empty space. This is one of the ideas that later becomes part of Maxwell's equations and the full theory of electromagnetic waves.
In beginner physics, the most important lesson is this. Magnetic fields do not need to be strong to induce an emf. They need to change the magnetic flux through a loop. That changing flux is the heart of electromagnetic induction.