We have already seen that a change in magnetic flux induces a voltage.
If the area enclosed by the conductor is constant, then the induced voltage depends only on the time variation of the magnetic field.
For a coil, the number of turns $n$ also matters — the induced voltage is proportional to $n$:
\[
U_\mathrm{ind} = -n \frac{dB}{dt} A
\]
For a current-carrying conductor, the magnetic field strength (flux density) is proportional to the current, according to Ampère’s law.
Therefore, the induced voltage in the conductor is proportional to the rate of change of the current:
\[
U_\mathrm{ind} \propto -\frac{dI}{dt}
\]
The proportionality constant is called the inductance $L$, measured in Henry (H).
Thus, the induced voltage can be written as:
\[
\boxed{U_\mathrm{ind} = -L \frac{dI}{dt}}
\]
In a long cylindrical coil, the magnetic field is nearly homogeneous (see figure).
We can calculate the flux density using Ampère’s law by choosing a rectangular integration path: one long side inside the coil, one long side far outside (where $B \approx 0$), and the two short sides perpendicular to $B$ (so their contribution vanishes).
This leaves only the path inside the coil:
\[
\oint \vec{B}\cdot d\vec{s} = \int_0^l B\,ds = Bl
\]
where $l$ is the length of the coil.
For a coil with $n$ turns, the magnetic flux density increases proportionally with $n$.
From Ampère’s law:
\[
Bl = \mu_0 \mu_r n I
\]
Dividing by $l$ gives the flux density inside the coil:
\[
\boxed{B = \mu_0 \mu_r n \frac{I}{l}}
\]
If we substitute this into the induction law, we find:
\[
U_\mathrm{ind} = -\mu_0 \mu_r \frac{n^2 A}{l} \frac{dI}{dt}
\]
Thus, the inductance of a long solenoid is:
\[
\boxed{L = \mu_0 \mu_r \frac{n^2 A}{l}}
\]
Here $A$ is the cross-sectional area of the coil.
As with the capacitor, edge effects are neglected, which is a good approximation for long coils.
Example problem: A cylindrical coil has 150 turns, a radius of $2\,$cm, and a length of $15\,$cm. Inside is an iron core with relative permeability $\mu_r = 5000$. If the current increases at a constant rate of $10\,\text{mA/s}$, what induced voltage appears across the coil? Result: 11.8 mV