By David Latchman
Updated Aug 30, 2022
A solenoid is a long, tightly wound coil of wire that generates a magnetic field when current flows through it. Typically wrapped around a metallic core, its field strength depends on coil density, current, and the core’s magnetic properties.
As a specialized electromagnet, a solenoid produces a controlled magnetic field useful for driving electric motors, acting as an inductor, or creating a uniform field for scientific experiments.
The field inside an ideal solenoid is derived from Ampère’s Law:
\(Bl=\mu_0 NI\)
Dividing by length gives the familiar form:
\(B=\mu_0\frac{N}{l}I\)
where B is magnetic flux density, l the solenoid length, N the number of turns, and I the current. The turns density N/l captures how tightly the wire is wound. The magnetic constant μ₀ equals 1.257 × 10⁻⁶ H/m.
Inserting a magnetic core multiplies the field by the core’s relative permeability μ_r:
\(\mu = \mu_r\mu_0\)
Consequently, the field becomes:
\(B=\mu\frac{N}{l}I\)
A high‑permeability core, such as iron, concentrates the field, markedly increasing B.
When current changes, a solenoid resists that change by inducing a voltage—a phenomenon known as electromagnetic induction. The ratio of induced voltage to the rate of current change defines the inductance L:
\(L=-\frac{v}{\frac{dI}{dt}}\)
Rearranging gives the classic expression:
\(v=-L\frac{dI}{dt}\)
Faraday’s Law relates the induced EMF to the time rate of change of magnetic flux:
\(v=-nA\frac{dB}{dt}\)
Substituting the solenoid field derivative dB/dt = \mu\frac{N}{l}\frac{dI}{dt} yields:
\(v=-\left(\frac{\mu N^2 A}{l}\right)\frac{dI}{dt}\)
Comparing with the definition of inductance gives the final formula:
\(L=\frac{\mu N^2 A}{l}\)
This shows that inductance depends on coil geometry—turns density and cross‑sectional area—and the core’s magnetic permeability.
