Last updated: 2026-06-15

Solenoid Magnetic Field

A solenoid is a coil of wire wound in a helix. When current flows through the wire, each loop contributes a small magnetic field, and the fields from all the loops add together constructively inside the coil. The result is a nearly uniform field directed along the axis of the solenoid. For a solenoid that is long relative to its diameter, the interior field is:

$B = \mu_0 n I$

where $\mu_0 = 1.2566 \times 10^{-6}\ \text{T·m/A}$ is the permeability of free space, $n = N/L$ is the turn density in turns per metre, and $I$ is the current in amperes. Outside an ideal infinite solenoid the field is exactly zero; in practice it is very small.

What Is Turn Density?

Turn density $n$ is the number of wire loops per unit length of solenoid:

$n = \frac{N}{L}$

where $N$ is the total number of turns and $L$ is the solenoid length in metres. A solenoid wound with 500 turns over 0.5 m has $n = 1000\ \text{turns/m}$, the same as 1000 turns over 1 m. Both produce the same field for the same current. What matters for the field strength is the turn density, not the total number of turns or the length separately.

Formula

The field is proportional to both $n$ and $I$. Doubling either one doubles the field.

Worked Example

A solenoid has 1000 turns wound over a length of 1 m and carries a current of 2 A. Find the interior field.

First calculate the turn density:

$n = \frac{N}{L} = \frac{1000}{1} = 1000\ \text{turns/m}$

Then the field:

Entering 1000 turns, 1 m, and 2 A in the calculator gives the same result. Note that 500 turns over 0.5 m with 2 A gives exactly the same field, because both configurations share the same turn density of 1000 turns/m.

Solenoid Compared with a Bar Magnet

The external magnetic field pattern of a solenoid is essentially identical to that of a bar magnet — a dipole field with identifiable north and south poles at the ends. The main practical difference is control: a solenoid's field can be switched on and off, reversed, and tuned continuously by adjusting the current or the number of active turns. This makes solenoids central to electric motors, transformers, MRI machines, and particle accelerators.

Inside a long solenoid the field is far more uniform than inside a bar magnet. This uniformity is exploited wherever a controlled and predictable field is needed over a region of space.

When the Approximation Breaks Down

The formula $B = \mu_0 n I$ holds for a solenoid whose length is much larger than its diameter. Near the ends the field weakens and begins to resemble a dipole. The field at the very end of a long solenoid is approximately half the interior value. For short solenoids or applications that demand high accuracy near the ends, the full Biot–Savart integral must be evaluated numerically.

Adding a ferromagnetic core (iron, for example) multiplies the interior field by the material's relative permeability $\mu_r$, which can reach several thousand for soft iron: $B = \mu_0 \mu_r n I$. This principle underlies the electromagnets used in electric motors and transformers.