5.6.8. Solenoids

Table of Contents

Magnetic field inside a solenoid

A solenoid is a long coil of wire with many turns. When electric current flows through the wire, the coil produces a magnetic field. Solenoids are important because they can create a strong and fairly uniform magnetic field in a controlled region of space.

A single circular loop creates a magnetic field, but many loops placed closely together make the field stronger. In a solenoid, the magnetic effects of the turns add together, especially along the central axis of the coil.

Structure and idea

A simple solenoid is made by winding insulated wire into a helical coil. The key geometric quantities are the length of the coil, the number of turns, and the current through the wire. If the solenoid has total number of turns $N$ and length $L$, then the number of turns per unit length is

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

This quantity $n$ is very important because the magnetic field of an ideal solenoid depends on it.

Rendering scientific sketch…

A simple solenoid

Direction of the magnetic field

The direction of the magnetic field inside a solenoid is found with the right hand rule. Curl the fingers of your right hand in the direction of the current around the turns. Your thumb then points in the direction of the magnetic field inside the solenoid.

This also tells you which end behaves like a north pole and which end behaves like a south pole. A solenoid acts in many ways like a bar magnet, with one end behaving as north and the other as south.

Rendering scientific sketch…

Right hand rule idea for a solenoid

Ideal solenoid

An ideal solenoid is very long compared with its diameter, and its turns are closely spaced. For such a solenoid, the magnetic field inside is approximately uniform and parallel to the axis of the coil. Outside, the field is much weaker and is often treated as nearly zero.

For an ideal solenoid in air or vacuum, the magnetic field magnitude inside is

$$
B = \mu_0 n I
$$

where $B$ is the magnetic field, $\mu_0$ is the permeability of free space, $n$ is the number of turns per unit length, and $I$ is the current.

Since $n = N/L$, this can also be written as

$$
B = \mu_0 \frac{N}{L} I
$$

For an ideal long solenoid,
$$
B = \mu_0 n I = \mu_0 \frac{N}{L} I
$$
The field is strongest and most uniform inside the solenoid, and much weaker outside.

Why the field is nearly uniform

Each turn of the coil contributes to the total magnetic field. Near the center of a long solenoid, the contributions from many turns combine in the same direction. Because the turns are evenly spaced, the field changes very little from one point to another inside the central region.

Near the ends, the field is less uniform. This is called the end effect. So the formula for an ideal solenoid works best well inside a long coil, not near its edges.

Use of Ampère's law

The standard derivation of the solenoid field uses Ampère's law. For a long solenoid, symmetry tells us that the field inside is almost constant and parallel to the axis, while outside it is very small. Applying Ampère's law gives

$$
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}
$$

If the rectangular path encloses part of the solenoid, then the enclosed current is the number of turns crossed times the current in each turn. This leads directly to

$$
B = \mu_0 n I
$$

A full treatment of Ampère's law belongs with that topic, but here it explains why the solenoid field has such a simple form.

Solenoids with a magnetic core

If the solenoid contains a magnetic material such as iron, the magnetic field can become much larger than in air. In a simple introductory description, the field is often written as

$$
B = \mu n I
$$

where $\mu$ is the permeability of the material inside the solenoid.

In many practical devices, inserting a ferromagnetic core greatly increases the field strength. This is why solenoids are used in electromagnets, relays, and actuators.

A core material can increase the magnetic field:
$$
B = \mu n I
$$
In vacuum or air, use $\mu = \mu_0$.

Comparison with a bar magnet

A solenoid and a bar magnet produce similar field patterns outside. Both have north and south poles, and magnetic field lines emerge from the north side and enter the south side outside the object. Inside the solenoid, however, the field is concentrated and approximately uniform if the solenoid is long.

Object	Field inside	Field outside	Poles
Long solenoid	Strong, nearly uniform	Weak	Yes, effective north and south ends
Bar magnet	Present but fixed by material	Clear dipole pattern	Yes

Practical examples

A solenoid can be used as an electromagnet. When current flows, it produces magnetism. When the current stops, the field weakens or disappears, especially if there is no permanent magnetic core. This makes solenoids useful when controllable magnetism is needed.

Examples include electric bells, door locks, relays, valves, magnetic lifting devices, and starter mechanisms in machines. In these devices, the magnetic field often pulls a piece of iron into the coil, converting electrical energy into mechanical motion.

Field pattern

The field lines of a long solenoid are dense and nearly parallel inside. Outside they loop back from one end to the other.

Rendering scientific sketch…

Magnetic field pattern of a solenoid

Important observations

The magnetic field becomes stronger if the current increases. It also becomes stronger if the turns are packed more closely, which means a larger value of $n$. For a fixed current, doubling the number of turns per unit length doubles the field. For a fixed coil geometry, doubling the current also doubles the field.

This simple proportional behavior makes solenoids easy to design and use in basic circuits and magnetic devices.

For a long solenoid, the field is directly proportional to current and turns per unit length:
$$
B \propto I
$$
and
$$
B \propto n
$$

Limits of the ideal model

Real solenoids are not infinite in length. Their field is not perfectly uniform, especially near the ends. The outside field is not exactly zero. Also, if a core material is used, the relationship between $B$ and $I$ may become more complicated because real magnetic materials do not always respond linearly.

Still, the ideal solenoid model is extremely useful and gives a very good first approximation in many physical situations.

Summary

A solenoid is a coil of many turns that produces a magnetic field when current flows through it. A long solenoid creates a strong, nearly uniform magnetic field inside. Its field direction is found by the right hand rule, and its magnitude in air or vacuum is

$$
B = \mu_0 n I = \mu_0 \frac{N}{L} I
$$

This makes the solenoid one of the most important devices for generating controlled magnetic fields in physics and engineering.

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5.6.8. Solenoids

Magnetic field inside a solenoid

Structure and idea

Direction of the magnetic field

Ideal solenoid

Why the field is nearly uniform

Use of Ampère's law

Solenoids with a magnetic core

Comparison with a bar magnet

Practical examples

Field pattern

Important observations

Limits of the ideal model

Summary

Comments

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