Magnetic Field of Solenoid
A solenoid is a long coil of wire wound into a series of closely spaced loops, or turns. When an electric current flows through the wire, each turn behaves like a small bar magnet, and together they produce a combined magnetic field. The remarkable result is that inside a long solenoid the magnetic field is very nearly uniform — it points in a single direction and has the same strength everywhere along the central axis. This makes the solenoid one of the most useful devices in electromagnetism, appearing in everything from MRI scanners to electric doorbells. Outside a long solenoid, the magnetic field is surprisingly weak — ideally zero for an infinitely long coil — because the field lines returning along the outside are spread over a vast area. This contrast between a strong, uniform interior field and a negligible exterior field is what sets the solenoid apart from a single current loop or a straight wire. When the wire is wound tightly and the length is much greater than the diameter, the solenoid approximates the ideal case very closely, and the formula B = μ₀nI gives an excellent prediction of the interior field strength. Adding a ferromagnetic core — typically soft iron — dramatically amplifies the field because the core's magnetic domains align with the applied field, increasing the net flux. This is the principle behind an electromagnet: a solenoid whose magnetism can be switched on and off with the current. Electromagnets are used in cranes, loudspeakers, particle accelerators, and medical imaging, making the solenoid one of the most practically important structures in modern technology.
- Why individual loops add up
- Applying Ampère's Law
- The number of turns per metre, n
- Effect of a ferromagnetic core
- End effects and ideal limits
Why individual loops add up
Each circular loop of the solenoid carries current and produces a magnetic field along its axis, just like a tiny bar magnet. When loops are stacked side by side, the fields from neighbouring turns reinforce each other inside the coil. Crucially, the sideways components from adjacent turns cancel, leaving only an axial component. Think of many weak magnets all pointing the same way — the collective effect is much stronger than any single one.
Applying Ampère's Law
The cleanest derivation uses Ampère's circuital law: ∮ B · dl = μ₀ I_enc. Choose a rectangular Amperian loop with one side of length L running inside the solenoid (parallel to the axis) and the opposite side outside, where B ≈ 0. The two short sides are perpendicular to B and contribute nothing. If the coil has n turns per metre, the enclosed current is I_enc = nLI. The equation then gives B × L = μ₀ × nLI, so B = μ₀nI.
The number of turns per metre, n
The parameter n is the turn density — the total number of turns N divided by the length ℓ of the solenoid: n = N/ℓ. A solenoid with 500 turns wound over 0.25 m has n = 2000 turns per metre. Doubling n (by winding more tightly or using more wire) doubles B. This is why high-field laboratory electromagnets use densely wound, water-cooled coils.
Effect of a ferromagnetic core
Inserting a soft-iron core replaces μ₀ with μ = μ_r μ₀, where μ_r is the relative permeability of the material (often 1000–5000 for soft iron). The field becomes B = μ_r μ₀ n I — potentially thousands of times stronger. Unlike a permanent magnet, this field disappears when the current is switched off, which is exactly the behaviour needed for electromagnets used in recycling yards or MRI quench systems.
End effects and ideal limits
The formula B = μ₀nI is exact only for an infinitely long solenoid. Real solenoids experience 'end effects': field lines diverge and the field weakens towards the ends, falling to roughly half the central value at the opening. Engineers design solenoids to be long compared to their diameter (ℓ/d > 10 is a common rule of thumb) to minimise this. Helmholtz coil pairs are used when even greater uniformity is required over a working volume.
The Key Equation
- B — Magnetic flux density (Tesla (T)): The strength and direction of the magnetic field inside the solenoid.
- \mu_0 — Permeability of free space (T·m·A⁻¹): A fundamental constant equal to 4π × 10⁻⁷ T m A⁻¹, describing how easily a magnetic field forms in a vacuum.
- n — Turn density (turns per metre (m⁻¹)): The number of wire turns per unit length of the solenoid; equal to N/ℓ where N is total turns and ℓ is solenoid length.
- I — Current (Ampere (A)): The electric current flowing through the solenoid wire.
Worked Example
Problem. A solenoid is 40 cm long and has 800 turns of wire. A current of 2.5 A passes through it. (a) Calculate the magnetic flux density inside the solenoid. (b) If a soft-iron core with relative permeability μ_r = 1200 is inserted, find the new flux density.
Answer. Air core: approximately 6.3 mT; with iron core: approximately 7.5 T.
Origins · 1 minAndré-Marie Ampère and how this was discovered
Ampère formulated Ampère's circuital law, the mathematical foundation used to derive the solenoid field formula B = μ₀nI. After hearing of Ørsted's discovery that currents deflect compass needles in 1820, Ampère developed a full mathematical theory of electromagnetism within weeks, showing how current-carrying conductors exert forces on each other and establishing the link between electricity and magnetism. He coined the term 'solenoid' from the Greek word for a pipe or channel. Ampère reportedly derived his laws of electrodynamics so rapidly after Ørsted's announcement that colleagues joked he had known the answer before the experiment was performed. The SI unit of electric current, the ampere, is named in his honour.
Related Concepts
- Magnetic Field of a Straight Wire — The solenoid's field is derived from superimposing fields of individual current loops, each governed by the same principles as a current-carrying wire, making this a foundational prerequisite.
- Magnetic Field of a Circular Loop — Each turn of a solenoid is a circular current loop; understanding a single loop's axial field builds directly to the solenoid result.
- Magnetic Flux Density — B is the key quantity calculated for a solenoid; a solid grasp of what magnetic flux density means is essential for interpreting the formula B = μ₀nI.
- Electromagnets — The electromagnet is the primary application of the solenoid, where a ferromagnetic core amplifies the field and the magnetism is controlled by switching the current.