The solenoid

What is a solenoid?

In plain words: one loop of wire makes a field that is strong at its centre and spreads out. Stack hundreds of loops side by side and their fields line up inside the coil into one strong, even field running along it, while outside they nearly cancel.

Field inside a long solenoid

Unless stated otherwise, these formulas describe an air/vacuum core and the ideal continuous winding. For a finite solenoid, B ≈ μ_₀nI applies well inside, away from the ends; a real discrete winding can have local variations close to the wire.

The picture (NCERT pages 121–122, Figures 4.15–4.16):

• A long solenoid is one whose length is large compared with its radius. It is wound with enamelled wire, so that the turns are insulated from each other (page 121).
• Between two neighbouring turns the local contributions from neighbouring wires oppose and cancel at symmetric points in the schematic (NCERT Figure 4.15(a)).
• For a sufficiently long finite solenoid, the field in the central interior region around P is approximately uniform, strong and along the axis. At the exterior mid-point Q it is weak and parallel to the axis, with no normal component; in NCERT Figure 4.15(b) it points the opposite way to the field inside (page 121).
• As the solenoid is made longer it looks like a long cylindrical metal sheet. The field outside approaches zero and is taken as zero, and the field inside becomes everywhere parallel to the axis (NCERT page 122, Figure 4.16).

Applying Ampere’s law to a rectangular amperian loop abcd (NCERT page 122):

• The side ab, of length h, runs inside the solenoid parallel to the axis, at any distance from it, where the field is B. The side cd is outside, where the field is zero.
• Along the sides bc and ad, which cross the winding at right angles to the axis, the field has no component along the path, so they add nothing. So the length that counts is L = h.
• With n turns per unit length, the loop encloses nh turns, each carrying the current I, so I_e = I(nh). Then Ampere’s law in the form BL = μ₀I_e gives Bh = μ₀I(nh), so (NCERT Eq. (4.16), page 122):

• Signs: go round a → b → c → d, anticlockwise in the figure below. With the right-hand fingers curled that way the thumb points out of the page, so the currents in the top row, which come out of the page, count as positive, and B along ab points along the path. Both sides of Ampere’s law are positive (NCERT Figure 4.16, page 121).
• Direction: by the right-hand rule. Curl the fingers along the current in the turns; the thumb points along the field inside (NCERT page 122).
• In the ideal infinite-solenoid model, B is the same everywhere inside: it does not depend on the distance from the axis or on the radius of the solenoid, only on n and I (reasoned from NCERT Eq. (4.16)).
• If the winding has several layers, n counts all the turns per unit length (reasoned; see Example 2).
• Inserting a soft iron core inside makes a much larger field possible; NCERT takes this up in the next chapter (page 122).
• At an end of a long solenoid the field on the axis is about half the inside value, μ₀nI/2 (beyond NCERT, reasoned by superposition: two such halves joined end to end make one long solenoid, and each half gives the same field at the join).
• NCERT slip: Example 4.8 says “we can use the long solenoid formula, namely, Eq. (4.20)” (page 122). The long-solenoid formula is Eq. (4.16); Eq. (4.20), on page 125, is the torque on a current loop.

Solved examples

Common mistakes to avoid

Quick revision summary

• A long solenoid: closely spaced turns, length much larger than radius; the fields of neighbouring turns cancel between them.
• Inside: uniform, strong and along the axis; outside: nearly zero.
• Ampere’s law on a rectangle with one side inside: Bh = μ_₀I(nh), so B = μ_₀nI, with n the turns per unit length.
• Direction by the right-hand rule: fingers along the current, thumb along the field.
• At an end, about μ_₀nI/2 (beyond NCERT); a soft iron core increases the field.

Questions students ask

Why is the magnetic field outside a long solenoid nearly zero?

The field outside a long solenoid is nearly zero because the field lines running along the inside must return through the unlimited space outside, where they spread ever more thinly as the solenoid is made longer (beyond the NCERT text, reasoned from NCERT page 122). Ampere’s law agrees: a rectangle wholly outside encloses no current, so the axial field is the same at every distance outside, and as it must die away far off, it is zero outside an ideal solenoid. Strictly, a helical winding also carries I once along its length, adding, before the return lead’s contribution is included, a weak field that circles the solenoid like a long wire’s, μ₀I/(2πr), far weaker than μ₀nI inside.

What is an ideal solenoid, and how long must a real one be to count as long?

An ideal solenoid is infinitely long and so closely wound that it acts as a continuous cylindrical sheet of current, NCERT’s idealised picture (NCERT page 122, with Figure 4.16 on page 121): its field is exactly μ₀nI everywhere inside and zero outside. A real solenoid comes close to this only well inside it, away from its ends. Beyond the NCERT text, adding the fields of the turns gives the field at the centre of a solenoid of length L and radius R as μ₀nI × L/√(L^² + 4R^²), so a length of 10 radii already gives 98% of the ideal value, but a length of 4 radii only 89%.

How can the magnetic field inside a solenoid be made stronger?

The field inside a long solenoid, B = μ₀nI, grows with the turns per metre and with the current, which the heating of the wire limits (NCERT Eq. (4.16), page 122). The biggest gain comes from a soft iron core, which NCERT notes makes a large field possible (page 122): the magnetised iron adds a far larger field of its own, because soft iron can have a large relative permeability; its value depends on field and magnetic history and decreases as saturation is approached (NCERT Chapter 5, page 149). A solenoid with a soft iron core is an electromagnet (NCERT Class 10 Science, Chapter 12, page 201).

What is the difference between a solenoid and a toroid?

A solenoid is a straight coil with two open ends, while a toroid is a solenoid bent round into a closed ring, so its field lines run round inside the winding: “Magnetic lines are completely confined within a toroid” (NCERT Chapter 5, Example 5.3(c), page 144). Beyond the NCERT text, Ampere’s law round a circle of radius r inside the ring gives B = μ₀NI/(2πr) for N turns, close to μ₀nI for a thin ring, and zero outside in the ideal continuous toroidal winding; real windings can have leakage fields. The current NCERT book has no toroid section, naming it only in Examples 5.1(c) and 5.3(c), and neither the JEE Main nor the NEET (UG) 2026 syllabus lists it.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A solenoid 40 cm long has 800 turns and carries 2.0 A. Find the field inside.

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   n = 800/0.40 = 2000 per metre, so B = 4π × 10⁻⁷ × 2000 × 2.0 ≈ 5.0 × 10⁻³ T (NCERT Eq. (4.16), page 122).

2. Q2. The number of turns per unit length of a solenoid is doubled and the current halved. What happens to the field inside?

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   It stays the same, since B = μ_₀nI and the product nI is unchanged (NCERT Eq. (4.16), page 122).

3. Q3. Why do the sides bc and ad of the amperian rectangle add nothing to the line integral?

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   They run at right angles to the axis, and the field inside is along the axis and zero outside, so B has no component along them (NCERT page 122).