Q.Obtain an expression for the magnetic field on the axis of a current carrying very long solenoid using Ampere's circuital law. Draw the necessary diagram.

A long current-carrying solenoid produces a uniform axial field inside, so its field lines are parallel and straight.

Each turn of a solenoid behaves like a small circular loop. When many turns are stacked closely, their fields add along the axis and largely cancel outside. The result inside a long solenoid is a nearly uniform magnetic field of magnitude

$B = \mu_0 n I$ …

Interior field of a long solenoid B = μ₀nI, option (i).

Using Ampere's circuital law for a long solenoid, the magnetic field well inside (away from the ends) is uniform and directed along the axis:

B = μ₀ n I,

…

At a solenoid's end the field is half the interior value: B_end = μ₀nI/2, option (iii).

For a long solenoid, the interior field is B = μ₀nI. At the very end of the solenoid, only 'half' of the contributing turns lie on one side, and the field on the axis drops to exactly half …

Ampère's circuital law gives the axial field of a long ideal solenoid as B = μ₀nI, which contains no radius term, so B does not depend on r. Option (d).

Step 1 — apply Ampère's law to a rectangular loop, partly inside and partly outside a long ideal solenoid (a standard NCERT/CBSE Class 12 Physics derivation in Moving Charges and Magnetism).

…

The field inside a solenoid is uniform and parallel to its axis.

A solenoid is a long coil of many turns. The circular fields of the individual turns add up so that, well inside a long solenoid, the resultant magnetic field is nearly uniform, strong, and directed along the axis (its direction given by the right-hand rule). Outside …

A long, tightly-wound solenoid is the standard example of a uniform magnetic field region: inside it (away from the ends), B is constant in both magnitude and direction.

Compare the field patterns:

• A straight current-carrying wire produces a field that varies as 1/r with distance and circles the wire (non-uniform).
• At the centre of a circular loop the field has a single fixed value only at that one point, not over a region. …

A current-carrying solenoid behaves like a bar magnet, with one end acting as a north pole and the other as a south pole.

When a steady current flows through a solenoid (a long, closely wound helical coil), the magnetic field lines emerge from one end and enter the other, exactly as they do for a bar magnet, and the field pattern outside the solenoid is indistinguishable from that of an equivalent bar magnet of the same magnet …

Electromagnets give a strong magnetic field only while current flows, so they are used where a switchable magnet is needed — e.g. lifting heavy iron loads in cranes.

Concept. When current passes through a solenoid wound on a soft-iron core, the core becomes strongly magnetised, producing an electromagnet. The magnetism can be switched on/off and its strength controlled by the current.

…