Question

Can an ideal solenoid, one with no magnetic field outside the solenoid, exist? If not, does that invalidate the derivation of the magnetic field inside the solenoid (Section 28.4 )?

Short answer

Answer: A truly ideal solenoid with no magnetic field outside the solenoid cannot exist. However, this fact does not invalidate the derivation of the magnetic field inside the solenoid for practical solenoids, where the length is much greater than the diameter, and the winding is tight. The assumption of an ideal solenoid with zero magnetic field outside simplifies the derivation, but the resulting formula for the magnetic field inside the solenoid remains valid for a wide range of practical solenoids.

Step-by-step solution

Understanding an ideal solenoid

An ideal solenoid has a uniform magnetic field inside and zero magnetic field outside. The magnetic field inside the solenoid is given by B = μ₀ * n * I , where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current passing through the solenoid. The condition of an ideal solenoid is only an approximation for practical solenoids in which the length is much greater than the diameter, and the winding is tight.

Magnetic field outside an ideal solenoid

According to Ampere's law, the closed line integral of the magnetic field around any closed loop is equal to μ₀ times the total current passing through the loop. For an ideal solenoid, the magnetic field outside the solenoid should be zero. However, this condition cannot hold for a finite solenoid because the magnetic field lines must return to their starting point, creating an external magnetic field. The approximation of an ideal solenoid with zero magnetic field outside is only valid when the solenoid's length is much larger than its diameter, and the observer is far away from the solenoid's ends. In this case, the external magnetic field becomes negligible compared to the magnetic field inside the solenoid.

The effect on the derivation of the magnetic field inside the solenoid

The derivation of the magnetic field inside the solenoid is based on Ampere's law and the symmetry of a long solenoid. The assumption of an ideal solenoid with zero magnetic field outside simplifies the derivation but does not entirely invalidate it as the external magnetic field is negligibly small in practical solenoids. If we accept the small error introduced by the assumption, the derivation for the magnetic field inside the solenoid remains valid for a wide range of practical solenoids. In conclusion, a truly ideal solenoid with no magnetic field outside the solenoid cannot exist. However, this fact does not invalidate the derivation of the magnetic field inside the solenoid for practical solenoids, where the length is much greater than the diameter, and the winding is tight. The assumption of an ideal solenoid with zero magnetic field outside simplifies the derivation, but the resulting formula for the magnetic field inside the solenoid remains valid for a wide range of practical solenoids.