The magnetic field inside a tightly wound, long solenoid is B = μ 0 ni. It suggests that the field does not depend on the total length of the solenoid, and hence if we add more loops at the ends of a solenoid the field should not increase. Explain qualitatively why the extra-added loops do not have a considerable effect on the field inside the solenoid.

Question

Philoid · Accepted Answer

Magnetic field on the axis of a circular current carrying loop is given by,

where a denotes the radius of the loop and x is the distance of the point on the axis at which the magnetic field is to be calculated. i is the current flowing in the loop. μ₀ is the magnetic permeability of free space.

And B = μ₀ni (where μ₀ is the magnetic permeability of free space, n is the number of turns of the wire per unit length of the solenoid, and i is current in the wire) holds true for a very long solenoid where at the terminal points x >> a. We can say that if we keep adding more loops at the ends, the total number of loops increase but with that length also increases so the turns per unit length n is unvaried. We can also think of it this way,

The value of B for loops added to the ends of the solenoid will have dropped to almost zero as for the condition x >> a,

This can be shown mathematically by taking the limit of B for a circular loop. So, let us take x to be tending to infinity and a to have a finite value.

(Dividing both sides by x³)

So if we add more loops at the ends of a very long solenoid, the magnetic field due to each one of them will be zero at the centre of the solenoid thus won’t have considerable effect.