magnetic field?

why magnetic fields at the ends of the solenoid is half of that of at centre of solenoid?

Answers:

1The answers above are incorrect. The magnetic field has nothing to do with some unexplained "positive and negative cancellation."
The real reason is actually far simpler!

For this to work, imagine an infinitely long solenoid.
The field at any point along the central axis of the solenoid is the same because it is infinitely long.

Now, take a real solenoid. If it is long enough, (say a few cm just to picture it) we can approximate this real solenoid to an infinite solenoid.

So at the centre of this real solenoid the field is what it would be for the infinite solenoid - the solenoid "appears" to extend infinitely in both directions, as far as the magnetic field is concerned - there is a slight inaccuracy because it is not REALLY infinite, but this inaccuracy is so small it hardly matters.

Now think about the end of the solenoid. If you were standing at the end of the real solenoid, it would appear to extend to infinity in ONLY ONE DIRECTION. Hence, through pure reasoning and physical modelling we can deduce that the magnetic field at the end of a solenoid must be precisely half of the magnetic field at the centre of the solenoid.

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