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## Magnetic fields of infinite solenoid and infinite current carrying plane

I am following the book 'Introduction to Electrodynamics' by Griffith. In that book,the magnetic field outside an infinite solenoid has been found as follows:

First of all,it is proved that the magnetic field is along the axis. (I have no problems with that proof).
Then,a rectangular loop is taken outside the loop,and since the current passing the loop is 0,it can be said that the magnetic fields along the two sides of the loop parallel to the axes are same.
By this,it can be inferred that magnetic field outside the solenoid is constant.

Now,the problem I have is with the next argument which says that since the magnetic field will decrease to 0 at very large distances from the solenoid,and since the magnetic field outside the solenoid is constant,that means the magnetic field everywhere outside the solenoid is 0.

Now why can't we make the same assumption in case of an infinite current carrying plane (that is,the magnetic field at infinite height from the plane will decrease to 0) ?
(This assumption is wrong because using this we get the wrong answer.)
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