(a)Using Ampere's circuital law, obtain the expression for the magnetic field due to a long solenoid at a point inside the solenoid on its axis.
(b )How is the magnetic field inside a given solenoid made strong?

Explanation

(a)Magnetic field due to solenoid:Consider a rectangular amperian loop `abcd` near the middle of solenoid as shown in fig.where `PQ=I`
Let the magnetic field along the path ab be `B` and in zero along `cd`.As the paths `bc` and `da` are perpendicular to the axis of solenoid, the magnetic field component along these path is zero.Therefore, the path `bc` and `da` will not contribute to the line integral of magnetic field `B`.
Total number of turns in length `l=Nl`
The line integral of magnetic field induction `B` over the closed path `abcd` is:
`underset(abcd)(ointvecB).vec(dl)=underset(a)overset(b)intvecB.vec(dl)+underset(b)overset(c)intvecB.vec(dl)+underset(c)overset(d)intvecB.vec(dl)+underset(d)overset(a)intvecB.vec(dl)`
`because underset(a)overset(b)intvecB.vec(dl)=underset(a)overset(b)intBdl cos 0^(@)=Bl`
and `underset(b)overset(c)intvecB.vec(dl)=underset(b)overset(c) int B dl cos 90^(@)=0=underset(d)overset(a)intvecB.vec(dl)`
Also `underset(c)overset(d)vecB.vec(dl)=0`
`(because ` Outside the solenoid `B=0`)
`therefore underset(abcd)(ointvecB).vec(dl)=vec(dl)=Bl+0+0=Bl` ..(i)
Using Ampere's circuital law
`underset(abcd)(ointvecB).vec(dl)=mu_(0)xx"total current in rectangle abcd"`
`mu_(0)xx"no . of turns in rectangle" xx "current" `
`=mu_(0)xxNIxxI=mu_(0) Nl`...(2)
From (1) and (2), we have
`Bl=mu_(0)N//l`
`therefore B=mu_(0)NI`
(b)The magnetic lines of force from a finite solenoid are shown:
The magnetic field due to the neighbouring turns add up along the axis of the solenoid and tend to cancel out of the perpendicular direction.
Thus the field at the exterior midpoint is weak and at the interior, it is uniform and strong.When current is passed in toroid, `M.F.` is produced inside the core.
Line integral of magnetic field around the circular path of radius `r`.
`ointvecB.vec(dl)=mu_(0)NI`
`ointB.dlcos 0^(@)=mu_(0)NI`
`B2pir=mu_(0)NI`
`B=(mu_(0)NI)/(2pir)`
`B=mu_(0)nl`
( c) The magnetic field inside a given solenoid made stronger because it becomes everywhere parallel to the axis.