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A current carrying solenoid is shown bel...

A current carrying solenoid is shown below. Show that the magnetic field intensity at point P be
`B=(mu_(0))/(4pi) (2m)/(r^(3))`
where, the symbols have their usual meaning.

Text Solution

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For circular element of thickness dx, at a distance x from the centre.
The magnitude of the field,
`dB=(mu_(0) Ia^(2) dxn)/(2((r-x)^(2)+a^(2))^(3//2))`
The magnitude of the total field is
`B=(mu_(0) nIa^(2))/(2) int_(-1)^(+1) (dx)/([(r-x)^(2)+a^(2)]^(3//2))`
Let for axial field `r gt gt a and r gt gt x`
So, `B=(mu_(0)NIa^(2))/(2) int_(-1)^(1) (dx)/(r^(3))`
`rArr B=(mu_(0) nIa^(2))/(2r^(3))2I rArr B=(mu_(0))/(4pi) (2m)/(r^(3))`
where, `m=2pinlIa^(2)`
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