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Assuming that the length the of the sole...

Assuming that the length the of the solenoid is large when compared to its diameter, find the edquation for its inductance.

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Let us consider a long solenoid of length l and cross-sectional area A. Let n be the number of turns per unit length (or turn density ) of the solenoid. When an electric current I is passed through the solenoid, a magnetic field is produced by it which is almost uniform and is directed along the axis of the solenoid figure.

The magnetic field at any poin inside the solenoid is given by
`B= mu_(0)ni`
As this magnetic field passes through the solenoid, the winding of the solenoid are linked by the field lines. The magnettic flux passing through each truns is
`phi_(B)=int_(A)vecB.dvecA=BAcostheta`
`=BA " " sincetheta=0^(0)`
`= (mu_(0)ni)A`
The total magnetic flux linked or flux linkage of the solenoid with N truns (the total numer of turns N is given by N = nl) is
`Nphi_(B)=(nl)(mu_0ni)A`
`Nphi_(B)=(mu_(0)n^(2)Al)i" "...(i) `
`Nphi_(B)=Li" " ...(2)`
Comparing equations (1) and (2)
be have `L=mu_(0)n^(2) Al` From the above equation, it is clear that inductance depends on the geometry of the solenoid ( truns density n, cross-sectional area A, lenght l ) and the medium present inside the solenoid. If the solenoid is filled with a dielectric medium of relative permeability `mu_(r)` then
`L=mu_(0)n^(2) Al`
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