We get different magnetic field inside and outside plates of capacitor by using ampere circuital law.
Hence some correction is to sought out in this law.
If area of plates of capacitor is A and charge is Q then electric field between plates of capacitor
`E=(sigma)/(in_(0))=(Q)/(A in_(0))`
where `sigma =` surface of charge density `=(Q)/(A)`
This field will be normal to closed loop considered and its magnitude will be equal on area A of plate of capacitor and outside it will be zero.
Flux linked with closed loop inside capacitor can be obtained by Gauss.s law,
`therefore Phi_(E )=vec(E ).vec(A)=EA cos theta`
but `theta = 0^(@)`
`therefore Phi_(E )=EA cos^(@)=EA`
`therefore Phi_(E )=(Q)/(A in_(0)).A=(Q)/(in_(0)) " "` .....(1)
If charge on capacitor changes with time then corresponding current `i=(dQ)/(dt)`
Differentiating equation (1) with respect to time,
`(d Phi_(E ))/(dt)=(1)/(in_(0)).(dQ)/(dt)=(i)/(in_(0))`
`therefore in_(0)(d Phi_(E ))/(dt)=i`
Thus by using ampere circuital law magnetic field inside and outside capacitor should be equal. For this in law total current i(t) should be replaced by `i_(c )+i_(d)` then magnetic field in both part will be equal.
`therefore i(t)=i_(c )+i_(d)=i_(c )+in_(0).(d Phi_(E ))/(dt)`
`I_(c )` is obtained due to flow of charge hence it is called conduction current.
`i_(d)` is obtained due to changing electric field hence it is called displacement current or Maxwell displacement current.
In any region product of permittivity `in_(0)` and rate of change of electric flux gives displacement current `i_(d)`.
SI unit of displacement current is ampere or coulomb/sec.
Thus, by writing `i(t)=i_(c )+i_(d)` in Ampere circuital law.
`oint vec(B).vec(d)l=mu_(0)i_(c )+mu_(0)i_(d)`
`=mu_(0)(i_(c )+i_(d))` where, `i(t)=i_(c )+i_(d)`
This means that in region outside only conduction current `i_(c )` is obtained `i_(c )=i(t)` and displacement current `i_(d)=0` and in inside capacitance `i_(c )=0` and current will be displacement current `i_(d)=i(t)`.
Thus missing term in Ampere circuital law is `i_(d)=in_(0)(d Phi_(E ))/(dt)`.
In broad sence law can be written as,
`oint vec(B).vec(d)l=mu_(0)i_(c )+mu_(0)in_(0)(d Phi_(E ))/(dt)`
By using this law magnetic field can be obtained inside as well as outside capacitor.
