A parallel plate capacitor made of circular plates each of radius `R= 6.0` cm has a capacitance `c = 100pF`. The capacitor is connected to a `230 V AC` supply with a ( angular) frequency of `300 rad//s`
(a) What is the rms value of the conduction current ?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of `B` at a point `3.0 cm` from the axis between the plates.
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Explanation

Radius of plate ,`r=6.0 cm = 6 xx 10 ^(-2) m` ,
capacitance `, C = 100 pF = 10 ^(-10) F,`
rms value of supply voltage `, E_(rms ) = 230 V `
Angular frequency ` omega = 300 rad s^(-1)` ,
Reactance `,X_(C ) = 1// omega C `
Area of the plate ` A = pi r^2 `
(a) Rms value of conduction current is given by
`I_(rms ) = (E_rms )/(X_C ) = (E_(rms ))/( 1 // omega C ) = E_(rms ) xx omega C `
` = 230 xx 300 xx 100 xx 0^(-12)`
`6.9 xx 10^(-6) A = 6.9 mu A `
(b) Displacement current is equal to conduction current as shown below :
` i_d = epsi_0 ( d theta phi )/( d t) = epsi_0 (E A) = epsi_0 A ( dE )/(dt )`
`E=(q) / ( epsi_0 A ) `
` therefore (dE )/( dt ) = (1 )/( epsi_0 A ) ( dq )/( dt ) = ( I )/(epsi_0 A ) `
`implies i_d = epsi_0 A ((I )/(epsi_0 A ))=I`
(c ) consider a loop of radius ` r. ` between the two circular plates .
THen area of the loop ` A. = pi r ^2 `
BY symmertry magnetic field induction ` vecB ` is equal in magnitude and is tangential to the circle at every point .
Here only displacement current will cross the loop for ` r lt R ` , using Ampere - Maxwell law , we get
` oint vecB . d vecl = mu _0 i d `
` or B xx 2 pi r. = mu _0 i_d `( through area A. )
`= mu _0 i _d ( pi r ^2)/( 2 pi r^2) [ :. i_d = I]`
since the capacitor is connnected to the alternating voltage , so the magnetic field within the capacitor plates is alternating
Let ` B=B_0 cos omega t`
` Therefore ` Amplitude of ` B=B_0 `
= maximum value of B
` = ( mu _0 I_0 r.)/( 2 pi r^2 )`
` [ I_0 ` is peak value of AC` ]`
` = ( 4 pi xx 10 ^(-7) xx sqrt( 2 ) xx I_( rms )xx r.)/( 2 pi r^2 )`
`= ( 2 sqrt( 2 ) I_(rms ) r. xx 10^(-7) )/( r^2)`
` = ( 2 xx 1.414 xx 6.9 xx 10^(-6 ) xx 3xx 10 ^(-2) xx 10^(-7))/( ( 6xx 10 ^(-2))^2)`
`= 1.626 xx 10 ^(-11) T ,`