A parallel plate capacitor made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad `s^(-1)`.

Determine the amplitude of B at a point 3.0 cm from the axis between the plates.

Explanation

Here for given point P, its distance from the axis of plates is r = 3 cm. But radius of plate of a capacitor is R = 6 cm
`rArr (r )/(R )=(3)/(6)=0.5`

Here, since the current density is same everywhere in the region between the plates, we can write,
`(i_(d))/(pi r^(2))=(I_(rms))/(pi R^(2))`
`therefore i_(d)=I_(rms)((r )/(R ))^(2)`
Above is the displacement current enclosed by circular Amperean loop of radius r, passing through given point P. Because of this current, magnitude of magnetic field is same (equal to B) at all points on this loop. Hence, according to Ampere - Maxwell law,
`int vec(B).vec(d)l=mu_(0)(i_(c )+i_(d))`
`therefore int B dlcos 0^(@)=mu_(0)(0+i_(d))`
(`because vec(B)"||"vec(d)l` and between the plates `i_(c )=0`)
`therefore B int dl = mu_(0)i_(d) " "` (`because` B = constant)
`therefore B (2pi r)=mu_(0)i_(d)`
`therefore B=(mu_(0)i_(d))/(2pi r)`
Since above value is the rms value of induced magnetic field,
`B_(rms)=(mu_(0)i_(d))/(2pi r)`
`therefore B_(rms)=(mu_(0))/(2pi r)xx I_(rms)((r )/(R ))^(2)`
`therefore B_(rms)=(4pi xx10^(-7))/(2pi xx 0.03)xx6.9xx10^(-6)xx(0.5)^(2)`
`therefore B_(rms)=115xx10^(-13)T`
`therefore (B_(0))/(sqrt(2))=115xx10^(-13)T` (Where `B_(0)` is the amplitude of induced manetic field at point P)
`therefore B_(0)=sqrt(2)xx115xx10^(-13)T`
`=1.414xx115xx10^(-13)T`
`=162.6xx10^(-13)T`
`therefore B_(0)=1.626xx10^(-11)T`
Second method :
From equation (3) and (4),
`B=(mu_(0))/(2pi).(r )/(R^(2))I_(rms)`
`therefore B_(0)=(mu_(0))/(2pi).(r )/(R^(2))I_(0) " "` ....(1)
(Where `B_(0)` and `i_(0)` are the amplitudes of oscillating magnetic field and current)
`I_(0)=sqrt(2)I_(rms)`
`=1.414xx6.9xx10^(-6)`
`=9.7566xx10^(-6)A`
`~~9.76 mu A`
`therefore_(0)=(4pi xx10^(-7))/(2pi)xx(3xx10^(-2)xx9.76xx10^(-6))/((6xx10^(-2))^(2)) " "` [`because r=3xx10^(-2)m` and `R=6xx10^(-2)m`]
`=1.6266xx10^(-11)T`
`therefore B_(0)~~1.63xx10^(-11)T`