A parallel plate capacitor with circular plates of radius 1m has a c apacitance of 1n F. At `t=0,`it is connected for charging in series with a resistance `R=1MOmega` across2V battery . Calculate the magnetic field at a point P, in between the plates and half way between the centre and the periphery of the plates after `10^(-3)s.`

Explanation

Here, `r=1m, C=0.5muF =0.5xx10^-6F,`
`R=2kOmega=2xx10^3Omega,V=4V,t=10^-3s`.
The charge Q on the capacitor at any time t during
charging is
`Q=Q_0(1-e^(-t//RC))=CV(1-e^(-t//RC))`
`:. (dQ)/(dt)=CVxx1/(RC) e^(-t//RC)=V/R e^(-t//tau) ( :' tau=RC)`
Electric field in between the plates at time t is
`E=Q/(in_0A) =Q/(in_0pir^2)`
`(dE)/(dt)=1/(in_0pir^2) (dQ)/(dt) =1/(in_0pir^2)xxV/R e^(-t//tau)`
Let us now consider a circular loop of radius
`r'=(=1/2m)`, parallel to the plates passing
through P. The magnetic field B at all points on
this loop is of the smae value and acting
tengentially to the loop.
Applying Ampere Maxwell's Law to this loop,
we get
`oint vecB.vec(dl)=mu_0[I+in_0 (dphi_E)/(dt)]`
or `B2pir'=mu_0(0+in_0pir' (dE)/(dt))`
or `B=mu_0(in_0r')/2 (dE)/(dt) =(mu_0in_0r')/2xx1/(in_0pir^2) V/R e^(-t/tau)`
`(mu_0r')/(2pir^2)xxV/R e^(-t/tau)`
`=((4pixx10^-7)xx(1//2))/(2pi(1)^2)xx4/((2xx10^3))e^-1`
`( :' t=10^-3s , tau=10^-3s)`
`=(2xx10^-10)/2.718=7.358xx10^-11T`