Figure shows a capacitor made of two circular plates each of radius `12 cm`, and separated by `5.0 cm`. the capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to `0.15 A`
(a) Calcuate the capacitance and the rate of change of potential difference between the plates
(b) Obtain the displacement current across the plates
(c) Is kirchhoff's first rule(junction rule) vaild at each plate of the capacitor ? Explain.
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Explanation

Radius of plates `r=0.12 m,`
distance between the plates ` d= 5.0 cm`
`= 5 xx 10^(-2)`m,
Charging current `,I = 0.15 A `
(a) Capacitance of the capacitor ,
` C= ( epis_0 A )/( d ) = ( 8.85 xx 10^(-12) xx pi r^2 )/( 5 xx 10^(-2))`
or ` C= ( 8.85 xx 10^(-12) xx 3.14 xx (0.12 )^2)/( 5 xx 10^(-2))`
`= 8xx 10^(-12) F = 8.0 pF `
since ` I= (dq )/(dt ) = (d)/( dt ) (CV ) = C = ( dV )/( dt )`
` ( dV )/(dt ) = (I )/(C ) = ( 0.15 )/( 8.0 xx 10^(-12))`
`= 1.87 xx 10^(10) Vs^(-1)`
(b) Displacement current is given by
`i_d = epsi_0 ( d phi )/( dt ) = epsi_0 (d )/( dt ) (EA ) = epsi_0 A (dE )/(dt )`
`E= (q )/(epsi_0 A )`
` therefore (dE )/( dt ) = (1 )/(epsi_0 A ) ( dq)/(dt ) = (I )/(epsi_0A )`
`implies i_d = epsi_0 A ((I )/(epis_0A )) =I`
The displacement current is equal to the conduction current and is equal to 0.15 A .
(c ) Yes , Kirchhoff.s first law holds good at each plate if we take both conduction and displacement current the conduction current arriving at a plate is equal to the displacement current leaving that plate .