Fig. 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 0.15A.
(a) Calculate 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 valid at each plate of the capacitor ? Explain.

Explanation

Radius of each circular plate, r=12cm =0.12m
Distance between the plates, d=5cm=0.05m
Changing current, I=0.15A
permittivity of free space, `epsilon_(0)=3.85xx10^(-12)C^(2)N^(-1)m^(-2)`
(a) Capacitance between the two plates is given by the relation
`=(epsilon_(0)A)/(d)`
`C`
where A=Area of each plate `=pir^(2)`
`C=(epsilon_(0)pir^(2))/(d)`
`=(8.85xx10pi^(12)xx0.12^(2))/(0.05)`
`=8.0032xx10^(-12)F=80.032pF`
Charge on each plate, q=CV
Where,
V=potential difference across the plates
Differentiation on both sides with respect to time (t) gives :
`(dq)/(dt)=C(dV)/(dt)`
But, `(Dq)/(dt)="current"(l)`
`therefore (dv)/(dt)=(1)/(C)`
`rArr(0.15)/(80.032xx10^(-12))=1.87xx10^(9)V//s`
Therefore, the change in potential difference between the plates is `1.87xx10^(9)V//s`.
(b) The displacement current across the plates is the same as the conduction current.
Hence, the displacement current `,i_(d)` is 0.15A.
(c) Yes
Kirchhoff's first rule is valid at each plate of the capacitor provided that we take the sum of conduction and displacement for current.