Question

A parallel plate capacitor (fig.) made of circular plates each of radius $R=6.0cm$ has a capacitance $C=100\mu F$. The capacitor is connected to a $230V$ ac supply with a (angular) frequency of $300$ rad $s^{-1}$.
Is the conduction current equal to the displacement current?

Answer 1

Given: A parallel plate capacitor made of circular plates each of radius R=6.0 cm has a capacitance $C=100\mu F$. The capacitor is connected to a 230 V ac supply with a (angular) frequency of $300rad{s}^{-1}.$

To find if the conduction current is equal to displacement current

Solution:

Radius of plates, $R=6cm=6\times {10}^{-2}m$

Capacitance of capacitor, $C=100\mu F=100\times {10}^{-6}F={10}^{-4}F$

Voltage of capacitor, $V=230V$

Frequency of capacitance, $\omega =300rad/s$

Yes, the conduction current is equal to displacement current

By definition of the displacement current,

$I_d={\epsilon }_0\frac{d{\varphi }_E}{dt}⟹I_d={\epsilon }_0\frac{d}{dt}\left(EA\right)⟹I_d=\epsilon A\frac{dE}{dt}$

But $E=\frac{\sigma }{{\epsilon }_0}=\frac{Q}{{\epsilon }_{0}A}$

Substituting this in the abve equation, we get

$I_d={\epsilon }_{0}A\frac{d}{dt}\left(\frac{Q}{{\epsilon }_{0}A}\right)⟹I_d={\epsilon }_{0}A.\frac{1}{{\epsilon }_{0}A}.\frac{dQ}{dt}⟹I_d=\frac{dQ}{dt}=I$

Hence the conduction current is equal to displacement current