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A capacitor, made of two parallel plates...

A capacitor, made of two parallel plates each of plate area A and separation d, is being charged by an external ac socurce. Show that the displacement current inside the same as the current charging the capacitor.

Text Solution

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Let the alternating emf charging the plates of the capacitor `epsilon=epsilon_(0) sin omegat.`
Charge on capacitor
`q=C_(epsilon)=C_(epsilon_(0))sin omegat`
Let Instantaneous current be I
`So," "I=(dq)/(dt)=(d)/(dt)C_(epsilon_(0)) sin omega t`
`=C_(epsilon_(0))omega cos t`
`=I_(0) cos omega t`
`"where, "I_(0)=C_(epsilon_(0))omega`
`I_(d)=in_(0)(dphi_(E))/(dt){phi_(E)=EA.`
`I_(d)=in_(0) A(DE)/(dt){E=(sigma)/(in_(0))=(q)/(A_(in_(0)))`
`I_(d)=in_(0)A(d)/(dt)((q)/(A_(in_(0))))`
`I_(d)=(d)/(dt)C_(epsilon_(0))sin omega t = C_(epsilon_(0))omega cos omegat= I_(0) cos omega t`
Hence, conduction current is equal to the displacement current.
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