Related Problems with Solution

Problem Statement : A parallel plate capacitor with plate area ( A ) and plate separation ( d ) is being charged by a current ( I ). Find the magnitude of the displacement current through a surface between the plates of the capacitor.

Given Data:

  • Plate area of the capacitor, ( A )
  • Separation between the plates, ( d )
  • Charging current, ( I )

Solution:

Step 1: Understand the Concept of Displacement Current

  • Displacement current arises in regions where the electric field is changing with time, such as in the space between the plates of a charging capacitor.

Step 2: Calculate the Electric Field in the Capacitor

  • The electric field ( E ) between the plates of a capacitor is given by (E=σε0),where(σ) is the surface charge density on the plates.

Step 3: Relate Surface Charge Density with Current

  • As the capacitor is charging, the surface charge density σ is changing with time. This change is related to the current ( I ) by (I=dQdt=Adσdt), where ( Q ) is the charge on the capacitor plates.

Step 4: Calculate the Rate of Change of Electric Field

  • Substitute σ from Step 3 into the equation from Step 2 to get (E=IAε0).
  • The rate of change of the electric field (dEdt=1Aε0dIdt).
  • For a constant charging current, (dIdt=0),andtherefore(dEdt=0).

Step 5: Calculate the Displacement Current

  • The displacement current Id is given by (Id=ε0AdEdt).
  • Substituting the values, we get (Id=ε0A×0=0).