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 $$( E = \frac{\sigma}{\varepsilon_0} ), where ( \sigma )$$ 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 $$( I = \frac{dQ}{dt} = A \frac{d\sigma}{dt} ),$$ 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 $$( E = \frac{I}{A\varepsilon_0} ).$$
• The rate of change of the electric field $$( \frac{dE}{dt} = \frac{1}{A\varepsilon_0} \frac{dI}{dt} ).$$
• For a constant charging current, $$( \frac{dI}{dt} = 0 ), and therefore ( \frac{dE}{dt} = 0 ).$$

Step 5: Calculate the Displacement Current

• The displacement current I_d is given by $$( I_d = \varepsilon_0 A \frac{dE}{dt} ).$$
• Substituting the values, we get $$( I_d = \varepsilon_0 A \times 0 = 0 ).$$