A parallel plate capacitor of area $A = 16 {\mathrm{cm}}^{2 }$and separation between the plates 10 cm, is charged by a DC current. Consider a hypothetical plane surface of area $A_0=3.2 {\mathrm{cm}}^2$ inside the capacitor and parallel to the plates. At an instant, the current through the circuit is 6A. At the same instant the displacement current through $A_0$ is ________ mA.

The displacement current ($I_d$) inside a capacitor is given by:

$$I_d = I$$

where $I$ is the conduction current charging the capacitor.

However, if we consider a hypothetical plane of area $A_0$ inside the capacitor, the displacement current through that specific area is proportional to the fraction of the total capacitor plate area it occupies:

$$I_{d0} = I \times \frac{A_0}{A}$$

Given values:

• Total plate area: $A = 16$ cm² = $16 \times 10^{-4}$ m²
• Hypothetical plane area: $A_0 = 3.2$ cm² = $3.2 \times 10^{-4}$ m²
• Conduction current: $I = 6$ A

Step 1: Calculate the displacement current through $A_0$

$$I_{d0} = 6 \times \frac{3.2 \times 10^{-4}}{16 \times 10^{-4}}$$

 $$I_{d0} = 6 \times \frac{3.2}{16}$$ $$I_{d0} = 6 \times 0.2$$ $$I_{d0} = 1.2 \text{ A}$$

Step 2: Convert to mA

$$I_{d0} = 1.2 \times 10^3 \text{ mA} = 1200 \text{ mA}$$