A parallel plate capacitor has circular plates each of radius 5.0 cm. It is being charged so that electric field in the gap between its plates rises steadily at the rate of `2 xx 10^(12) Vm^(-1)s^(-1)`What is the displacement current.

Answer

To find the displacement current in a parallel plate capacitor with circular plates, we can follow these steps:

### Step 1: Understand the Concept of Displacement Current
Displacement current is defined as the rate of change of electric field in the capacitor multiplied by the permittivity of free space (ε₀) and the area of the plates. It is given by the formula:

\[ I_d = \epsilon_0 \frac{dE}{dt} A \]

where:
- \(I_d\) is the displacement current,
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \, \text{F/m}\)),
- \(\frac{dE}{dt}\) is the rate of change of electric field,
- \(A\) is the area of the plates.

### Step 2: Calculate the Area of the Plates
The area \(A\) of a circular plate is given by the formula:

\[ A = \pi r^2 \]

Given the radius \(r = 5.0 \, \text{cm} = 0.05 \, \text{m}\):

\[ A = \pi (0.05)^2 = \pi (0.0025) \approx 0.00785 \, \text{m}^2 \]

### Step 3: Substitute the Values into the Displacement Current Formula
We are given that the electric field is changing at a rate of:

\[ \frac{dE}{dt} = 2 \times 10^{12} \, \text{V/m/s} \]

Now, substituting the values into the displacement current formula:

\[ I_d = \epsilon_0 \frac{dE}{dt} A \]

Substituting the known values:

\[ I_d = (8.85 \times 10^{-12} \, \text{F/m}) \times (2 \times 10^{12} \, \text{V/m/s}) \times (0.00785 \, \text{m}^2) \]

### Step 4: Calculate the Displacement Current
Now, we can calculate \(I_d\):

\[ I_d = (8.85 \times 10^{-12}) \times (2 \times 10^{12}) \times (0.00785) \]

Calculating the product:

1. Calculate \(8.85 \times 2 = 17.7\)
2. Then, \(17.7 \times 10^{0} = 17.7\)
3. Finally, \(17.7 \times 0.00785 \approx 0.139845\)

Thus,

\[ I_d \approx 0.14 \, \text{A} \]

### Final Answer
The displacement current is approximately:

\[ I_d \approx 0.14 \, \text{A} \]

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