A parallel plate condenser has conducting plates of radius 12 cm separated by a distance of 5mm. It is charged with a constant charging current of 0.16 A, the rate at which the potential difference between the plate change is

Options

A. `1 xx 10^(9)V s^(-1)`
B. `2 xx 10^(10) Vs^(-1)`
C. `3 xx 10^(12) Vs^(-1)`
D. `2 xx 10^(9) V s^(-1)`

Explanation

To find the rate at which the potential difference (V) between the plates of a parallel plate condenser changes when charged with a constant current, we can follow these steps:

### Step 1: Identify the given values
- Radius of the plates (R) = 12 cm = 0.12 m
- Distance between the plates (d) = 5 mm = 5 x 10^-3 m
- Charging current (I) = 0.16 A

### Step 2: Calculate the capacitance (C) of the parallel plate condenser
The formula for the capacitance of a parallel plate capacitor is given by:
\[ C = \frac{\varepsilon_0 \cdot A}{d} \]
where:
- \( \varepsilon_0 \) (the permittivity of free space) = \( 8.85 \times 10^{-12} \, \text{F/m} \)
- \( A \) (the area of the plates) = \( \pi R^2 \)

Calculating the area:
\[ A = \pi (0.12)^2 = \pi \cdot 0.0144 \approx 0.04524 \, \text{m}^2 \]

Now substituting the values into the capacitance formula:
\[ C = \frac{(8.85 \times 10^{-12}) \cdot (0.04524)}{5 \times 10^{-3}} \]
\[ C \approx \frac{3.999 \times 10^{-13}}{5 \times 10^{-3}} \]
\[ C \approx 8 \times 10^{-11} \, \text{F} \]

### Step 3: Relate the current to the rate of change of potential difference
The relationship between the current (I), capacitance (C), and the rate of change of potential difference (dV/dt) is given by:
\[ I = C \frac{dV}{dt} \]

Rearranging this gives:
\[ \frac{dV}{dt} = \frac{I}{C} \]

### Step 4: Substitute the known values into the equation
Now substituting the known values:
\[ \frac{dV}{dt} = \frac{0.16}{8 \times 10^{-11}} \]

### Step 5: Calculate the rate of change of potential difference
Calculating the above expression:
\[ \frac{dV}{dt} = \frac{0.16}{8 \times 10^{-11}} = 2 \times 10^{9} \, \text{V/s} \]

### Final Answer
The rate at which the potential difference between the plates changes is:
\[ \frac{dV}{dt} = 2 \times 10^{9} \, \text{V/s} \]

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