To establish an instantaneous displacement current of I ampere in the space between the plates of a parallel plate capacitor of `(1)/(2)` farad, the value of `(d V)/(dt)` is

Options

A. `2I`
B. `I/2`
C. `1/2I`
D. `I`

Answer

To find the value of \(\frac{dV}{dt}\) that establishes an instantaneous displacement current of \(I\) amperes in a parallel plate capacitor with a capacitance of \(\frac{1}{2}\) farad, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Displacement Current**:
The displacement current \(I_D\) in a capacitor is given by the formula:
\[ I_D = C \frac{dV}{dt} \]
where \(C\) is the capacitance and \(\frac{dV}{dt}\) is the rate of change of voltage across the capacitor.

2. **Identify Given Values**:
- The capacitance \(C = \frac{1}{2}\) farad.
- The instantaneous displacement current \(I = I\) amperes (where \(I\) is a variable representing the current).

3. **Set Up the Equation**:
Substitute the known values into the displacement current equation:
\[ I = C \frac{dV}{dt} \]
This becomes:
\[ I = \frac{1}{2} \frac{dV}{dt} \]

4. **Solve for \(\frac{dV}{dt}\)**:
Rearranging the equation to solve for \(\frac{dV}{dt}\):
\[ \frac{dV}{dt} = \frac{2I}{1} = 2I \]

5. **Conclusion**:
Thus, the value of \(\frac{dV}{dt}\) that establishes an instantaneous displacement current of \(I\) amperes in the capacitor is:
\[ \frac{dV}{dt} = 2I \text{ volts per second} \]