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 \(\frac{dV}{dt}\) required to establish an instantaneous displacement current of \(I\) amperes in the space between the plates of a parallel plate capacitor with a capacitance of \(\frac{1}{2}\) farad, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between charge, capacitance, and voltage**:
The charge \(Q\) on a capacitor is given by the formula:
\[ Q = C \cdot V \]
where \(C\) is the capacitance and \(V\) is the voltage across the capacitor.

2. **Differentiate the charge with respect to time**:
To find the displacement current, we differentiate both sides with respect to time \(t\):
\[ \frac{dQ}{dt} = C \cdot \frac{dV}{dt} \]
Here, \(\frac{dQ}{dt}\) represents the current flowing through the capacitor, which is the displacement current \(I_d\).

3. **Set the displacement current equal to the given current**:
We know that the displacement current \(I_d\) is equal to \(I\) (the instantaneous current given in the problem):
\[ I = C \cdot \frac{dV}{dt} \]

4. **Substitute the value of capacitance**:
Given that the capacitance \(C\) is \(\frac{1}{2}\) farad, we can substitute this into the equation:
\[ I = \left(\frac{1}{2}\right) \cdot \frac{dV}{dt} \]

5. **Rearrange the equation to solve for \(\frac{dV}{dt}\)**:
To isolate \(\frac{dV}{dt}\), we rearrange the equation:
\[ \frac{dV}{dt} = \frac{2I}{1} = 2I \]

### Final Answer:
Thus, the value of \(\frac{dV}{dt}\) is:
\[ \frac{dV}{dt} = 2I \]