If a variable frequency ac source is connected to a capacitor then with decrease in frequency the displacement current will

Options

A. increase
B. decrease
C. remains constant
D. first decrease then increase

Explanation

To solve the question regarding the behavior of displacement current when a variable frequency AC source is connected to a capacitor, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Displacement Current:**
Displacement current is a term used to describe the current that flows in a capacitor when it is charged or discharged. It is defined by the changing electric field in the capacitor, which is proportional to the rate of change of voltage across the capacitor.

2. **Identify the Relationship Between Current and Frequency:**
The current (I) flowing through a capacitor in an AC circuit is related to the capacitive reactance (Xc). The capacitive reactance is given by the formula:
\[ X_c = \frac{1}{\omega C} \]
where:
- \( \omega \) is the angular frequency (in radians per second),
- \( C \) is the capacitance.

3. **Express Angular Frequency in Terms of Frequency:**
The angular frequency \( \omega \) is related to the frequency (f) by the equation:
\[ \omega = 2\pi f \]
Substituting this into the expression for capacitive reactance gives:
\[ X_c = \frac{1}{2\pi f C} \]

4. **Determine the Relationship Between Current and Frequency:**
The current through the capacitor can be expressed as:
\[ I \propto \frac{1}{X_c} \propto 2\pi f C \]
This indicates that the current is directly proportional to the frequency (f).

5. **Analyze the Effect of Decreasing Frequency:**
Since the current (I) is directly proportional to the frequency (f), if the frequency decreases, the current will also decrease. Therefore, the displacement current will decrease as the frequency of the AC source decreases.

6. **Conclusion:**
Thus, with a decrease in frequency, the displacement current will decrease.

### Final Answer:
The displacement current will **decrease** with a decrease in frequency.

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