Which one of the following graphs below represents variation of reactance `'X_(c)'` of a capacitor with frequency 'f' of an a.c. supply?

Options

A.
B.
C.
D.

Explanation

To determine the variation of reactance \(X_c\) of a capacitor with frequency \(f\) of an alternating current (a.c.) supply, we can follow these steps:

### Step 1: Understand the formula for capacitive reactance
The capacitive reactance \(X_c\) is given by the formula:
\[ X_c = \frac{1}{\omega C} \]
where \(\omega\) is the angular frequency and \(C\) is the capacitance.

### Step 2: Relate angular frequency to frequency
The angular frequency \(\omega\) can be expressed in terms of frequency \(f\) as:
\[ \omega = 2\pi f \]
Substituting this into the reactance formula gives:
\[ X_c = \frac{1}{2\pi f C} \]

### Step 3: Analyze the relationship between \(X_c\) and \(f\)
From the equation \(X_c = \frac{1}{2\pi f C}\), we can see that \(X_c\) is inversely proportional to the frequency \(f\). This means:
- As the frequency \(f\) increases, the reactance \(X_c\) decreases.
- As the frequency \(f\) decreases, the reactance \(X_c\) increases.

### Step 4: Determine the shape of the graph
Since \(X_c\) decreases as \(f\) increases, the graph of \(X_c\) versus \(f\) will be a hyperbola that approaches the horizontal axis (the x-axis) as \(f\) increases. This indicates that the reactance decreases with increasing frequency.

### Step 5: Identify the correct graph
Given the options, the graph that shows a decreasing trend of \(X_c\) as \(f\) increases represents the correct relationship. This is typically a downward sloping curve.

### Conclusion
The correct option is the one that shows \(X_c\) decreasing as \(f\) increases.

---