As the frequency of an ac circuit increases, the current first increases and then decreases. What combination of circuit elements is most likely to comprise the circuit ?

Options

A. incuctor and capacitor
B. Resistor and inductor
C. Resistor and capacitor
D. Resistor, inductor and capacitor

Explanation

To solve the question of how the current in an AC circuit behaves as the frequency increases, we need to analyze the relationship between the circuit elements (resistor, inductor, and capacitor) and their reactances. Here's a step-by-step solution:

### Step 1: Understand the Impedance Formula
The total impedance \( Z \) of an LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance.

### Step 2: Define Reactances
The reactances are defined as:
- Inductive reactance: \( X_L = 2\pi f L \)
- Capacitive reactance: \( X_C = \frac{1}{2\pi f C} \)

### Step 3: Analyze Frequency Effects
As the frequency \( f \) increases:
- \( X_L \) (inductive reactance) increases because it is directly proportional to \( f \).
- \( X_C \) (capacitive reactance) decreases because it is inversely proportional to \( f \).

### Step 4: Determine Impedance Behavior
Substituting \( X_L \) and \( X_C \) into the impedance formula:
\[ Z = \sqrt{R^2 + (2\pi f L - \frac{1}{2\pi f C})^2} \]
As \( f \) increases, \( X_L \) increases and \( X_C \) decreases. This means that the difference \( (X_L - X_C) \) will initially increase, leading to an increase in \( Z \).

### Step 5: Current Behavior
The current \( I \) in the circuit is given by:
\[ I = \frac{V}{Z} \]
Where \( V \) is the voltage. As \( Z \) increases, the current \( I \) will initially increase until a certain frequency where the impedance is minimized.

### Step 6: Find the Resonance Condition
At a specific frequency, the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \implies 2\pi f L = \frac{1}{2\pi f C} \]
At this point, the impedance \( Z \) is minimized to \( R \) (the resistance), and the current is maximized.

### Step 7: Conclusion on Circuit Elements
The combination of circuit elements that allows the current to first increase and then decrease with increasing frequency is most likely an L-C (inductor-capacitor) circuit. If a resistor is also present, it does not affect the frequency response significantly.

### Final Answer
The circuit is most likely comprised of an inductor and a capacitor (L-C circuit). A combination of all three elements (R, L, and C) can also exhibit this behavior, but the L-C combination is the primary reason for the described current behavior.

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