In a series LCR circuit containing an AC voltage source of frequency `omega`, current and voltage are measured. Then for the resistance, consider the statements-
[a] current is maximum at `omega^(2)=1//LC`
[b] current is minimum at `omega^(2)=1//LC`
[c] voltage across R is maximum at `omega^(2) =1//LC`
[d] voltage across R is minimum at `omega^(2)=1//LC`
the voltage statement is

Options

A. a and c are both correct
B. b and d are both correct
C. a and d are both correct
D. none of these

Answer

To analyze the statements regarding the behavior of current and voltage in a series LCR circuit at resonance, we can follow these steps:

### Step 1: Understand Resonance in LCR Circuit
In a series LCR circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC). This condition can be mathematically expressed as:
\[ \omega^2 = \frac{1}{LC} \]
where \( \omega \) is the angular frequency, \( L \) is the inductance, and \( C \) is the capacitance.

### Step 2: Analyze Current at Resonance
At resonance, the impedance of the circuit is minimized and is equal to the resistance \( R \) only. This results in the maximum current flowing through the circuit:
\[ I = \frac{V}{R} \]
where \( V \) is the voltage across the circuit. Therefore, the statement:
- **[a] Current is maximum at \( \omega^2 = \frac{1}{LC} \)** is **true**.

### Step 3: Analyze Current Minimum Statement
The statement:
- **[b] Current is minimum at \( \omega^2 = \frac{1}{LC} \)** is **false** because, as established, the current is maximum at this frequency.

### Step 4: Analyze Voltage Across Resistor
At resonance, the voltage across the resistor \( V_R \) is also at its maximum because the entire source voltage appears across the resistor when the circuit is in resonance. Therefore, the statement:
- **[c] Voltage across R is maximum at \( \omega^2 = \frac{1}{LC} \)** is **true**.

### Step 5: Analyze Voltage Minimum Statement
The statement:
- **[d] Voltage across R is minimum at \( \omega^2 = \frac{1}{LC} \)** is **false** as the voltage across the resistor is maximum at resonance.

### Conclusion
From the analysis, the correct statements are:
- **[a] Current is maximum at \( \omega^2 = \frac{1}{LC} \)**
- **[c] Voltage across R is maximum at \( \omega^2 = \frac{1}{LC} \)**

Thus, the voltage statement is **true** for options [a] and [c].

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