A resistor `R`, an inductor `L` and a capacitor `C` are connected in series to an oscillator of frequency `n`. If the resonant frequency is `n_r,` then the current lags behind voltage, when

Options

A. `n=0`
B. `n lt n_(r)`
C. `n=n_(r)`
D. `n gt n_(r)`

Explanation

To solve the problem step by step, we need to analyze the conditions under which the current lags behind the voltage in an RLC series circuit.

### Step 1: Understand Resonance in RLC Circuit
In a series RLC circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC). This can be expressed mathematically as:
\[ X_L = X_C \]
Where:
- \( X_L = \omega L \) (Inductive reactance)
- \( X_C = \frac{1}{\omega C} \) (Capacitive reactance)

At resonance, the frequency is given by:
\[ \omega_r = \frac{1}{\sqrt{LC}} \]
And the resonant frequency \( n_r \) is:
\[ n_r = \frac{1}{2\pi\sqrt{LC}} \]

### Step 2: Determine Current and Voltage Relationship
In an RLC circuit, the phase relationship between current and voltage depends on the relative values of \( X_L \) and \( X_C \):
- If \( X_L > X_C \), the circuit is inductive, and the current lags behind the voltage.
- If \( X_L < X_C \), the circuit is capacitive, and the current leads the voltage.

### Step 3: Condition for Current Lagging Voltage
From the above, we can conclude that for the current to lag behind the voltage, the condition must be:
\[ X_L > X_C \]
Substituting the expressions for reactance, we get:
\[ \omega L > \frac{1}{\omega C} \]
Multiplying both sides by \( \omega \) (assuming \( \omega > 0 \)):
\[ \omega^2 L > \frac{1}{C} \]
This can be rearranged to:
\[ \omega^2 > \frac{1}{LC} \]

### Step 4: Relate to Resonant Frequency
Since the resonant frequency \( n_r \) is defined as:
\[ n_r = \frac{1}{2\pi\sqrt{LC}} \]
We can express this condition in terms of \( n_r \):
\[ \omega^2 > \frac{1}{LC} \implies n^2 > n_r^2 \]
Thus, we find:
\[ n > n_r \]

### Conclusion
Therefore, the current lags behind the voltage when the frequency \( n \) is greater than the resonant frequency \( n_r \).

### Final Answer
The current lags behind the voltage when:
\[ n > n_r \]