Resonance occures in a series LCR circuit when the frequency of the applied emf is 1000 Hz.

Options

A. when frequency = 900Hz then the current through the voltage source will be ahead of emf of the source
B. the impedance of the circuit is minimum at `f=1000 Hz`
C. at only resonance the voltages across L and C differ in phase by `180^(@)`
D. if the value of C is double, resonance occurs at `f=2000 Hz`.

Explanation

To solve the problem regarding resonance in a series LCR circuit, we can follow these steps:

### Step 1: Understand the Condition for Resonance
In a series LCR circuit, resonance occurs when the inductive reactance (XL) is equal to the capacitive reactance (XC). Mathematically, this is expressed as:
\[ XL = XC \]

### Step 2: Express Reactances in Terms of Frequency
The inductive reactance (XL) and capacitive reactance (XC) can be expressed as:
\[ XL = \omega L = 2\pi f L \]
\[ XC = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]
where:
- \( \omega \) is the angular frequency,
- \( L \) is the inductance,
- \( C \) is the capacitance,
- \( f \) is the frequency of the applied EMF.

### Step 3: Set Up the Resonance Condition
At resonance, we set the two reactances equal:
\[ 2\pi f L = \frac{1}{2\pi f C} \]

### Step 4: Solve for Frequency
Rearranging the equation gives:
\[ (2\pi f)^2 = \frac{1}{LC} \]
Taking the square root of both sides, we find:
\[ 2\pi f = \frac{1}{\sqrt{LC}} \]
Thus, the frequency at resonance is:
\[ f = \frac{1}{2\pi \sqrt{LC}} \]

### Step 5: Given Frequency
The problem states that resonance occurs at a frequency of 1000 Hz. Therefore, we can conclude:
\[ f = 1000 \text{ Hz} \]

### Step 6: Analyze the Options
Now, we can analyze the options provided in the question based on the conditions of resonance and the behavior of the circuit:
1. If the frequency is less than 1000 Hz, the circuit becomes capacitive, and the current leads the voltage.
2. If the frequency is equal to 1000 Hz, the impedance is at its minimum and equals the resistance (R).
3. If the frequency is greater than 1000 Hz, the circuit becomes inductive, and the voltage leads the current.

### Conclusion
From the analysis:
- Option 1 is correct: At 900 Hz (less than 1000 Hz), the current leads the EMF.
- Option 2 is correct: The impedance is minimum at 1000 Hz (the resonant frequency).
- Option 3 is incorrect: The phase difference between the voltages across L and C is always 180 degrees, not only at resonance.
- Option 4 is incorrect: Doubling the capacitance does not lead to a frequency of 2000 Hz.

### Final Answer
The correct options are 1 and 2.

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