In `L-C-R` series circuit

Options

A. current may achieve same value for two different value of frequency
B. current can not achieve same value for two different values of frequency
C. current is maximum for a maximum value of frequency
D. none of these

Explanation

To solve the problem regarding the behavior of current in an L-C-R series circuit, we need to analyze how the impedance (Z) and current (I) are affected by frequency (ω). Here’s a step-by-step solution:

### Step 1: Understand the Expression for Current
The current in an L-C-R series circuit is given by:
\[ I = \frac{E \sin(\omega t - \phi)}{Z} \]
where \( Z \) is the impedance of the circuit.

### Step 2: Define Impedance
The impedance \( Z \) in an L-C-R circuit is defined as:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( X_L = L \omega \) (inductive reactance)
- \( X_C = \frac{1}{\omega C} \) (capacitive reactance)

### Step 3: Analyze the Condition for Same Current at Different Frequencies
For the current \( I \) to be the same at two different frequencies \( \omega_1 \) and \( \omega_2 \), the impedance \( Z \) must also be the same at these frequencies. Thus, we have:
\[ Z(\omega_1) = Z(\omega_2) \]

### Step 4: Set Up the Equation
From the definition of impedance:
\[ \sqrt{R^2 + (L \omega_1 - \frac{1}{\omega_1 C})^2} = \sqrt{R^2 + (L \omega_2 - \frac{1}{\omega_2 C})^2} \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square root gives:
\[ R^2 + (L \omega_1 - \frac{1}{\omega_1 C})^2 = R^2 + (L \omega_2 - \frac{1}{\omega_2 C})^2 \]

### Step 6: Cancel Out Common Terms
Since \( R^2 \) appears on both sides, we can cancel it out:
\[ (L \omega_1 - \frac{1}{\omega_1 C})^2 = (L \omega_2 - \frac{1}{\omega_2 C})^2 \]

### Step 7: Expand and Rearrange
Expanding both sides gives:
\[ L^2 \omega_1^2 - 2L \omega_1 \frac{1}{\omega_1 C} + \frac{1}{\omega_1^2 C^2} = L^2 \omega_2^2 - 2L \omega_2 \frac{1}{\omega_2 C} + \frac{1}{\omega_2^2 C^2} \]

### Step 8: Simplify the Equation
This leads to a more complex equation, but essentially, we will find that:
\[ L(\omega_1 - \omega_2) = \frac{1}{C}(\frac{1}{\omega_1} - \frac{1}{\omega_2}) \]
This indicates a relationship that would not hold for two different frequencies unless certain conditions are met.

### Step 9: Conclusion
The analysis shows that it is impossible for the current to achieve the same value at two different frequencies in an L-C-R series circuit. Therefore, the statement that current can achieve the same value for two different frequencies is incorrect.