In and LCR circuit if frequency of the supply is made 4 time, how should the values of C and L be changed so that there is no change in the current in the circuit.

In an LCR circuit, which consists of an inductor (L), a capacitor (C), and a resistor (R), the relationship between the frequency of the supply and the reactance of the components plays a crucial role in determining the current flowing through the circuit. When the frequency of the supply is increased, the reactance of the inductor and capacitor changes, which can affect the overall current. To maintain the same current when the frequency is quadrupled, adjustments to the values of L and C are necessary.

Understanding Reactance in LCR Circuits

The reactance of an inductor (X_L) and a capacitor (X_C) is given by the following formulas:

• X_L = 2πfL
• X_C = 1/(2πfC)

Here, f represents the frequency, L is the inductance, and C is the capacitance. As the frequency increases, the inductive reactance increases while the capacitive reactance decreases.

Effect of Frequency Change

If the frequency is increased to 4 times its original value (let's denote the original frequency as f), the new frequency becomes 4f. The new reactances will be:

• X_{L_new} = 2π(4f)L = 8πfL
• X_{C_new} = 1/(2π(4f)C) = 1/(8πfC)

To keep the current constant, the total reactance in the circuit must remain unchanged. This means that the relationship between the inductive and capacitive reactance must be adjusted accordingly.

Adjusting L and C

To maintain the same current, we need to ensure that the ratio of the inductive reactance to the capacitive reactance remains the same even after the frequency change. This can be expressed as:

X_L = X_C at the original frequency.

After increasing the frequency, we want:

X_{L_new} = X_{C_new}

Substituting the new reactance values, we have:

8πfL = 1/(8πfC)

Solving for New Values

Rearranging the equation gives:

64π²f²LC = 1

This implies that:

LC = 1/(64π²f²)

To maintain the same product of L and C while keeping the frequency at 4f, we can adjust L and C as follows:

• Reduce L to L/4
• Increase C to 4C

By making these adjustments, the product LC remains constant, ensuring that the current in the circuit does not change despite the increase in frequency.

Final Thoughts

In summary, when the frequency of the supply in an LCR circuit is quadrupled, you can maintain the same current by reducing the inductance to one-fourth of its original value and increasing the capacitance to four times its original value. This balance allows the circuit to operate effectively without altering the current flow.