A series LCR circuit contains $L=1 mH, C=10 \mu F$ and $R=500 \Omega$. If the inductive and capacitive reactance are equal, then the value of the reactance is

Question

Accepted Answer

To determine the correct value of the reactance in a series LCR circuit where the inductive and capacitive reactance are equal, we need to understand the concepts of inductive reactance ($X_L$) and capacitive reactance ($X_C$).

1. **Inductive Reactance ($X_L$)**: This is given by the formula $X_L = 2\pi f L$, where $f$ is the frequency and $L$ is the inductance. For $L = 1 \, \text{mH} = 1 \times 10^{-3} \, \text{H}$, the inductive reactance depends on the frequency.

2. **Capacitive Reactance ($X_C$)**: This is given by the formula $X_C = \frac{1}{2\pi f C}$, where $C$ is the capacitance. For $C = 10 \, \mu F = 10 \times 10^{-6} \, \text{F}$, the capacitive reactance also depends on the frequency.

Given that the inductive and capacitive reactance are equal ($X_L = X_C$), we can set the equations equal to each other:

$$ 2\pi f L = \frac{1}{2\pi f C} $$

Solving for the frequency $f$, we get:

$$ f^2 = \frac{1}{(2\pi)^2 LC} $$

$$ f = \frac{1}{2\pi \sqrt{LC}} $$

Substituting the given values:

$$ f = \frac{1}{2\pi \sqrt{(1 \times 10^{-3})(10 \times 10^{-6})}} $$

$$ f = \frac{1}{2\pi \sqrt{10 \times 10^{-9}}} $$

$$ f = \frac{1}{2\pi \times 10^{-4.5}} $$

$$ f = \frac{1}{2\pi \times 3.16 \times 10^{-5}} $$

$$ f \approx 1591.55 \, \text{Hz} $$

Now, substituting this frequency back into the formula for either $X_L$ or $X_C$:

$$ X_L = 2\pi f L = 2\pi \times 1591.55 \times 1 \times 10^{-3} $$

$$ X_L \approx 10 \, \Omega $$

Thus, the value of the reactance when the inductive and capacitive reactance are equal is indeed 10 Ω, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (50 Ω)**: This value is incorrect because, as calculated, the reactance when $X_L = X_C$ is 10 Ω, not 50 Ω. The calculations based on the given inductance and capacitance do not support this value.

- **Option 3 (100 Ω)**: This value is incorrect for the same reason as Option 2. The calculated reactance is 10 Ω, and 100 Ω is significantly higher than the correct value.

- **Option 4 (5 Ω)**: This value is also incorrect because it is lower than the calculated reactance of 10 Ω. The calculations clearly show that the reactance is 10 Ω, not 5 Ω.

In conclusion, the correct value of the reactance when the inductive and capacitive reactance are equal in this LCR circuit is 10 Ω, which is why Option 1 is the correct answer.