Question

A box contains L, C and R When 250V dc is applied to the terminals of the box a current of 0.1A flows in the circuit When an ac source of 250V rms at 2250 rad//sec is connected a current of 1.25 A rms flows it is observed that the current rises with frequency and becomes maximum at 4500rad/sec Find the value of L, C and R Draw the circuit diagram.

Answer 1

Contradictory information, problem not solvable

Answer 2

The problem describes an electrical circuit within a "box" containing an inductor (L), a capacitor (C), and a resistor (R). We analyze its behavior under DC and AC conditions.

1. DC Analysis

When a DC voltage of 250 V is applied, a current of 0.1 A flows. For DC ($\omega = 0$):

• An inductor (L) acts as a short circuit ($X_L = \omega L = 0$).
• A capacitor (C) acts as an open circuit ($X_C = 1/(\omega C) = \infty$).

If L, C, and R were in a simple series configuration, the capacitor would block the DC current, resulting in zero current. Since a current of 0.1 A flows, the capacitor cannot be in series with the main DC current path. A common configuration that allows DC current and exhibits resonant behavior with AC is a resistor (R) in series with a parallel combination of an inductor (L) and a capacitor (C).

Circuit Diagram: R in series with (L || C)

For this configuration under DC conditions:

• The inductor (L) acts as a short circuit.
• The capacitor (C) acts as an open circuit.
• Therefore, the parallel combination (L || C) effectively becomes a short circuit (as L shorts C). The total DC resistance of the circuit is simply R.

Using Ohm's Law for DC: $R = \frac{V_{dc}}{I_{dc}} = \frac{250 \text{ V}}{0.1 \text{ A}} = 2500 \text{ } \Omega$

2. AC Analysis

An AC source of 250 V (rms) at $\omega_1 = 2250 \text{ rad/s}$ is connected, and a current of 1.25 A (rms) flows. The magnitude of the impedance at this frequency is: $|Z_1| = \frac{V_{rms}}{I_{rms}} = \frac{250 \text{ V}}{1.25 \text{ A}} = 200 \text{ } \Omega$

The problem states that "the current rises with frequency and becomes maximum at 4500 rad/sec". This characteristic is typical of a series RLC circuit at resonance. For the proposed circuit (R in series with L || C), the total impedance is $Z = R + \frac{j\omega L}{1 - \omega^2 LC}$. For this circuit, the current is minimum (ideally zero) when the parallel LC branch is in resonance (i.e., $1 - \omega^2 LC = 0$). This contradicts the statement that the current becomes maximum at 4500 rad/s.

This fundamental contradiction suggests that the assumed circuit configuration (R in series with L || C) is inconsistent with the AC behavior described, or more likely, the problem statement contains contradictory numerical values for a simple series RLC circuit.

Re-evaluation assuming a Series RLC Circuit (most common for resonance problems)

Let's assume the circuit is a simple series RLC circuit, as implied by the resonance characteristic ("current rises with frequency and becomes maximum").

Circuit Diagram: Series RLC Circuit (most likely intended for AC resonance)

1. Resistance (R): From the DC analysis, we found $R = 2500 \text{ } \Omega$.
2. Resonant Frequency ($\omega_0$): The problem states that the current becomes maximum at $\omega_0 = 4500 \text{ rad/s}$.
3. Impedance at $\omega_1 = 2250 \text{ rad/s}$: We calculated $|Z_1| = 200 \text{ } \Omega$.

For a series RLC circuit, the impedance magnitude is given by: $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$ This equation implies that the impedance magnitude $|Z|$ must always be greater than or equal to the resistance R (i.e., $|Z| \ge R$).

However, we have $R = 2500 \text{ } \Omega$ and $|Z_1| = 200 \text{ } \Omega$. This leads to $200 < 2500$, which violates the fundamental property $|Z| \ge R$ for a series RLC circuit.

Conclusion on Problem Flaw

The problem statement contains contradictory information for a standard series RLC circuit. It is mathematically impossible to find values of L, C, and R that satisfy all given conditions simultaneously. This strongly indicates a typo in the numerical values provided in the question.

Without further clarification or correction of the numerical values, a consistent solution for L, C, and R cannot be derived.

The problem is unresolvable as stated.

Subject, Chapter and Topic

Subject: Physics Chapter: Alternating Current Topic: Series RLC Circuit, Resonance in RLC Circuits

Difficulty Level

Hard (due to the inherent contradiction in the problem statement)

Question Type

Descriptive