Question

In an LCR series circuit, the source of emf is E=30sin(100t), R=120Ω, L=100mH, C=100μF.
(A) The numerical value of impedance 
(B) The numerical value of resistance R 
(C) The numerical value of capacitive reactance 
(D) The numerical value of inductive reactance.
Arrange the values of quantities mentioned in (A, B, C, D) in increasing order.

• A. (A), (B), (C), (D)
• B. (A), (C), (B), (D)
• C. (D), (C), (B), (A)
• D. (B), (A), (D), (C)

Answer 1

The Correct Option is C

To solve the problem of finding the impedance, resistance, capacitive reactance, and inductive reactance in an LCR series circuit, we follow these steps:

1. Given data: \(E=30\sin(100t)\), \(R=120\Omega\), \(L=100\text{ mH}\), \(C=100\mu\text{F}\). Frequency \(\omega=100\text{ rad/s}\).
2. Calculate inductive reactance \((X_L)\):
   \(X_L=\omega L=100 \times 0.1=10\Omega\).
3. Calculate capacitive reactance \((X_C)\):
   \(X_C=\frac{1}{\omega C}=\frac{1}{100 \times 100 \times 10^{-6}}=100\Omega\).
4. Calculate impedance \((Z)\):
   \(Z=\sqrt{R^2+(X_L-X_C)^2}=\sqrt{120^2+(10-100)^2}=\sqrt{120^2+(-90)^2}=\sqrt{14400+8100}=\sqrt{22500}=150\Omega\).

Thus, the values are:
(A) Impedance = 150Ω,
(B) Resistance = 120Ω,
(C) Capacitive Reactance = 100Ω,
(D) Inductive Reactance = 10Ω.

Arranging the values in increasing order: (D), (C), (B), (A).

Answer 2

Given LCR circuit parameters:

E = 30sin(100t) V, R = 120 Ω, L = 100 mH, C = 100 μF

1. Angular frequency (ω): \[ ω = 100\ \text{rad/s} \] (from the source equation)

2. Inductive reactance (X_L): \[ X_L = ωL = 100 × 0.1 = 10\ \Omega \]

3. Capacitive reactance (X_C): \[ X_C = \frac{1}{ωC} = \frac{1}{100 × 100×10^{-6}} = 100\ \Omega \]

4. Impedance (Z): \[ Z = \sqrt{R^2 + (X_C - X_L)^2} = \sqrt{120^2 + (100 - 10)^2} \] \[ Z = \sqrt{14400 + 8100} = \sqrt{22500} = 150\ \Omega \]

Values obtained:

A. Impedance (Z) = 150 Ω

B. Resistance (R) = 120 Ω

C. Capacitive reactance (X_C) = 100 Ω

D. Inductive reactance (X_L) = 10 Ω

Increasing order: D (10 Ω) < C (100 Ω) < B (120 Ω) < A (150 Ω)

Final answer: The values in increasing order are D < C < B < A.