Q4. Assertion: at resonance in a series LCR circuit, the source voltage and the current are in phase. Reason: at resonance XL = XC, so tan φ = 0. Are both true, and does the reason explain the assertion?
Both are true, and the reason explains the assertion (NCERT Eqs. (7.27)–(7.28), pages 188–189).
Q5. A series LCR circuit has R = 10 Ω, L = 0.10 H and C = 100 µF, on a 200 V rms, 50 Hz supply. Find XL, XC, Z, the current and the phase angle.
XL = 2π × 50 × 0.10 ≈ 31.4 Ω and XC = 1/(2π × 50 × 100 × 10−6) ≈ 31.8 Ω. Keeping unrounded reactances gives XC − XL ≈ 0.4151 Ω. Z ≈ 10.0086 Ω, so I ≈ 19.98 A, and tan φ ≈ 0.4151/10, φ ≈ +2.4°: almost at resonance, the current very slightly leading (NCERT Eqs. (7.26)–(7.27), pages 187–188).
Q6. What capacitance must be used with a 0.25 mH coil to tune a radio to a station at 1.0 MHz?
ω0 = 2π × 106 rad/s and C = 1/(ω02L) = 1/((2π × 106)2 × 0.25 × 10−3) ≈ 1.0 × 10−10 F, about 100 pF (NCERT Eq. (7.28), page 189).
Q7. In a series LCR circuit the rms voltages across R, L and C are 40 V, 90 V and 60 V. What is the source voltage?
V = √(402 + (90 − 60)2) = √(1600 + 900) = 50 V (NCERT page 187).