#phase relationship

Resonance in LCR Circuit and Q-Factor

In an ideal AC circuit with only a capacitor, what is the relationship between the rates of change of voltage and curren

**At resonance** V_L = I X_L = I X_C = V_C, may be larger than source voltage Q times, Q-factor = ω₀ L/R =1/(ω₀ C R)= V_L/V = V_C/V, measures sharpness, higher Q sharper resonance, bandwidth Δω = R/L = ω₀/Q, resonant frequency independent of R. In a capacitor, I = C (dV/dt) , meaning the current is directly proportional to the rate of change of voltage. Conversely, the rate of change of current relates to the second derivative of voltage, but the primary relationship is that current depends on (dV/dt) . Applying X_L = ωL, X_C = 1/ωC, Z = √(R² + (X_L -

Ref: NCERT > Physics Book > Alternating Currents > Resonance in LCR Circuit and Q-Factor

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AC Fundamentals - RMS, Average and Peak Values

In an AC circuit with only an inductor, what is the phase relationship between the current and the voltage?

**Peak current** I_peak = V_peak/R for resistor, I_rms = V_rms/R, V_peak = √2 V_rms, for 200 V rms, V_peak=282.8 V, I_peak=282.8/80=3.535 A, rms I=200/80=2.5 A, average over complete cycle zero because positive and negative halves cancel. In a purely inductive AC circuit, the current lags the voltage by 90°. This is because the inductor opposes changes in current, causing the current to reach its peak after the voltage. Applying X_L = ωL, X_C = 1/ωC, Z = √(R² + (X_L - X_C)²), I_rms = V_rms/Z, P = V_rms I_rms cosφ, V_s/V_p = N_s/N_p, calculation gives Current lags voltage by 90°, consistent wit

Ref: NCERT > Physics Book > Alternating Currents > AC Fundamentals - RMS, Average and Peak Values

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AC Through Resistor - Phasor and Power

In an AC circuit with only a resistor, how does the current behave relative to the applied voltage?

**Current relative to voltage** in resistor in phase, φ=0°, power factor cos φ=1, maximum power, unlike inductor/capacitor where average power zero due to 90° phase shift, explaining why resistor heats while pure L/C does not. In a purely resistive AC circuit, the current and voltage oscillate in phase, meaning they reach their peak, zero, and minimum values simultaneously. This occurs because a resistor does not introduce any phase shift between voltage and current. Applying X_L = ωL, X_C = 1/ωC, Z = √(R² + (X_L - X_C)²), I_rms = V_rms/Z, P = V_rms I_rms cosφ, V_s/V_p = N_s/N_p, calculation g

Ref: NCERT > Physics Book > Alternating Currents > AC Through Resistor - Phasor and Power

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Periodic Motion and SHM Basic Concepts

In SHM, what is the phase relationship between displacement and acceleration?

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). Displacement ( x = A cos (ω t + Φ) ) and acceleration ( a = -ω² A cos (ω t + Φ) ) are 180° ( π radians) out of phase, as acceleration is the negative of displacement. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² =

Ref: NCERT > Physics Book > Oscillations > Periodic Motion and SHM Basic Concepts

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Transverse and Longitudinal Waves

In a progressive wave, what is the phase relationship between two points separated by one full wavelength?

**Wave classification** depends on particle vibration relative to propagation. Longitudinal waves have particle oscillation parallel to propagation, creating compressions and rarefactions as in sound in air; transverse have perpendicular oscillation. Tuning fork generates longitudinal sound because air cannot sustain shear. For a progressive wave, phase difference Δ Φ = k Δ x , where k = (2π/λ) . If Δ x = λ , then Δ Φ = (2π/λ) × λ = 2π , meaning they are in phase. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields In phase, illustrating frequency

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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