Alternating Current (AC) – Class 12 Physics

✅ Definition:

Alternating current (AC) is a current that changes its magnitude and direction periodically with time.

• Most household electricity is AC.

• Represented as:

i(t)=I0sin⁡(ωt+ϕ)i(t) = I_0 \sin(\omega t + \phi)i(t)=I0sin(ωt+ϕ)

Where:

• I0I_0I0 = peak current (A)

• ω=2πf\omega = 2\pi fω=2πf = angular frequency (rad/s)

• fff = frequency (Hz)

• ϕ\phiϕ = phase constant

🔹 Key Features of AC

1. Current changes direction periodically.

2. Instantaneous value: i=I0sin⁡(ωt)i = I_0 \sin(\omega t)i=I0sin(ωt)

3. Time period: T=1fT = \frac{1}{f}T=f1

4. RMS (Root Mean Square) value:

Irms=I02,Vrms=V02I_\text{rms} = \frac{I_0}{\sqrt{2}}, \quad V_\text{rms} = \frac{V_0}{\sqrt{2}}Irms=2I0,Vrms=2V0

5. Average value over half cycle:

Iavg=2I0π,Vavg=2V0πI_\text{avg} = \frac{2 I_0}{\pi}, \quad V_\text{avg} = \frac{2 V_0}{\pi}Iavg=π2I0,Vavg=π2V0

🔹 AC in Pure Resistor, Inductor, and Capacitor

🔹 Impedance in AC Circuits

• Impedance ZZZ is the AC equivalent of resistance.

Z=R2+(XL−XC)2Z = \sqrt{R^2 + (X_L - X_C)^2}Z=R2+(XL−XC)2

Where:

• XL=ωLX_L = \omega LXL=ωL = inductive reactance

• XC=1ωCX_C = \frac{1}{\omega C}XC=ωC1 = capacitive reactance

AC current:

Irms=VrmsZI_\text{rms} = \frac{V_\text{rms}}{Z}Irms=ZVrms

🔹 Resonance in AC Circuits

• In series LCR circuit, resonance occurs when:

XL=XCorωL=1ωCX_L = X_C \quad \text{or} \quad \omega L = \frac{1}{\omega C}XL=XCorωL=ωC1

• At resonance:

  • Impedance Z=RZ = RZ=R

  • Current is maximum

  • Voltage across L and C can be much higher than applied voltage

🔹 Power in AC Circuits

• Instantaneous power: p(t)=v(t)⋅i(t)p(t) = v(t) \cdot i(t)p(t)=v(t)⋅i(t)

• Average power:

P=VrmsIrmscos⁡ϕP = V_\text{rms} I_\text{rms} \cos \phiP=VrmsIrmscosϕ

Where ϕ\phiϕ = phase difference between voltage and current.

• Power factor = cos⁡ϕ\cos \phicosϕ

  • For pure R → cos φ = 1

  • For pure L → cos φ = 0

  • For pure C → cos φ = 0

🔹 Important Formulas Summary