FREE UNDAMPED ELECTROMAGNETIC OSCILLATIONS The charge current and

FREE UNDAMPED ELECTROMAGNETIC OSCILLATIONS The charge, current, and potential difference vary sinusoidally with time (period T and angular frequency ω) The circuit is said to oscillate The resulting oscillations of the capacitor's electric field and the inductor's magnetic field are said to be electromagnetic oscillations

STAGES OF THE OSCILLATIONS IN A SIMPLE LC CIRCUIT Green bar is stored magnetic energy. Blue bar is stored electric energy (a) Capacitor with maximum charge, no current. (b) Capacitor discharging, current increasing. (c) Capacitor fully discharged, current maximum. (d) Capacitor charging but with polarity opposite that in (a), current decreasing. (e) Capacitor with maximum charge having polarity opposite that in (a), no current. (f) Capacitor discharging, current increasing with direction opposite that in (b). (g) Capacitor fully discharged, current maximum. (h) Capacitor charging, current decreasing.

DIFFERENTIAL EQUATION OF FREE UNDAMPED ELECTROMAGNETIC OSCILLATIONS R = 0, IR = 0 qm is amplitude of charge oscillations φ0 is initial phase

The oscillations of potential difference across the capacitor are The potential difference across the capacitor is in the phase with charge The current is The current leads the charge by π/2 The potential difference across the coil is The potential difference across the coil leads the charge by π

ENERGY OF ELECTRIC FIELD IS THE ENERGY OF CHARGED CAPACITOR ENERGY OF MAGNETIC FIELD IS THE ENERGY, CONCENTRATED IN A COIL TOTAL ENERGY OF ELECTROMAGNETIC OSCILLATIONS

DAMPED ELECTROMAGNETIC OSCILLATIONS A circuit containing resistance, inductance, and capacitance is called an RLC circuit. With a resistance R present, the total electromagnetic energy E of the circuit (the sum of the electric energy and magnetic energy) is no longer constant; instead, it decreases with time as energy is transferred to thermal energy in the resistance. Because of this loss of energy, the oscillations of charge, current, and potential difference continuously decrease in amplitude, and the oscillations are said to be damped A series RLC circuit

DIFFERENTIAL EQUATION OF FREE DAMPED ELECTROMAGNETIC OSCILLATIONS

CHARACTERISTICS OF DAMPED OSCILLATIONS 1. Damping factor is the rate of damping 2. Relaxation time is the time for the amplitude to diminish e times 3. Damping decrement is the ratio of two successive amplitudes 4. Logarithmic damping decrement is logarithm of the ratio of two successive amplitudes 5. Q-factor (quality factor), the ratio of energy stored in the oscillating system and energy lost during one cycle is called wave resistance.

FORCED ELECTROMAGNETIC OSCILLATIONS When the external alternating emf is connected to an RLC circuit, the oscillations of charge, potential difference, and current are said to be forced oscillations or driven oscillations. These oscillations always occur at the forcing (driving) angular frequency . q is the phase shift between q and EMF

AMPLITUDE CHARACTERISTICS Amplitude characteristics is dependence between amplitude qm and forcing frequency ω = ω0 and R = 0 res = res= qres=

PHASE CHARACTERISTICS Phase characteristics is relationship between phase difference (phase of displacement and phase of external periodic force) and forcing frequency for discrete values of damping factor β q q is the phase shift between q and EMF = 0 then = π/2 for all values of R

ALTERNATING CURRENT is the phase shift between current and EMF , Im→ 0, qm→qo At any R res= 0=1/

OHM’S LAW for ALTERNATING CURRENT is impedance R is resistance is reactance is inductive reactance is capacative reactance

THE POTENTIAL DIFFERENCE ACROSS DIFFERENT ELEMENTS OF A CIRCUIT THE POTENTIAL DIFFERENCE ACROSS A RESISTOR is in phase with a current THE POTENTIAL DIFFERENCE ACROSS A CONDENSER lags π/2 behind the current THE POTENTIAL DIFFERENCE ACROSS A COIL leads by π/2 on the current

VECTOR DIAGRAM for THE POTENTIAL DIFFERENCE ACROSS DIFFERENT ELEMENTS OF A CIRCUIT

POWER in ALTERNATING CURRENT CIRCUIT