1. D.C. circuit : P = VI

Not true of AC circuits, in general

2. Power delivered to R

Active power can do useful work measured in W or KW

3. Capacitors and inductor $\Delta \phi=\frac{\pi}{2}$

4. For LCR circuit current may lead or lag depending on reactance values

5. $\cos \phi =$ Power factor

6. Power Triangle

Multiply by $I^2$

Active power $=P=I^2R$(Walts)

Reactive power $=Q=I^2X$(VAr)

Apparent Power $=S=I^2Z$(VA)

$S=VI$

$P=VI \cos \phi$

$Q=VI \sin \phi$

$X=X_L$

Lags the voltage

Capacitive voltage

$I=0.5A(rms)$

Lags the voltage $75^o$

Apparent Power $=220 \times 0.5 =110 V-A$

True Power $=110 \cos (75^o)=28.47 W$

Reactive Power $=110 \sin (75^o)=106.25 V-Ar$

$t=0$

$v=0$

spring energy

$U=U_{max}=\frac{1}{2}Kx_0^2$

$K=0$

from t = 0 to $t=\frac{T}{4},x>0$

$U=\frac{1}{2}Kx^2$

$K=\frac{1}{2}mv^2$

$t=\frac{T}{4}$

$U=0$

$K=\frac{1}{2}mv_{max}^2=K_{max}$

from $t=\frac{T}{4} t<11$

$t=\frac{T}{2}x<0$

$K=\frac{1}{2}mv^2$

$U=\frac{1}{2}kx^2$

At $t=\frac{T}{2}$

Compression is max $=x_0$

$U=U^{max}=\frac{1}{2}Kx_0^2$

$K=0$

From $t=\frac{T}{2}$

$t<11$

$t=\frac{3T}{4}$

$x<0$ , $v\neq 0$

$K=\frac{1}{2}mv^2$

$U=\frac{1}{2}kx^2$

At $t=\frac{3T}{4}$

$K=\frac{1}{2}mv_{max}^2$

$U=0$

At $t=T$

$U=\frac{1}{2}Kx_0^2$

$K=0$

$F=-Kx$

$m\frac{d^2x}{dt^2}=-Kx$

$x=x_0 \cos (\omega t),$ where

$\omega=\sqrt{\frac{x}{m}}$

Since at t = 0

$x=x_0$

Spring mass system executes SHM

1 - 2 connected capacitor will be charged at t = 0 $Q=Q_{max}$ I = 0

Disconnected 1 - 2 but Connect 1 - 3

Currenr flows $\frac{dI}{dt}>0$

$I=-\frac{dQ}{dt}$

$-L\frac{dI}{dt}+\frac{Q}{C}=0$

$\frac{d^2 x}{dt}+\frac{Q}{LC}=0 \Leftrightarrow \frac{d^2x}{dt^2}+\frac{k}{m}x=0$

$\omega = \frac{1}{\sqrt{LC}} \Leftrightarrow \omega_{mech}=\sqrt{\frac{k}{m}}$

$Q \leftrightarrow x$

At $t = \frac{T}{4}$

Capacitors are discharged electrical energy has been completely transferred to its magnetic energy associated will inductor

$U_s=\frac{1}{2}LI^2$

$U_E=\frac{Q^2}{2C}$

$U=U_s + U_E = \frac{Qm^2}{2C}=\frac{1}{2}LI_m^2$

Kinetic energy of spring mass

$\leftrightarrow$ magnetic energy (L - C)

$\frac{1}{2}mv^2 \leftrightarrow \frac{1}{2}LI^2$

$v \leftrightarrow I$, $x \leftrightarrow Q$, $m \leftrightarrow L$

$\frac{1}{2}kx^2 \leftrightarrow \frac{Q^2}{2C}$

$k \leftrightarrow \frac{1}{C}$

Total mechanical energy

$P.E. + x.E \Leftrightarrow U_E + U_B =$ Electromaganetic

L = 50 mH, C = 20 $\mu F$

Inactually current is maximum How long does it take to fully charge the capacitor?

$\omega =\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{5 \times 10^{-2} \times 2 \times 10^{-5}}}$

$=10^{2} rad/s$

$T=\frac{2 \pi}{\omega}=6.3 ms$

$T/4 \cong 1.6 ms$

Radio tuner

800 KHz -1200 KHz

Inductance $= 200 \times 10^{-6}$

$=2 \times 10^{-4}H$

$\omega:$ 800 KHz $\Rightarrow$ $5.03 \times 10^{6}$ rad\s

1200 KHz $\Rightarrow$ $7.54 \times 10^{6}$ rad\s

$\omega = \frac{1}{\sqrt{LC}} \Rightarrow c = \frac{1}{L \omega^2}$

If $\omega = 5.03 \times 10^6$

$c=\frac{1}{2 \times 10^{-4} \times 25 \times 10^{-12}}$

$\approx 200 PF$

$\omega = 7.54 \times 10^6$

$c \approx 90 PF$

In an LC circuit C = 64 $\mu F$

$I(t) = 2 \sin (500 t + 0.4)\Delta$

(a) At what time t does the current reach maximum?

$I = I_m$ when $500t + 0.4 = \frac{\pi}{2}$

$t = 2.34 \times 10^{-3}s$

(b) The value of L?

$\omega =500 = \frac{1}{\sqrt{LC}} \Rightarrow L=\frac{1}{16}H$

(c) Total Energy

$\frac{1}{2}LI_m^2 = \frac{1}{2} \times \frac{1}{16} \times 4$

$=\frac{1}{8}$

LC circuit L = 2mH

$c=80 \mu F$

Initial charge $4 /mu C$

$\omega =\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{2 \times 10^{-3} \times 80 \times 10^{-6}}}=2500 rad/s$

$U_E = \frac{Qm^2}{2C}=\frac{16 \times 10^{-12}}{2 \times 8 \times 10^{-5}}$

$I_m=Qm \omega$

$=4 \times 10^{-6} \times 2500 = 10^{-2}A$

$U_{mag}^{max}=\frac{1}{2}LI_m^2=\frac{1}{2}\times 2 \times 10^{-3} \times 10^{-4}$

$=10^{-7}J$

1. Considered LC circuit

2. Charge and current oscillation

$\omega=\frac{1}{\sqrt{LC}}$

3. Established a parallel between L-c circuit and a mechanical circuit

$Q \leftrightarrow x$, $I \leftrightarrow v$, $L \leftrightarrow m$, $c \leftrightarrow 1/x$

electrical $\leftrightarrow$ spring energy

magnetic $\leftrightarrow$ kinetic energy