PHYS 208 Lecture 21

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Exam next Tuesday (11/15) over Ch. 27-30

RL-Circuits

Now we add our new inductor elements to circuits

Remember that $V=L\frac{\mathrm{d}I}{\mathrm{d}t}$, so using kirchoff's Rule, we can use the voltage drop across a simple circuit with a battery, an inductor, and a resistor:

${ℰ}\frac{\mathrm{d}I}{\mathrm{d}t}-IR=0$

This differential equation looks remarkably similar to the equation for charging a capacitor:

$V-\frac{Q}{C}-IR=V-\frac{Q}{C}-R\frac{\mathrm{d}Q}{\mathrm{d}t}$

So the equation for the current in an RL-circuit is

$\frac{{ℰ}}{R}(1-{\mathrm{e}}^{-\frac{R}{L}t})$

For RL time-constant $\tau =L/R$,
$\frac{{ℰ}}{R}(1-{\mathrm{e}}^{-\frac{t}{\tau }})$

LC-Circuits

When we take a charged capacitor and put it into a circuit with an inductor, replacing $I=\frac{\mathrm{d}Q}{\mathrm{d}t}$, the equation for the voltage becomes

$-\frac{Q(t)}{C}-L\frac{{\mathrm{d}}^{2}Q(t)}{\mathrm{d}{t}^2}=0$

Recall Simple Harmonic Motion from PHYS 218, the current in this circuit will oscillate. The angular frequency is given by

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

mechanical frequency [Hz] of system given by:

$f=\frac{\omega }{2\pi }$

Thus the harmonic equation for an LC circuit is

$\begin{array}{rl}q& =Q_{max}\cos (\omega t+\varphi )\\ i& =-\omega Q_{max}\sin (\omega t+\varphi )\end{array}$

Note:  $A\cos (\omega t+\varphi )=\alpha \sin \omega t+\beta \cos \omega t$ (alternate forms of each other)

Energy in Circuit

$U_E=\frac{1}{2}\frac{Q_{max}^2}{C}=\frac{1}{2}L{I}_{max}^2=U_B$

Conservation of energy between conversion from electric field to magnetic field.

LRC Damping Circuit

If we put a resistor into the circuit, it dissipates power, producing a damping effect:

$-L\frac{{\mathrm{d}}^{2}Q}{\mathrm{d}{t}^2}-R\frac{\mathrm{d}Q}{\mathrm{d}t}-\frac{Q}{C}=0$

The solution for a damped differential equation is

$\begin{array}{rl}q=& A{\mathrm{e}}^{-\frac{R}{2L}t}\cos ({\omega }^{\prime }t+\varphi )\\ {\omega }^{\prime }& =\sqrt{\frac{1}{LC}-\frac{R^2}{4L^2}}\end{array}$