Class Discussion Activity 19

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Exercises

(1)

Which CANNOT be the current $I(t)$ in an RLC circuit with a constant voltage source?

(a): $I(t)=(5-t)e^{-2t}$
(b): $I(t)=e^{-t}\left (\cos {(800t)}-\sin {(800t)}\right )$
(c): $I(t)=e^{-50t}-e^{-100t}$
(d): $I(t)=1+(5-t)e^{-2t}$
(e): $I(t)=e^{-15t}+5e^{-2t}$
(f): $I(t)=e^{-20t}\cos {(100t)}$

(2)

Suppose an RLC circuit with a resistor of resistance $10$ ohms, an inductor of inductance $1$ henry and a capacitor of capacitance $0.04$ farads is hooked-up to a $9$-volt battery. Suppose the current in the circuit, $I(t)$ (in amps) at $t\geq 0$ seconds, satisfies $I(0)=5$, $I'(0)=0$. Find $I(0.1)$. Round to the nearest hundredth.

(3)

Suppose an RLC circuit plugged into the wall, consists of an inductor of inductance $L$ henries, a resistor of resistance $R=80$ ohms, a capacitor of capacitance is $C=4\times 10^{-5}$ farads, and suppose a transient solution for the current in the circuit is $I(t)=0.75 \cos {(120\pi t)}+12te^{-625t}$. Find $L$ to the nearest hundredth.

(4)

An ideal LC circuit (without a voltage source) assumes the resistance of the conducting wire is $0$, and has just two components, an inductor (of inductance $L$ henries in SI units) and a capacitor (of capacitance $C$ farads in SI units). In any practical application, the resistance of the conducting wire can’t be $0$, but it can be small enough that it takes a very LONG time for current to dissipate. An ideal LC circuit is modeled by $LI''(t)+I(t)/C=0$ where $I(t)$ is the current at time $t$ seconds, and is a simple harmonic oscillator (where the “oscillation" is the alternating current). Suppose an ideal LC circuit has a natural frequency of $12$ hz. Find the product $LC$ (in SI units) to one significant figure.

(5)

Let $\omega >0$ be a constant. Suppose a voltage source of $V(t)=120 \sin (\omega t)$ volts at time $t\geq 0$ seconds is applied to an RLC circuit, where the inductor has inductance $0.1$ henries, the resistor has resistance $2$ ohms and the capacitor has capacitance $0.01$ farads. Find $\omega $ such that the steady state solution has the largest possible amplitude. Round to the nearest hundredth.