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# 6.3E: The RLC Circuit (Exercises)

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In Exercises [exer:6.3.1} -[exer:6.3.5} find the current in the $$RLC$$ circuit, assuming that $$E(t)=0$$ for $$t>0$$.

[exer:6.3.1] $$R=3$$ ohms;   $$L=.1$$ henrys;   $$C=.01$$ farads; $$Q_0=0$$ coulombs;  $$I_0=2$$ amperes.

[exer:6.3.2] $$R=2$$ ohms;   $$L=.05$$ henrys;   $$C=.01$$ farads’; $$Q_0=2$$ coulombs;  $$I_0=-2$$ amperes.

[exer:6.3.3] $$R=2$$ ohms;   $$L=.1$$ henrys;   $$C=.01$$ farads; $$Q_0=2$$ coulombs;  $$I_0=0$$ amperes.

[exer:6.3.4] $$R=6$$ ohms;   $$L=.1$$ henrys;   $$C=.004$$ farads’; $$Q_0=3$$ coulombs;  $$I_0=-10$$ amperes.

[exer:6.3.5] $$R=4$$ ohms;   $$L=.05$$ henrys;   $$C=.008$$ farads; $$Q_0=-1$$ coulombs;  $$I_0=2$$ amperes.

[exer:6.3.6] $${1\over10}Q''+3Q'+100Q=5\cos10t-5\sin10t$$

[exer:6.3.7] $${1\over20}Q''+2Q'+100Q=10\cos25t-5\sin25t$$

[exer:6.3.8] $${1\over10}Q''+2Q'+100Q=3\cos50t-6\sin50t$$

[exer:6.3.9] $${1\over10}Q''+6Q'+250Q=10\cos100t+30\sin100t$$

[exer:6.3.10] $${1\over20}Q''+4Q'+125Q=15\cos30t-30\sin30t$$

[exer:6.3.11] Show that if $$E(t)=U\cos\omega t+V\sin\omega t$$ where $$U$$ and $$V$$ are constants then the steady state current in the $$RLC$$ circuit shown in Figure [figure:6.3.1} is $I_p={\omega^2RE(t)+(1/C-L\omega^2)E'(t)\over\Delta},$ where $\Delta=(1/C-L\omega^2)^2+R^2\omega^2.$

[exer:6.3.12] Find the amplitude of the steady state current $$I_p$$ in the $$RLC$$ circuit shown in Figure [figure:6.3.1} if $$E(t)=U\cos\omega t+V\sin\omega t$$, where $$U$$ and $$V$$ are constants. Then find the value $$\omega_0$$ of $$\omega$$ maximizes the amplitude, and find the maximum amplitude.

[exer:6.3.13] $${1\over10}Q''+3Q'+100Q=U\cos\omega t+V\sin\omega t$$

[exer:6.3.14] $${1\over20}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t$$

[exer:6.3.15] $${1\over10}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t$$

[exer:6.3.16] $${1\over10}Q''+6Q'+250Q=U\cos\omega t+V\sin\omega t$$

[exer:6.3.17] $${1\over20}Q''+4Q'+125Q=U\cos\omega t+V\sin\omega t$$