
# 6.3E: The RLC Circuit (Exercises)


## Q6.3.1

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

1. $$R=3$$ ohms;   $$L=.1$$ henrys;   $$C=.01$$ farads; $$Q_0=0$$ coulombs;  $$I_0=2$$ amperes.

2. $$R=2$$ ohms;   $$L=.05$$ henrys;   $$C=.01$$ farads’; $$Q_0=2$$ coulombs;  $$I_0=-2$$ amperes.

3. $$R=2$$ ohms;   $$L=.1$$ henrys;   $$C=.01$$ farads; $$Q_0=2$$ coulombs;  $$I_0=0$$ amperes.

4. $$R=6$$ ohms;   $$L=.1$$ henrys;   $$C=.004$$ farads’; $$Q_0=3$$ coulombs;  $$I_0=-10$$ amperes.

5. $$R=4$$ ohms;   $$L=.05$$ henrys;   $$C=.008$$ farads; $$Q_0=-1$$ coulombs;  $$I_0=2$$ amperes.

## Q6.3.2

In Exercises 6.3.6-6.3.10 find the steady state current in the circuit described by the equation.

6. $${1\over10}Q''+3Q'+100Q=5\cos10t-5\sin10t$$

7. $${1\over20}Q''+2Q'+100Q=10\cos25t-5\sin25t$$

8. $${1\over10}Q''+2Q'+100Q=3\cos50t-6\sin50t$$

9. $${1\over10}Q''+6Q'+250Q=10\cos100t+30\sin100t$$

10. $${1\over20}Q''+4Q'+125Q=15\cos30t-30\sin30t$$

## Q6.3.3

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 6.3.1 is $I_p={\omega^2RE(t)+(1/C-L\omega^2)E'(t)\over\Delta},$ where $\Delta=(1/C-L\omega^2)^2+R^2\omega^2.$

12. Find the amplitude of the steady state current $$I_p$$ in the $$RLC$$ circuit shown in 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.

## Q6.3.4

In Exercises 6.3.13-6.3.17 plot the amplitude of the steady state current against $$ω$$. Estimate the value of $$ω$$ that maximizes the amplitude of the steady state current, and estimate this maximum amplitude. HINT: You can confirm your results by doing Exercise 6.3.12.

13. $${1\over10}Q''+3Q'+100Q=U\cos\omega t+V\sin\omega t$$

14. $${1\over20}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t$$

15. $${1\over10}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t$$

16. $${1\over10}Q''+6Q'+250Q=U\cos\omega t+V\sin\omega t$$

17. $${1\over20}Q''+4Q'+125Q=U\cos\omega t+V\sin\omega t$$