An LR circuit (see Figure 1) contains an inductor (1 H) and a resistor (0.5 Ω) linked in s...

An LR circuit (see Figure 1) contains an inductor (1 H) and a resistor (0.5 Ω) linked in series with an electromotive force E(t) = cos π/t. Use a numerical solver to sketch the graph of the current in the circuit versus time for each of the initial conditions 7(0) = −1, −0.75, −0.5,…, 1 on the time interval [0, 20], What is the frequency of the emf? What appears to be the frequency of the steady-state response (see Exercise)?

An RC circuit (see Figure 2) contains a resistor (10 Ω) and a capacitor (0.2 F) linked in series with an electromotive force E(t) = 10 sin 2πt.

(a) Use a numerical solver to sketch the graph of the charge on the capacitor versus time for each of the initial conditions Q(0) = − 1, −0.75, −0.5,…, 1 on the time interval [0, 15]. What appears to happen to each of the solutions with the passage of time? Explain what might be meant by a steady-state response.

(b) What is the frequency of the emf? What appears to be the frequency of the steady-state response? Hint: Recall that in the function f(t) = sin wt, w is called the angular frequency, the period is given by T = 2π/w, and the frequency is the reciprocal of the period. Think of the frequency as the number of cycles per second, a unit known as Hertz (Hz).

Figure 1 An LC circuit

Figure 2 An RC circuit