Problem 4:

1. In some alternating current circuit (not the one shown below), a 0.5 H inductor has a reactance of 400 in the circuit it is in. What is the oscillatory angular frequency in the circuit? What is the frequency (not angular?)
2. Another circuit, shown here, has a 10 V battery, resistors R₁ = 10 and R₂ = 20 and a 0.3 H inductor. Say the switch is closed and left closed a long time. What then are the currents through each of the resistors and the inductor?

1. All we have to recall is the definition of inductive reactance, X_L = L

   Thus    X_L = 400 = (0.5 H)

      = 400 /(0.5 H) = 800 rad/sec

   The frequency f is found from = 2f

      f = /2 = 127.3 Hz

2. This is not an AC system, but rather it is a DC system. With steady currents (DC) an inductor is just a piece of wire, with no resistance ( = 0 so X_L = 0). Thus the parallel combination of R₂ and L has an equivalent resistance given by

   1/R_eq = 1/0 + 1/R₂

   or R_eq = 0 !

   This means the circuit is equivalent to R₁ alone, in series with the battery, so we have

   I₁ = 10V/10 = 1 A

   All of this current also goes through the inductor, and none goes through R₂ (since it can pass through the inductor without resistance).