Chapter 30 Problem 26

Problem

In this circuit, determine the current in each resistor $I_{1}$ , $I_{2}$ , $I_{3}$ at the moment

(a) the switch is closed,

(b) a long time after the switch is closed.

After the switch has been closed for a long time, and then reopened, what is each current

(c) just after it is opened,

(d) after a long time?

Solution

(a)

At the moment the switch is closed, no current will flow through the inductor. Therefore, the resistors $R_{1}$ and $R_{2}$ can be treated as in series.

${\mathcal {E}}=I(R_{1}+R_{2})\rightarrow I_{1}=I_{2}={\frac {\mathcal {E}}{R_{1}+R_{2}}},I_{3}=0$

(b)

A long time after the switch is closed, there is no voltage drop across the inductor so resistors $R_{2}$ and $R_{3}$ can be treated as parallel resistors in series with $R_{1}$ .

$I_{1}=I_{2}+I_{3}$

${\mathcal {E}}=I_{1}R_{1}+I_{2}R_{2}$

$I_{2}R_{2}=I_{3}R_{3}$

${\frac {{\mathcal {E}}-I_{2}R_{2}}{R_{1}}}=I_{2}+{\frac {I_{2}R_{2}}{R_{3}}}$

$I_{2}={\frac {{\mathcal {E}}R_{3}}{R_{2}R_{3}+R_{1}R_{3}+R_{1}R_{2}}}$

$I_{3}={\frac {I_{2}R_{2}}{R_{3}}}={\frac {{\mathcal {E}}R_{2}}{R_{2}R_{3}+R_{1}R_{3}+R_{1}R_{2}}}$

$I_{1}=I_{2}+I_{3}={\frac {{\mathcal {E}}(R_{3}+R_{2})}{R_{2}R_{3}+R_{1}R_{3}+R_{1}R_{2}}}$