Question

Figure 30-29 shows three circuits with identical batteries, inductors, and resistors. Rank the circuits, greatest first, according to the current through the resistor labeled R (a) long after the switch is closed, (b) just after the switch is reopened a long time later, and (c) long after it is reopened

Short answer

The ranks of the circuits according to the current through the resistor R,

1. Long after the switch is closed (1), (2), and (3) all tie(zero).
2. Just after the switch is reopened a long time later are (1), (2) tie, and then (3).
3. Long after it is reopened are (1), (2), and (3) all tie(zero).

Step-by-step solution

Given

1. Fig.30-29.
2. Three circuits with identical batteries, inductors, and resistors.

Determining the concept

Using the properties of resistors in parallel and series, the properties of the inductors and applying to the given Fig.30-29, find the ranks of the circuits according to the current through the resistor R; long after the switch is closed, just after the switch is reopened a long time later, and long after it is reopened.

1. Formula:

The current through the circuits is,

$i=\frac{\epsilon }{R}$

(a) Determining the ranks of the circuits according to the current through the resistor R Long after the switch is closed

Long after the switch is closed, then the circuits reach their steady-state$\left(\epsilon /R\right)$ due to the inductor and the inductor starts to act like the conducting wire, that is, its resistance is minimum or zero. So, the current from the battery and its neighboring resistor prefers to pass through the inductor, and hence, the current through the other resistors is zero. This happens similarly in all the given circuits.

Therefore, the ranks of the circuits according to the current through the resistor R are (1), (2), and (3) all tie.

Hence, the ranks of the circuits according to the current through the resistor R. Long after the switch is closed (1), (2), and (3) all tie(zero).

(b) Determining the ranks of the circuits according to the current through the resistor R Just after the switch is reopened a long time later, and

Just after reopening the switch, the current start dropping a little bit from the steady-state current$\epsilon /R$ , and in the (1) circuit the inductor is in series with the central resistor. Also, in the (2) circuit the inductor is in series with two central resistors. As they are in series and all the given circuits with identical batteries, inductors, and resistors, the current is the same. But in the (3) circuit, the resistors are in parallel. Therefore, the current through the inductor splits between both resistors. Hence the current through R is less as compared with (1) and (2).

Therefore, the ranks of the circuits according to the current through the resistor R are (1), (2) tie, and (3).

Hence, The ranks of the circuits according to the current through the resistor R. Just after the switch is reopened a long time later are (1), (2) tie, and then (3).

(c) Determining the ranks of the circuits according to the current through the resistor R Long after it is reopened

A long time after the reopening, all three circuits are in steady-state with zero current.

Therefore, the ranks of the circuits according to the current through the resistor R are (1), (2), and (3) all tie.

Hence, the ranks of the circuits according to the current through the resistor R. Long after it is reopened are (1), (2), and (3) all tie(zero).