# RL Circuits

RL Circuits

In Section 28-8 we saw that if we suddenly introduce an emf’l!; into a single-loop circuit containing a resistor R and a capacitor C, the charge on the capacitor does  not build up immediately to its final equilibrium value C’l!; but approaches it in an exponential fashion:

The rate at which the charge builds up is determined by the capacitive time constant “c,

If we suddenly remove the emf from this same circuit, the charge does not immediately fall to zero but approaches zero in an exponential fashion

The time constant TC describes the fall of the charge as well as its rise. An analogous slowing of the rise (or fall) of the current occurs if we introduce  an emf ~ into (or remove it from) a single-loop circuit containing a resistor Rwandan inductor L. When the switch S in is closed on a, for example, the current in the resistor starts to rise. If the inductor were not present, the current would rise rapidly to a steady value ‘l!;IR. Because of the inductor, however, a elfinduced mf ‘l!;L appears in the circuit; from Lenzs law. this emf opposes the rise of the current, which means that it opposes the battery emf ‘l!; in polarity. Thus, the current in the resistor responds to the difference between two emfs. a constant one  ‘l!; due to the battery and a variable one ‘f!:L (= -L dildt; due to self-induction. Aslong as ‘€L is present, the current in the resistor will be less than <&IR. As time goes on, the rate at which the current increases becomes less rapid and  the magnitude of the self-induced emf, which is proportional to dilute, becomes smaller. Thus, the current in the circuit approaches ‘l!;IR asymptotically. We can generalize these results as follows:

~ Initially, an inductor acts to oppose changes in the current through it. A long time later, it acts like ordinary connecting wire.  Now let us analyze the situation quantitatively. With the switch S in Fig. 31-17thrown to a, the circuit is equivalent to that of Fig. 31-18. Let us apply the loop rule. starting at point x in this figure and moving clockwise around the loop along with c urrent i.1. Resistor. Because we move through the resistor in the direction of current i, the electric potential decreases by iR. Thus, as we move from point x to point y, we encounter a potential change of -iR.  . Inductor. Because current i is changing. there is a .self-induced emf <&L’ in the inductor. The  magnitude of <&L is given by Eq. 31-37 as L dildt. The direction of  ‘€,L is upward in Fig. 31-18 because current i is downward through the inductor and increasing. Thus, as we move from point y to point :, opposite the direction
of ‘l!;L’ we encounter a potential change of – L dildt, 3. Battery. As we move from point: back to starting point x, we encounter a potential  change of +’€, due to the battery’s emf.