The R-L Circuit: Responding to Changes Learning Goal: To understand the behavior of an inductor in a series R-L circuit. In a circuit containing only resistors, the basic (though not necessarily explicit) assumption is that the current reaches its steady-state value instantly. This is not the case for a circuit containing inductors. Due to a fundamental property of an inductor to mitigate any "externally imposed" change in current, the current in such a circuit changes gradually when a switch is closed or opened. Consider a series circuit containing a resistor of resistance R and an inductor of inductance L connected to a source of emf & and negligible internal resistance. The wires (including the ones that make up the inductor) are also assumed to have negligible resistance. (Figure 1) Let us start by analyzing the process that takes place after switch S₁ is closed (switch S2 remains open). In our further analysis, lowercase letters will denote the instantaneous values of various quantities, whereas capital letters will denote the maximum values of the respective quantities. Note that at any time during the process, Kirchhoff's loop rule holds and is, indeed, helpful: gure ER-L-0 ww-oooo ** 1 of 2 > ▾ Review I Now that we have a feel for the state of the circuit in its steady state, let us obtain the expression for the current in the circuit as a function of time. Note that we can use the logelagoing around counterclockwise): EUR-L=0 Note as well that R=iR and L=L.Using these equations, we can get, after some rearranging of the variables and making the subsitution == Integrating both sides of this equation yields i(t) = Use this last expression to obtain an expression for i(t). Remember that z=-i and that io-i(0) = 0. Express your answer in terms of E, R, and L. You may or may not need all these variables. Use the notation exp(x) for ². Part J E Submit Previous Answers Request Answer E RR x Incorrect; Try Again; 5 attempts remaining as 15: ? Rr i(6r) max =-dt Just as in the case of R-C circuits, the steady state here is never actually reached: The exponential functions approach their limits asymptotically as t→∞ However, it usually does not take very long for the value of; to get very close to its presumed limiting value. The next several questions illustrate this point. Find the ratio of the current : (t) at time t = 6r to the maximum current Imax= Express your answer numerically, using three significant figures. > G3 ? z=20-Rt/L Note that the quantity L/R has dimensions of time and is called the time constant (you may recall similar terminology applied to R-C circuits). The time constant is often denoted by . Using T, one can write the expression i(t)=(1-e-Rt/ i(t)=(1-e-t/) 9 of 15