Exercise:
The circuit in the circuit diagram below consists of three resistors R_RaO R_RbO R_RcO and an inductor LLO connected to a VO voltage supply. center includegraphicswidth.mm#image_path:lr-circuit# center abcliste abc What are the currents through the three resistors I_ I_ and I_ and the voltage V_L across the inductor immediately after closing the switch? abc What are the currents and the voltage V_L after a very long time? abc What are the currents and the voltage V_L immediately after reopening the switch? abcliste

Solution:
abcliste abc Immediately after closing the switch there is no current through the inductor Lenz's law i.e. resultI_. It follows that I_I_. For the left loop containing the voltage supply and the resistors R_ and R_ we find V_ R_ I_+R_ I_ R_+R_ I_ I_ I_ IaF fracVRa+Rc resultIaP The voltage across R_ and L in series is the same as the voltage across R_ we find for the induced voltage across the inductor V_L R_ I_ - R_ I_ R_ I_-R_ VLF RctimesIa resultVLP abc After a long enough time the system has reached a steady state i.e. the current does not change any more. This means that the voltage across the inductor has disappeared: resultV_L. The circuit can be considered to be an ordinary resistor circuit and the inductur can be ignored. Resistors R_ and R_ are in parallel to each other and together in series to R_. The total resistance of the circuit is sscRtot R_+leftfracR_+fracR_right^- R_+fracR_ R_R_+R_ RtF The total current I_ is then I_ fracV_sscRtot IafF fracVtimesRb+RcRatimesRb+RatimesRc+RbtimesRc resultIafP The total current is split up between R_ and R_ in the inverse ratio of the resistances: I_ IbfF IaftimesfracRcRb+Rc resultIbfP I_ IcfF IaftimesfracRbRb+Rc resultIcfP abc Immediately after reopening the switch the current through the inductor is still equal to the steady-state current I_IbfP. The right loop is still closed but the left loop is open so we must have I_ result I_ I_ resultIbfP The induced voltage across the inductor is V_L R_ I_+R_ I_ VLroF Rb+RctimesIbfP resultVLroP abcliste