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When you have learned to integrate, try to analyze the process of shorting a circuit made up of a coil and a resistor connected to a power supply with constant e.m.f., i.e. the dependence of the current on time. Assume the coil to be without a ferromagnetic core. |
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Answer» Divide by R and introduce the notation the stationary current, and `L//R=tau`, the relaxation time Me obtain the equation `I_(M)-tau (di)/(dt)= or -tau (dI)/(dt)=i-I_(M)` Multiplying by dt, we obtain `I-I_(M) dt, or (di)/(i-I_(M))=-(dt)/(tau)` INTEGRATING we obtain `int (dt)/(i-I_(M))=-1/tau int dt,"YIELDING ln "(i-I_(M))=-t/tau, ln C`. Where C is the INTEGRATION constant. Taking antilogarithms, we obtain `(t-I_(M))/(C)=` When l= 0, the current is l= (0), so `C =I_M.` After some simple transformations we obtain `i=I_(M) (1-e^(-t//tau))` The graph of this function is shown in 43.12, Fig. 43.6. |
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