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Question

Switch S in Figure is closed at time t = 0, initiating the buildup of current in the 15.0 mH inductor and the 20.0Ω resistor. At what time is the emf across the inductor equal to the potential difference across the resistor?
1782810_1570841bf6634a5093aa61644251247d.PNG

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Solution

Equation i=ER(1et/τL) (rise of current) applies, and the problem requires

iR=Ldidt=εiR

at some time t (where Eq. Ldidt+Ri=E (RL circuit)
has been used in that last step). Thus, we have 2iR=ε or

ε=2iR=2[εR(1et/τL)]R=2ε(1et/τL)

where Eq. τL=LR (time constant) gives the inductive time constant as
τL=L/R. We note that the emf ε cancels out of that
final equation, and we are able to rearrange (and take the natural log) and
solve. We obtain t=0.520ms


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