P31_055 - the switch is closed, the inductor prevents the...

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55. (a) We assume i is from left to right through the closed switch. We let i 1 be the current in the resistor and take it to be downward. Let i 2 be the current in the inductor, also assumed downward. The junction rule gives i = i 1 + i 2 and the loop rule gives i 1 R L ( di 2 /dt ) = 0. According to the junction rule, ( di 1 /dt )= ( di 2 /dt ). We substitute into the loop equation to obtain L di 1 dt + i 1 R =0 . This equation is similar to Eq. 31-48, and its solution is the function given as Eq. 31-49: i 1 = i 0 e Rt/L , where i 0 is the current through the resistor at t = 0, just after the switch is closed. Now just after
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Unformatted text preview: the switch is closed, the inductor prevents the rapid build-up of current in its branch, so at that moment i 2 = 0 and i 1 = i . Thus i = i , so i 1 = ie Rt/L and i 2 = i i 1 = i 1 e Rt/L . (b) When i 2 = i 1 , e Rt/L = 1 e Rt/L = e Rt/L = 1 2 . Taking the natural logarithm of both sides (and using ln(1 / 2) = ln 2) we obtain Rt L = ln 2 = t = L R ln 2 ....
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