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Consider the circuit shown in Figure P20.43. Take $\varepsilon=6.00 \mathrm{V}, L=8.00 \mathrm{mH},$ and $R=4.00 \Omega$ (a) What is the inductive time constant of the circuit? (b) Calculate the cur- rent in the circuit 250 . $\mu$ s after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0$\%$ of its maximum value?

a. 0.002 \mathrm{s}

b. 0.176 A

c. 1.50 \mathrm{A}

d. 3.22 \mathrm{ms}

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for part A were asked to figure out what the induced time constant of the circuit is. Well, the induced time Constant towel is equal to the induct in CE, divided by the resistance. Since we know both values for that, we'll go ahead and plug them into this expression, and we find that this is equal to two times 10 to the minus three in the units. Here are seconds we'll Box said it is a solution for part A for part B were asked to find the current after a given amount of time here. Well, the current is equal to in, uh, the IMF or the battery here, which is six volts divided by the resistance. Multiply by one minus E to the minus t over town, which we just found in part A. We're told that the time that is elapsing here is 250 microseconds, which is 250 times 10 to the minus six seconds. So plugging that value into this expression, as well as the six bolts for that six bolts for the battery and the four homes for the resistance, we find that the current here is zero 0.176 and the units here and pierce we can box. It is our solution. Your part B part, see, wants to know the maximum current. That's pops. That is possible. Here. We'll call this I'm Max. The maximum current is equal to Absalon, divided by resistance. That's just owns law plugging those values family find that this is equal to 1.5 and peers weakened box. And then is her solution for parts seat And then finally, part D. It wants us to figure out how long it will take for the current to reach 80% of it to maximum value. So we want to figure out what it 0.8% of my next. So this is going to be equal to the maximum value, which is, uh, when this we'll call it. I'm Max again because it's absolute over our right, which we were, uh, found to be I maxim part see, multiplied by one minus e to the minus t over the time of constant town. So solving for tea time I'm axes are gonna cancel here that cancels with that, we find that this is equal to well life so Let's do this one step at a time just so we can be explicit here with what we're doing. So e to the minus t over Towel, huh? I was here to this tea here, so we have enough room, so e to the minus t over tell it's equal to would minus 0.8. Okay. So we can take the natural log of both sides to get rid of e to the minus t over towel that will, just to the natural log of e to the minus. C over. Tower is minus two over towel. And then the right side will be the natural log of, uh 0.2. So we're gonna have minus t over. Towel is equal to the natural log of 0.2. Let's go ahead and start a new page here so we can solve for tea. We find that the time here is equal to minus towel times the national log 0.2. Okay. Carrying out this expression, we find that the time is equal to 3.22 times 10 to the minus three seconds Come. Bak said it is the solution for party