At t=0 s, the current in the circuit in FIGURE EX30.35 is I0. At ...

Question

At t=0 s, the current in the circuit in FIGURE EX30.35 is I0. At what time in μs is the current 1/ 2 I₀?

Accepted Answer

Welcome back. Everyone. In this problem. A circuit is constructed from resistors and an inductor as follows. The current at T equals zero seconds is I calculate the time after T equals zero seconds when the current is a quarter of IO for our answer choices, A is 5.5 microseconds. B is 0.23 milliseconds. C is 29 microseconds and D is 55 microseconds. Now, what, what are we trying to figure out here? Well, notice that in our circuit we don't see any EMF connected. So we'll have an exponential decay of current. OK? Because we have our doctor here. So in other words, our current is going to be decaying looking something like that. And the way that the current is being decayed can be related to the formula. Recall that it decays according to the expression that says the current at any point in time is going to be equal to our initial current and not multiplied by E to the power of negative T divided by Tau where T is our time and Tau is the time constant. In this case, for an aerial circuit is equal to the ratio of the inductance to the resistance. And it's basically a measure of how quickly the current approach is the steady state. Now, we're trying to find the time when the current is a quarter of our initial current. In other words, our instantaneous current equals a quarter of I kno So when, let's make a note of that, when are instantaneous current drops to 1/4 of its initial value. In other words, it equals a quarter of a knot. Then if we put that into our formula that tells us, then that a quarter of I nat is going to be equal to a knot multiplied by E to the negative T divided by ta. Now, here's the thing that means then if we can use this to try and figure out what our time is, our time T is then we should be able to solve. But the problem here is that we don't know what Tau is, we don't know what that value is yet. So first let's try to work with our equation a bit to get to get it in a simpler form to solve for the time. And then we'll try to find the value of Tau. So solving for tea, OK? Because that's essentially what we really want oving for tea. Then here in our equation, notice both sides have our initial value. So we can cancel that out to find that a quarter is going to be equal to E raised to the power of negative T divided by Tau. Now using what we know about the definition of a logarithm, that means that means then, OK, let me write this properly. That means then that negative T is going to be equal to Tau multiplied by the natural logarithm of a quarter. OK. By definition, no, we can solve because we all, we all we have left to find this talk. OK? So now when we do that, OK. So for to recall that Tau is the ratio of inductance to resistance and that is the equivalent resistance in our circuit. Notice that in our circuit, the resistors are connected in series. So the equivalent resistance would be the sum of both resistances. So therefore, Tau is going to be equal to 20 million henries 20 times 10 to the negative three henries divided by 200 arms plus 300 arms. And when we work that out, if we were to put that into our calculator, that's 20 million henries divided by 500 ohms which equals four times 10 to the negative fifth seconds. OK. So that's the value for our talk. So what does that tell us? Then that tells us then that our time is going to be equal to Tau divided by Ln multiplied by a quarter. If we're solving for a time can divide both sides by negative one. So that means positive T is going to be equal to negative Tau multiplied by L of a quarter. But again, we can't have a negative time, right? Because it's, it's the, the time must be after zero seconds. So the time is going to be equal to positive Tau multiplied by L of a quarter, which is going to then be four times 10 to the negative fifth seconds multiplied by of a quarter. And when we put that into our calculator, OK, we find that the time is equal to 5.5 times 10 to the negative fifth seconds. In standard form, we can rewrite that as 55 microseconds. Therefore, the time after T equals zero seconds when the current is a quarter of its initial value is 55 microseconds. When we look back on our answer choices, that would have been answer choice. D Thanks a lot for watching everyone. I hope this video helped.

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At t=0 s, the current in the circuit in FIGURE EX30.35 is I0. At what time in μs is the current 1/ 2 I₀?

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