Question

# A circuit is constructed with four resistors, one inductor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 48 Ω, R3 = 100 Ω and R4 = 130 Ω.

A circuit is constructed with four resistors, one inductor, one battery and a switch as shown. The values for the resistors are: R1 = R2 = 48 Ω, R3 = 100 Ω and R4 = 130 Ω. The inductance is L = 330 mH and the battery voltage is V = 12 V. The positive terminal of the battery is indicated with a + sign.

1)

The switch has been open for a long time when at time t = 0, the switch is closed. What is I4(0), the magnitude of the current through the resistor R4 just after the switch is closed?

2)

What is I4(?), the magnitude of the current through the resistor R4 after the switch has been closed for a very long time?

3)

What is IL(?), the magnitude of the current through the inductor after the switch has been closed for a very long time?

4)

After the switch has been closed for a very long time, it is then opened. What is I3(topen), the current through the resistor R3 at a time topen = 5.6 ms after the switch was opened? The positive direction for the current is indicated in the figure.

5)

What is VL,max(closed), the magnitude of the maximum voltage across the inductor during the time when the switch is closed?

6)

What is VL,max(open), the magnitude of the maximum voltage across the inductor during the time when the switch is open?

Something we need to understand before solving the problem is that when we have an inductor and the problem have a switch that will be open/closed for long/short periods of time the inductor acts like a huge resistor that basically drains all the energy and dont let it flow when the switch is recently opened (t=0) and will act like a wire (as if it was not even on the picture) when the switch was opened a long time ago (t=infinity). This will help us solve some of the questions.

Question 1

To calculate the current in resistor 4 we have

$I_4=\frac{V}{R_T}$

And RT is the sum of all the resistors, but, since t=0 no current will flow after the inductor so we ignore R2

Question 2

Same as last question but this time we do consider R2 since the inductor is acting as just a wire (we ignore it)

Question 3

Again like last question the inductor acts like a wire so the current throught it will be the same as the current on R2 (because they are in series) so we basically need I2 as if there was no inductor there. In order to find it we need the voltage that is going on the second loop that splits (parallel) between R3 and R2 and the current for this case is the same as last problem since the conditions are the same

Now we calculate I2 since is the same current as the current in the inductor and remember that we calculated the voltage that goes to both R3 and R2 so V23=V2=V3

Question 4

Now to find current with a certain specific time we go by:

I made it bigger so you can see it better. Now we need since Imax is the current we calculated before since is the current with the switch open for the maximum time and t is the given time (in seconds). We have then

And R is the sum of the resistors in the loop which are R2 and R3 and this time they are in series (only for this loop not the circuit) so we have

This is a complex number so i will keep it for the formula to avoid having a serious difference in the result due to using less significant numbers.

Is negative because it goes in the opposite direction as of what the problem says is positive

Question 5

For this question we have that

And since is max closed we take as if it were t=0 (same current as question 1) and the resistor R3 is parallel to the inductor the voltage in both of them will be the same and like i said in question 1 we ignore R2, so we have.

Question 6

This time we follow the same procedure as last question but this time we have max open, so, we take the I from question 3 (same conditions) and the resistance now includes R2 so we need the total resistance R23 that are in parallel for this loop so we have

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This was a really long question, i hope you could understand everything.

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