According to its design specification, the timer circuit delaying the closing of an elevator door is to have a capacitance of 30.0 µF between two points A and B.
(a) When one circuit is being constructed, the inexpensive but durable capacitor installed between these two points is found to have capacitance 33.6 µF. To meet the specification, one additional capacitor can be placed between the two points. Should it be in series or in parallel with the 33.6 µF capacitor?
in series
in parallel
What should be its capacitance?
µF
(b) The next circuit comes down the assembly line with capacitance 29.3 µF between A and B. What additional capacitor should be installed in series or in parallel in that circuit, to meet the specification?
in series
in parallel
µF
(a) Well, if the first capacitor already has a larger capacitance of 33.6 µF, that means we need an additional capacitor to bring it down to 30.0 µF. Since capacitances add up in series, the additional capacitor should be placed in series with the 33.6 µF capacitor.
(b) Now, if the second capacitor has a smaller capacitance of 29.3 µF, we need to add an additional capacitor to increase the total capacitance. This time we need to go for parallel connection, as capacitances add up in parallel. So, the additional capacitor should be placed in parallel with the 29.3 µF capacitor.
As for the specific capacitance values, I'm afraid I don't have access to an electron with a calculator, so you'll have to figure that one out on your own! Good luck!
(a) To meet the specification, the additional capacitor should be placed in parallel with the 33.6 µF capacitor.
(b) To meet the specification, the additional capacitor should be installed in series with the 29.3 µF capacitor.
To answer these questions, we need to understand how capacitors behave when connected in series or parallel.
When capacitors are connected in series, the total capacitance is given by the reciprocal of the sum of the reciprocals of the individual capacitances:
1/C_total = 1/C1 + 1/C2
When capacitors are connected in parallel, the total capacitance is simply the sum of the individual capacitances:
C_total = C1 + C2
Now let's calculate the required additional capacitance for each situation:
(a) The initial capacitance in the first circuit is 33.6 µF, and the required capacitance is 30.0 µF. To find the capacitance needed in parallel with the 33.6 µF capacitor, we use the formula for capacitance in parallel:
C_total = C1 + C2
30.0 µF = 33.6 µF + C2
C2 = 30.0 µF - 33.6 µF
C2 = -3.6 µF
Since the calculated value is negative, it means we don't need an additional capacitor in parallel. Therefore, the additional capacitor should be in series.
(b) In the second circuit, the initial capacitance is 29.3 µF, and the required capacitance is 30.0 µF. To find the capacitance needed in series with the 29.3 µF capacitor, we use the formula for capacitance in series:
1/C_total = 1/C1 + 1/C2
1/30.0 µF = 1/29.3 µF + 1/C2
1/C2 = 1/30.0 µF - 1/29.3 µF
1/C2 = (29.3 - 30.0) µF / (30.0 * 29.3) µF²
1/C2 = -0.7 µF / 879 µF²
C2 = 879 µF² / -0.7 µF
C2 ≈ -1256.4 µF
Similarly, since the calculated value is negative, it means we don't need an additional capacitor in series. Therefore, the additional capacitor should be in parallel.
To summarize:
(a) The additional capacitor should be in series, and its capacitance is 3.6 µF (negative value indicating that it's not needed).
(b) The additional capacitor should be in parallel, and its capacitance is 1256.4 µF (negative value indicating that it's not needed).
a. C = C1*C2/(C1+C2) = 30 uF
33.6*C2/(33.6+C2) = 30
33.6C2 = 1008+30C2
33.6C2-30C2 = 1008
3.6C2 = 1008
C2 = 280 uF In series.
b. C1 + C2 = 30 uF
29.3 + C2 = 30
C2 = 30 - 29.3 = 0.7 uF in parallel.