Three identical capacitors are connected so that their maximum equivalent
capacitance is 15 μF. There are three other ways to combine all three capacitors in a circuit. What is the total of the equivalent capacitances for each arrangement?
in parallel: C = C1 + C2 + C3
in series: 1/C = 1/C1 + 1/C2 + 1/C3
two in parallel: 1/C = 1/(C1+C2) + 1/C3
you can see what C1,C2,C3 must be, and then get the others.
oh yeah. There is a 3rd way:
two in series, in parallel with the third.
C = C1 + 1/(1/C2 + 1/C3)
To find the total of the equivalent capacitances for each arrangement, we need to consider the different ways the capacitors can be combined.
Arrangement 1: All three capacitors are connected in parallel. In this case, the equivalent capacitance is the sum of the individual capacitances. Since all three capacitors are identical, the equivalent capacitance is 3 times the capacitance of each capacitor. Thus, the equivalent capacitance for this arrangement is 3 * 15 μF = 45 μF.
Arrangement 2: Two capacitors are connected in series, and this combination is then connected in parallel with the third capacitor. In this case, the two capacitors in series can be represented by a single equivalent capacitance using the formula:
1/Ceq = 1/C1 + 1/C2
Since all three capacitors are identical, we can substitute the value of C for all three capacitors. This gives:
1/Ceq = 1/C + 1/C
1/Ceq = 2/C
Ceq = C/2 = 15 μF / 2 = 7.5 μF
Now, the combined capacitance in parallel with the third capacitor is simply the sum of these two capacitances. Thus, the equivalent capacitance for this arrangement is 7.5 μF + 15 μF = 22.5 μF.
Arrangement 3: Two capacitors are connected in parallel and this combination is then connected in series with the third capacitor. In this case, the two capacitors in parallel can be represented by a single equivalent capacitance using the formula:
Ceq = C1 + C2
Since all three capacitors are identical, the equivalent capacitance for the two capacitors in parallel is 2 times the capacitance of each capacitor. Thus, the equivalent capacitance for this combination is 2 * 15 μF = 30 μF.
Now, this combined capacitance is connected in series with the third capacitor, so we can add them up. The equivalent capacitance for this arrangement is 30 μF + 15 μF = 45 μF.
Therefore, the total equivalent capacitances for each arrangement are:
Arrangement 1: 45 μF
Arrangement 2: 22.5 μF
Arrangement 3: 45 μF
To find the total equivalent capacitance for each arrangement, we need to consider the different ways the capacitors can be connected.
1. Series Combination:
In this arrangement, the capacitors are connected end-to-end, one after another. The total equivalent capacitance can be calculated by adding the reciprocals of the individual capacitances.
Let's denote the capacitance of each individual capacitor as C.
1/Ceq = 1/C + 1/C + 1/C
1/Ceq = 3/C
Ceq = C/3
Since we are given that the maximum equivalent capacitance is 15 μF, we can set up an equation:
Ceq = C/3 = 15 μF
Solving for C:
C = 15 μF * 3 = 45 μF
Therefore, in the series combination, the total equivalent capacitance is 45 μF.
2. Parallel Combination:
In this arrangement, the capacitors are connected side by side, with their plates connected together. The total equivalent capacitance can be found by summing the individual capacitances.
Ceq = C + C + C
Ceq = 3C
Using the same equation as before:
Ceq = 3C = 15 μF
C = 15 μF / 3 = 5 μF
Therefore, in the parallel combination, the total equivalent capacitance is 5 μF.
3. Combination of Series and Parallel:
In this arrangement, two capacitors are connected in series, and then the third capacitor is connected in parallel to the combination of the series capacitors.
Let's denote the capacitance of each individual capacitor as C.
Ceq1 = C + C = 2C (Series combination of the first two capacitors)
Ceq2 = C (Equivalent capacitance of the third capacitor)
The total equivalent capacitance can be found by summing the capacitances of Ceq1 and Ceq2:
Ceq = Ceq1 + Ceq2
Ceq = 2C + C
Ceq = 3C
Using the same equation as before:
Ceq = 3C = 15 μF
C = 15 μF / 3 = 5 μF
Therefore, in the combination of series and parallel, the total equivalent capacitance is 5 μF.
To summarize:
- In series combination: Total equivalent capacitance = 45 μF
- In parallel combination: Total equivalent capacitance = 5 μF
- In combination of series and parallel: Total equivalent capacitance = 5 μF