Certainly! Let's tackle each part of the problem one by one:
### Part (a): Effective Capacitance When Connected in Series
When capacitors are connected in series, the reciprocal of the total capacitance [tex]\( C_{\text{series}} \)[/tex] is the sum of the reciprocals of the individual capacitances:
[tex]\[
\frac{1}{C_{\text{series}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}
\][/tex]
Given the values:
- [tex]\( C_1 = 1 \, \text{F} \)[/tex]
- [tex]\( C_2 = 2 \, \text{F} \)[/tex]
- [tex]\( C_3 = 3 \, \text{F} \)[/tex]
Substitute these values into the formula:
[tex]\[
\frac{1}{C_{\text{series}}} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}
\][/tex]
Calculate the individual fractions:
[tex]\[
\frac{1}{C_{\text{series}}} = 1 + 0.5 + 0.333\overline{3}
\][/tex]
Adding these together:
[tex]\[
\frac{1}{C_{\text{series}}} = 1.833\overline{3}
\][/tex]
Now find the reciprocal of this sum to get [tex]\( C_{\text{series}} \)[/tex]:
[tex]\[
C_{\text{series}} = \frac{1}{1.833\overline{3}} \approx 0.545 \, \text{F}
\][/tex]
So, the effective capacitance when the capacitors are connected in series is approximately:
[tex]\[
C_{\text{series}} \approx 0.545 \, \text{F}
\][/tex]
### Part (b): Effective Capacitance When Connected in Parallel
When capacitors are connected in parallel, the total capacitance [tex]\( C_{\text{parallel}} \)[/tex] is the sum of the individual capacitances:
[tex]\[
C_{\text{parallel}} = C_1 + C_2 + C_3
\][/tex]
Again, using the given values:
- [tex]\( C_1 = 1 \, \text{F} \)[/tex]
- [tex]\( C_2 = 2 \, \text{F} \)[/tex]
- [tex]\( C_3 = 3 \, \text{F} \)[/tex]
Substitute these values into the formula:
[tex]\[
C_{\text{parallel}} = 1 + 2 + 3
\][/tex]
Add these together:
[tex]\[
C_{\text{parallel}} = 6 \, \text{F}
\][/tex]
So, the effective capacitance when the capacitors are connected in parallel is:
[tex]\[
C_{\text{parallel}} = 6 \, \text{F}
\][/tex]
### Summary
(a) The effective capacitance when the capacitors are connected in series is approximately [tex]\( 0.545 \, \text{F} \)[/tex].
(b) The effective capacitance when the capacitors are connected in parallel is [tex]\( 6 \, \text{F} \)[/tex].
