To calculate the equivalent capacitance $C_{ab}$ between points $a$ and $b$, let's break down the circuit step by step:

Step 1: Combine capacitors in series or parallel

1. Left loop:

   • The 6 µF and 1 µF capacitors are in series. For capacitors in series, the equivalent capacitance $C_s$ is given by: $\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2}$ Substituting values: $\frac{1}{C_s} = \frac{1}{6} + \frac{1}{1} = \frac{1}{6} + \frac{6}{6} = \frac{7}{6}$ So: $C_s = \frac{6}{7} \, \mu\text{F}$

2. Right loop:

   • The 9 µF and 4 µF capacitors are also in series. Using the same formula: $\frac{1}{C_s} = \frac{1}{9} + \frac{1}{4}$ Compute $\frac{1}{9} + \frac{1}{4}$ as: $\frac{1}{C_s} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}$ So: $C_s = \frac{36}{13} \, \mu\text{F}$

Step 2: Combine with middle capacitor

Now, the equivalent capacitances of the left loop ($\frac{6}{7} \, \mu\text{F}$) and right loop ($\frac{36}{13} \, \mu\text{F}$) are in parallel with the middle 1 µF capacitor. For capacitors in parallel, the equivalent capacitance is: $C_p = C_1 + C_2 + C_3$ Substitute the values: $C_{ab} = \frac{6}{7} + 1 + \frac{36}{13}$

Step 3: Compute final result

Simplify the terms:

1. Convert $\frac{6}{7}$ and $\frac{36}{13}$ to a common denominator:
   • Least common multiple of 7 and 13 is 91.
   • $\frac{6}{7} = \frac{78}{91}$, and $\frac{36}{13} = \frac{252}{91}$.
2. Add these along with $1 = \frac{91}{91}$: $C_{ab} = \frac{78}{91} + \frac{91}{91} + \frac{252}{91} = \frac{421}{91} \, \mu\text{F}$

Finally: $C_{ab} \approx 4.63 \, \mu\text{F}$

Final Answer:

The equivalent capacitance $C_{ab}$ is approximately 4.63 µF.

Would you like additional clarification or details about any of the steps?

Relative Questions:

1. How do capacitors combine in parallel versus series circuits?
2. What are some common applications of capacitance in electrical circuits?
3. How does the value of capacitance affect energy storage in a circuit?
4. What happens if additional capacitors are added to this circuit in parallel?
5. Can you explain the physical meaning of equivalent capacitance?

Tip:

Always double-check units when solving problems involving capacitance to ensure consistency across calculations.