Question

Determine the equivalent capacitance of an infinite circuit formed by repeating identical capacitors of capacitance \(C\).

• A. \(C\)
• B. \(\dfrac{C}{2}\)
• C. \(\dfrac{C}{3}\)
• D. \(2C\)

Hint:

For infinite repeating circuits, assume the equivalent value is \(X\). Attach one more repeating unit and use the series/parallel rules to form an equation for \(X\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the equivalent capacitance of an infinite network of identical capacitors. The specific arrangement of the capacitors (e.g., a ladder network) is not shown, which makes the problem ambiguous. However, we can analyze the provided answer to infer the likely intended structure or principle.
Step 2: Key Formula or Approach:
For an infinite repeating circuit, the main principle is self-similarity. If we let the equivalent capacitance of the entire network be \(C_{eq}\), then adding one more repeating unit to the network will not change the overall equivalent capacitance. This allows us to set up an equation for \(C_{eq}\). The formulas for series and parallel capacitors are:
Series: \(\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2}\)
Parallel: \(C_{parallel} = C_1 + C_2\)
Step 3: Detailed Explanation:
The structure of the infinite circuit is not specified. Standard infinite ladder networks typically lead to a quadratic equation for \(C_{eq}\), often involving irrational numbers (like the golden ratio). The answer provided, \(C/2\), is the result of two capacitors of capacitance \(C\) connected in series.
\[ C_{series} = \frac{C \times C}{C+C} = \frac{C^2}{2C} = \frac{C}{2} \] It is highly probable that the question is either flawed or refers to a non-standard, simplified model of an infinite network where the overall behavior simplifies to that of two capacitors in series. The reasoning provided in the original source, which uses the incorrect equation \(\frac{1}{C_{eq}}=\frac{1}{C}+\frac{1}{C_{eq}}\), is mathematically unsound as it implies \(\frac{1}{C}=0\).
Step 4: Final Answer:
Without a diagram, we must infer the intent from the answer. The result \(C_{eq} = C/2\) corresponds to the equivalent capacitance of two identical capacitors \(C\) connected in series. We assume this is the intended, albeit poorly formulated, problem.