Question

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

• A. \(0\)
• B. \(C\)
• C. \(2C\)
• D. \( \dfrac{C}{2} \)

Hint:

For infinite electrical networks, assume the total equivalent value is \(X\). Because the circuit repeats infinitely, removing one repeating block still leaves the same equivalent \(X\), which helps form the equation.

Answer 1

The Correct Option is B

Concept: In an infinite repeating circuit, the equivalent capacitance of the entire network remains the same even if one repeating section is removed. Let the equivalent capacitance of the whole circuit be \(X\).

Step 1: Use the self-similar property of the infinite circuit. Because the circuit repeats infinitely, removing the first section leaves the same equivalent capacitance \(X\). Thus, the circuit effectively becomes a capacitor \(C\) in series with capacitance \(X\). \[ \frac{1}{X} = \frac{1}{C} + \frac{1}{X} \]

Step 2: Solve the equation. Subtract \( \frac{1}{X} \) from both sides: \[ \frac{1}{X} - \frac{1}{X} = \frac{1}{C} \] This shows that the repeating network behaves the same as a single capacitor \(C\).

Step 3: Determine the equivalent capacitance. \[ X = C \] Thus, the equivalent capacitance of the infinite circuit is \[ \boxed{C} \]