Question

By connecting two given capacitors, a technician was able to make two new capacitors having the effective capacitance \( 12.5\mu F \) and \( 2\mu F \). What would be the capacitance of the given capacitors?

• A. \( 8.5\mu F \) and \( 4\mu F \)
• B. \( 10\mu F \) and \( 2.5\mu F \)
• C. \( 6.5\mu F \) and \( 6\mu F \)
• D. \( 10.5\mu F \) and \( 2\mu F \)

Hint:

Parallel gives maximum capacitance, series gives minimum — use both to form equations.

Answer 1

The Correct Option is B

Step 1: Identify combinations.
Two capacitors can form:
Parallel → maximum capacitance
Series → minimum capacitance

Step 2: Write equations.
Parallel:
\[ C_1 + C_2 = 12.5 \]
Series:
\[ \frac{C_1 C_2}{C_1 + C_2} = 2 \]

Step 3: Substitute sum value.
\[ \frac{C_1 C_2}{12.5} = 2 \Rightarrow C_1 C_2 = 25 \]

Step 4: Form quadratic equation.
\[ x^2 - 12.5x + 25 = 0 \]

Step 5: Solve equation.
\[ x = \frac{12.5 \pm \sqrt{156.25 - 100}}{2} \]
\[ x = \frac{12.5 \pm \sqrt{56.25}}{2} \]

Step 6: Final values.
\[ x = 10, \; 2.5 \]

Step 7: Final Answer.
\[ \boxed{10\mu F \text{ and } 2.5\mu F} \]