Consider the combination of 2 capacitors `C_1 and C_2` with `C_2gtC_1`, when connected in parallel, the equivalent capacitance is `15/4` time the equivalent capacitance of the same connected in series. Calculate the ratio of capacitors,`C_2/C_1`

Options

A. `(15)/(11)`
B. `(111)/(80)`
C. `(29)/(15)`
D. `(15)/4`

Answer

To solve the problem, we need to find the ratio of the two capacitors \( C_2 \) and \( C_1 \) given that when connected in parallel, the equivalent capacitance is \( \frac{15}{4} \) times the equivalent capacitance when connected in series.

### Step 1: Write the equations for equivalent capacitance

1. **Parallel Capacitance**:
\[ C_{parallel} = C_1 + C_2 \]

2. **Series Capacitance**:
\[ C_{series} = \frac{C_1 C_2}{C_1 + C_2} \]

### Step 2: Set up the relationship given in the problem

According to the problem, the equivalent capacitance in parallel is \( \frac{15}{4} \) times the equivalent capacitance in series:
\[ C_1 + C_2 = \frac{15}{4} \left( \frac{C_1 C_2}{C_1 + C_2} \right) \]

### Step 3: Cross-multiply to eliminate the fraction

Multiply both sides by \( (C_1 + C_2) \):
\[ (C_1 + C_2)^2 = \frac{15}{4} C_1 C_2 \]

### Step 4: Expand and rearrange the equation

Expanding the left side:
\[ C_1^2 + 2C_1C_2 + C_2^2 = \frac{15}{4} C_1 C_2 \]

Rearranging gives:
\[ C_1^2 + 2C_1C_2 + C_2^2 - \frac{15}{4} C_1 C_2 = 0 \]

### Step 5: Multiply through by 4 to eliminate the fraction

Multiplying everything by 4:
\[ 4C_1^2 + 8C_1C_2 + 4C_2^2 - 15C_1C_2 = 0 \]

### Step 6: Rearranging the equation

This simplifies to:
\[ 4C_1^2 + 4C_2^2 - 7C_1C_2 = 0 \]

### Step 7: Divide by \( C_1^2 \)

Let \( x = \frac{C_2}{C_1} \). Then \( C_2 = xC_1 \):
\[ 4 + 4x^2 - 7x = 0 \]

### Step 8: Rearranging into standard quadratic form

This can be rearranged to:
\[ 4x^2 - 7x + 4 = 0 \]

### Step 9: Use the quadratic formula to find \( x \)

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]
\[ x = \frac{7 \pm \sqrt{49 - 64}}{8} \]
\[ x = \frac{7 \pm \sqrt{-15}}{8} \]

### Step 10: Analyze the discriminant

Since the discriminant is negative (\(-15\)), there are no real solutions for \( x \). This means that under the given conditions, the ratio \( \frac{C_2}{C_1} \) does not exist.

### Final Result

Thus, the ratio \( \frac{C_2}{C_1} \) cannot be determined under the given conditions.

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