To find the ratio between the charges on capacitors P and Q, we can follow these steps:
### Step 1: Understand the Configuration
Capacitors P and Q are connected in series. In a series connection, the charge (Q) on each capacitor is the same.
### Step 2: Identify the Capacitance Values
- Capacitance of capacitor P, \( C_P = 10 \, \mu F \)
- Capacitance of capacitor Q, \( C_Q = 20 \, \mu F \)
### Step 3: Apply the Series Capacitance Formula
The equivalent capacitance \( C_{eq} \) for capacitors in series is given by the formula:
\[
\frac{1}{C_{eq}} = \frac{1}{C_P} + \frac{1}{C_Q}
\]
### Step 4: Calculate the Equivalent Capacitance
Substituting the values:
\[
\frac{1}{C_{eq}} = \frac{1}{10 \, \mu F} + \frac{1}{20 \, \mu F}
\]
Finding a common denominator (20):
\[
\frac{1}{C_{eq}} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20}
\]
Thus,
\[
C_{eq} = \frac{20}{3} \, \mu F
\]
### Step 5: Calculate the Total Charge
The total charge \( Q \) stored in the series combination when connected to a voltage \( V \) is given by:
\[
Q = C_{eq} \times V
\]
Substituting the values:
\[
Q = \left(\frac{20}{3} \, \mu F\right) \times 12 \, V = \frac{240}{3} \, \mu C = 80 \, \mu C
\]
### Step 6: Determine the Charge on Each Capacitor
Since the charge on capacitors in series is the same:
- Charge on capacitor P, \( Q_P = 80 \, \mu C \)
- Charge on capacitor Q, \( Q_Q = 80 \, \mu C \)
### Step 7: Find the Ratio of Charges
The ratio of the charges on capacitors P and Q is:
\[
\frac{Q_P}{Q_Q} = \frac{80 \, \mu C}{80 \, \mu C} = 1
\]
### Final Answer
The ratio of the charges on capacitors P and Q is \( 1:1 \).
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