Consider the electrical circuit shown.A potential difference V exists between A and B. The charges on various capacitors are shown. Select the correct relationship (s) .

Options

A. `q_(1)= q_(2)`
B. `q_(3)= q_(4)`
C. `q_(1) = q_(3)+q_(4)`
D. `q_(3)= (q_(4))/(2)`

Answer

To solve the problem, we need to analyze the given electric circuit with capacitors and the relationships between the charges on them. Here’s a step-by-step breakdown:

### Step 1: Understand the Circuit Configuration
We have a circuit with capacitors connected between points A and B. The charges on the capacitors are denoted as \( q_1, q_2, q_3, q_4 \), etc. Capacitors in series have the same charge, while capacitors in parallel have the same voltage across them.

**Hint:** Identify which capacitors are in series and which are in parallel.

### Step 2: Identify Relationships in Series Capacitors
For capacitors in series, the charge on each capacitor is the same. Therefore, we can write:
\[ q_1 = q_2 \]

**Hint:** Remember that in series, the charge remains constant across each capacitor.

### Step 3: Analyze the Parallel Capacitors
In the circuit, we have capacitors that are in parallel. The total charge on the capacitors in parallel is the sum of the individual charges. If \( q_1 \) is the charge on the first capacitor and \( q_3 \) and \( q_4 \) are the charges on the parallel capacitors, we can write:
\[ q_1 = q_3 + q_4 \]

**Hint:** For parallel capacitors, the total charge is the sum of the individual charges.

### Step 4: Relate Charges Using Capacitance
Using the relationship \( Q = C \cdot V \), we can express the potential differences across the capacitors. For the upper branch with capacitors of capacitance \( 3C \):
\[ V_{xy} = \frac{q_3}{3C} \]
For the lower branch with capacitance \( 2C \):
\[ V_{xy} = \frac{q_4}{2C} \]

**Hint:** Use the formula \( Q = C \cdot V \) to relate charge and voltage.

### Step 5: Set the Voltage Equations Equal
Since the potential difference across both branches must be equal, we can set the equations from Step 4 equal to each other:
\[ \frac{q_3}{3C} = \frac{q_4}{2C} \]

**Hint:** Equating the voltages gives us a relationship between the charges.

### Step 6: Solve for One Charge in Terms of Another
From the equation \( \frac{q_3}{3C} = \frac{q_4}{2C} \), we can simplify and solve for \( q_3 \):
\[ q_3 = \frac{3}{2} q_4 \]

**Hint:** Rearranging the equation helps to express one variable in terms of another.

### Step 7: Compile the Relationships
From the analysis, we have established the following relationships:
1. \( q_1 = q_2 \)
2. \( q_1 = q_3 + q_4 \)
3. \( q_3 = \frac{3}{2} q_4 \)

### Conclusion
Based on the relationships derived:
- \( q_1 = q_2 \) (True)
- \( q_1 = q_3 + q_4 \) (True)
- \( q_3 = \frac{3}{2} q_4 \) (True)

Thus, the correct relationships are:
- \( Q_1 = Q_2 \)
- \( Q_1 = Q_3 + Q_4 \)
- \( Q_3 = \frac{3}{2} Q_4 \)

### Final Answer
The correct options are A, C, and D.