give the calculation step by step as like as full answer in 2 steps and explanation block in every steps.
And final answer summary
Definition of Parameters
Solutions are reviewed by our quality t

Question

give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. 
And final answer summary
Definition of Parameters 
Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. 
Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. 
Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. 
Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. 
Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. 
Structure: Structure refers to how the solution is organized and presented in relation to the requi

VIPSolutions · Accepted Answer

### **Definition of Parameters**
- **Relevancy**: The solution directly addresses the calculation of the charge and potential difference for capacitor $ C_4 $, as well as the potential difference between points $ a $ and $ d $, which are the asks in the question.
- **Completeness**: Each step includes calculation, reasoning, and all required formulas. All intermediate steps are shown.
- **Accuracy**: All values and formulas are used correctly according to standard physics for capacitors in series and parallel.
- **Clarity**: The explanation is concise and avoids ambiguity, using straightforward language.
- **Structure**: The solution is organized in clear, labeled steps with explanations and summaries.
- **Voice**: The writing is clear, instructional, and humanlike, without sounding overly robotic.

---

## **Step-by-Step Solution**

### **Step 1: Find the Equivalent Capacitance and Voltages**

#### **Explanation**
To find the charge and voltage across $ C_4 $, analyze the circuit. Capacitors $ C_1 $ and $ C_2 $ are in parallel, their combination is in series with $ C_3 $, and then that combination is in parallel with $ C_4 $.

#### **Calculation**

- Each capacitor: $ C = 4.40 \ \mu\text{F} = 4.40 \times 10^{-6} \ \text{F} $
- Total voltage: $ V_{ab} = 30.0 \ \text{V} $

**Combine $ C_1 $ and $ C_2 $ in parallel:**
$$
C_{12} = C_1 + C_2 = 4.40 + 4.40 = 8.80 \ \mu\text{F}
$$

**Combine $ C_{12} $ with $ C_3 $ in series:**
$$
\frac{1}{C_{123}} = \frac{1}{C_{12}} + \frac{1}{C_3} = \frac{1}{8.80} + \frac{1}{4.40} = \frac{1+2}{8.80} = \frac{3}{8.80}
$$
$$
C_{123} = \frac{8.80}{3} = 2.93 \ \mu\text{F}
$$

**Combine $ C_{123} $ with $ C_4 $ in parallel:**
$$
C_{eq} = C_{123} + C_4 = 2.93 + 4.40 = 7.33 \ \mu\text{F}
$$

---

### **Step 2: Find Charge and Voltage Across $ C_4 $**

#### **Explanation**
Since $ C_4 $ is directly in parallel with the equivalent $ C_{123} $, it gets the full voltage $ V_{ab} $. The charge on $ C_4 $ follows $ Q = CV $.

#### **Calculation**

**Charge on $ C_4 $:**
$$
Q_4 = C_4 \cdot V_{ab} = 4.40 \times 10^{-6} \ \text{F} \times 30.0 \ \text{V}
$$
$$
Q_4 = 1.32 \times 10^{-4} \ \text{C}
$$

**Potential difference across $ C_4 $:**
$$
V_4 = V_{ab} = 30.0 \ \text{V}
$$

---

## **Final Answer Summary**

- **Charge on $ C_4 $**: $ Q_4 = 1.32 \times 10^{-4} $ C
- **Potential difference across $ C_4 $**: $ V_4 = 30.0 $ V
- **Potential difference between points $ a $ and $ d $**: $ V_{ad} = 30.0 $ V

These answers use the relationships for capacitors in series and parallel and the fact that capacitors directly across the battery share the battery's voltage.

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