Let's solve each part step by step: --- ## **1. Based on the number of free (valence) electrons, rank the following materials by their electrical conductivity:** **Step 1:** List the valence electrons for each element. - **Aluminum (Al):** 3 valence electrons (3s²3p¹) - **Silver (Ag):** 1 valence electron (5s¹) - **Gold (Au):** 1 valence electron (6s¹) - **Magnesium (Mg):** 2 valence electrons (3s²) - **Copper (Cu):** 1 valence electron (4s¹) - **Titanium (Ti):** 4 valence electrons (4s²3d²) - **Platinum (Pt):** 1 or 2 valence electrons (6s¹, 5d⁹) **Step 2:** More free electrons generally means higher conductivity, but **the actual conductivity also depends on how freely those electrons move** (electron mobility, atomic structure, etc.). In practice, the actual electrical conductivity (σ) is known for these elements: - **Silver (Ag):** Highest conductivity (~6.3 × 10⁷ S/m) - **Copper (Cu):** Second highest (~5.9 × 10⁷ S/m) - **Gold (Au):** Third (~4.1 × 10⁷ S/m) - **Aluminum (Al):** Next (~3.8 × 10⁷ S/m) - **Magnesium (Mg):** Lower (~2.3 × 10⁷ S/m) - **Platinum (Pt):** Lower (~9.4 × 10⁶ S/m) - **Titanium (Ti):** Lowest among these (~2.4 × 10⁶ S/m) **Step 3:** **Ranking (highest to lowest conductivity):** 1. **Silver (Ag)** 2. **Copper (Cu)** 3. **Gold (Au)** 4. **Aluminum (Al)** 5. **Magnesium (Mg)** 6. **Platinum (Pt)** 7. **Titanium (Ti)** **Final order:** **b > e > c > a > d > g > f** --- ## **2. Maximum current drawn by household appliances** **Step 1:** Use the formula: \( I = \dfrac{P}{V} \) Where \( I \) is current (A), \( P \) is power (W), and \( V \) is voltage (typically 120 V in North America, 230 V in many other places). *Typical power ratings (actual values may vary; check your appliance labels):* a. **Refrigerator/Freezer**: 100–800 W (average running ~200 W, startup up to 800 W) b. **Microwave Oven**: 600–1200 W (use 1200 W for max) c. **Toaster**: 800–1500 W (use 1500 W) d. **Hair Dryer**: 800–1800 W (use 1800 W) e. **Dishwasher**: 1200–1500 W (use 1500 W) f. **Electric Oven**: 2000–5000 W (use 5000 W) g. **Television**: 50–400 W (use 400 W for large TV) Assume **V = 120 V** (for North America). **Step 2:** Calculate current for each. - **a. Refrigerator/Freezer:** \( I = \dfrac{800}{120} = 6.67 \) A - **b. Microwave Oven:** \( I = \dfrac{1200}{120} = 10 \) A - **c. Toaster:** \( I = \dfrac{1500}{120} = 12.5 \) A - **d. Hair Dryer:** \( I = \dfrac{1800}{120} = 15 \) A - **e. Dishwasher:** \( I = \dfrac{1500}{120} = 12.5 \) A - **f. Electric Oven:** \( I = \dfrac{5000}{240} = 20.8 \) A (Most ovens use 240 V, so use 240 V.) - **g. Television:** \( I = \dfrac{400}{120} = 3.33 \) A **Step 3:** **Maximum total current** (if all run at once, not typical): Add all currents (using correct voltage for each): - Refrigerator: 6.67 A (120 V) - Microwave: 10 A (120 V) - Toaster: 12.5 A (120 V) - Hair Dryer: 15 A (120 V) - Dishwasher: 12.5 A (120 V) - Oven: 20.8 A (240 V) (If your panel is split-phase, this is on a separate line) - TV: 3.33 A (120 V) If you add up **120 V appliances** only: 6.67 + 10 + 12.5 + 15 + 12.5 + 3.33 = **60 A** (for 120 V circuits) Plus **20.8 A at 240 V** for the oven. **Final Answers:** a. **Refrigerator/Freezer:** ~6.67 A b. **Microwave Oven:** ~10 A c. **Toaster:** ~12.5 A d. **Hair Dryer:** ~15 A e. **Dishwasher:** ~12.5 A f. **Electric Oven:** ~20.8 A (at 240 V) g. **Television:** ~3.33 A h. **Maximum total current (all at once):** - 120 V circuit: **~60 A** - 240 V circuit: **~20.8 A** --- ## **3. Explanation** ### a. **Why do all hydraulic, wind, and fuel-based power plants generate AC voltage?** - **AC is easier to transmit over long distances:** Using transformers, AC voltage can be stepped up for transmission (reducing current and power loss), and stepped down for safe use. - **Generator design:** Rotating machines (generators) naturally produce AC due to the way magnetic fields interact with spinning coils. - **Historical reasons:** The electrical grid is based on AC distribution, so power plants produce AC to match the grid. ### b. **What type of voltage do solar collectors generate, and why?** - **Solar collectors (solar panels) generate DC (Direct Current) voltage.** - **Reason:** The photovoltaic effect in solar cells causes electrons to flow in one direction, resulting in DC. To use this power in homes (which use AC), an inverter is needed to convert DC to AC. --- ## **4. AC Signal: \( Q(t) = 120 \sin(2\pi ft),\ f = 60\,Hz \). Find the current.** **Step 1:** **Current** is the time-derivative of charge: \[ I(t) = \frac{dQ}{dt} \] **Step 2:** Plug in the equation: \[ Q(t) = 120 \sin(2\pi ft) \] \[ I(t) = \frac{d}{dt} [120 \sin(2\pi ft)] \] **Step 3:** Differentiate: - The derivative of \( \sin(ax) \) is \( a \cos(ax) \): \[ I(t) = 120 \cdot (2\pi f) \cos(2\pi f t) \] \[ = 120 \cdot 2\pi \cdot 60 \cos(2\pi \cdot 60 t) \] \[ = 120 \cdot 2 \cdot \pi \cdot 60 \cos(120\pi t) \] \[ = 120 \cdot 120\pi \cos(120\pi t) \] \[ = 14,400\pi \cos(120\pi t) \] **Final Answer:** \[ \boxed{I(t) = 14,400\pi \cos(120\pi t)\ \text{(in amperes, if Q in coulombs)}} \] Or, numerically: \[ 14,400\pi \approx 45,239.6 \] So, \[ I(t) \approx 45,240\ \cos(120\pi t)\ \text{A} \] --- ### **Summary of Final Answers** 1. **Conductivity ranking:** **Silver > Copper > Gold > Aluminum > Magnesium > Platinum > Titanium** 2. **Maximum currents (approx.):** a. Refrigerator: ~6.67 A b. Microwave: ~10 A c. Toaster: ~12.5 A d. Hair Dryer: ~15 A e. Dishwasher: ~12.5 A f. Electric oven: ~20.8 A (240 V) g. Television: ~3.33 A h. Max total: **~60 A (120 V)**, **~20.8 A (240 V)** 3. **Why AC for most power plants?** - AC is easily transformed for efficient transmission and matches grid requirements. - Rotating generators naturally produce AC. **Why solar panels generate DC?** - Photovoltaic effect produces unidirectional (DC) current. 4. **Current for given Q(t):** \[ \boxed{I(t) = 14,400\pi \cos(120\pi t)} \] Or numerically: \[ I(t) \approx 45,240\ \cos(120\pi t)\ \text{A} \]

Certainly! Here is a properly organized and detailed version of your answer for **question 2**: --- ## **2. Maximum current drawn by household appliances** ### **Step 1: Determine the formula** The current \( I \) drawn by an appliance can be estimated using: \[ I = \frac{P}{V} \] where: - \( P \) = power rating of the appliance (in watts, W), - \( V \) = voltage supply (in volts, V). *Note:* Power ratings are approximate; always check the appliance label for exact values. Assume household voltage \( V = 120\,V \) for North America unless specified otherwise. --- ### **Step 2: List typical power ratings** | Appliance | Power Rating \( P \) | Notes | |------------------------|----------------------|--------------------------------------------------| | Refrigerator/Freezer | 100–800 W | Average ~200 W; startup may be higher | | Microwave Oven | 600–1200 W | Use 1200 W for maximum load | | Toaster | 800–1500 W | Use 1500 W for maximum load | | Hair Dryer | 800–1800 W | Use 1800 W for maximum load | | Dishwasher | 1200–1500 W | Use 1500 W for maximum load | | Electric Oven | 2000–5000 W | Use 5000 W for maximum load | | Television | 50–400 W | Use 400 W for large or high-definition TVs | --- ### **Step 3: Calculate maximum currents** Using \( I = P / V \), with \( V = 120\,V \) (North America): | Appliance | Power \( P \) | Calculated Current \( I = P/120 \) | Result (A) | |------------------------|--------------|----------------------------------|------------------------| | **Refrigerator/Freezer** | 800 W | \( 800/120 \approx 6.67 \) | **6.67 A** | | **Microwave Oven** | 1200 W | \( 1200/120 = 10.0 \) | **10 A** | | **Toaster** | 1500 W | \( 1500/120 = 12.5 \) | **12.5 A** | | **Hair Dryer** | 1800 W | \( 1800/120 = 15.0 \) | **15 A** | | **Dishwasher** | 1500 W | \( 1500/120 = 12.5 \) | **12.5 A** | | **Electric Oven** | 5000 W | \( 5000/240 \approx 20.8 \) (uses 240 V) | **20.8 A** (at 240 V) | | **Television** | 400 W | \( 400/120 \approx 3.33 \) | **3.33 A** | --- ### **Step 4: Compute the maximum total current (if all appliances operate simultaneously)** - **On 120 V circuits:** Sum of currents for appliances powered at 120 V: \[ 6.67 + 10 + 12.5 + 15 + 12.5 + 3.33 \approx \boxed{60\,A} \] - **On 240 V circuits:** The oven consumes approximately 20.8 A at 240 V (since it's typically connected to a dedicated 240 V circuit). *Other appliances* are generally connected to 120 V circuits, so their combined current remains about 60 A. --- ### **Final Summary:** | Appliance | Max Current (A) | |------------------------|----------------------------| | Refrigerator/Freezer | **6.67 A** | | Microwave Oven | **10 A** | | Toaster | **12.5 A** | | Hair Dryer | **15 A** | | Dishwasher | **12.5 A** | | Electric Oven | **20.8 A** (at 240 V) | | Television | **3.33 A** | | **Maximum total (all at once)** | **~60 A** (120 V) + **20.8 A** (240 V oven) | --- This breakdown provides a clear, step-by-step calculation and understanding of household appliance currents.