Unit 2: Physics Part 2 - Electric Circuits, Energy, and Power

Introduction to Electric Circuits and Analogies

• The study of electricity moves from electrostatics (stationary charges) to circuits, which emphasize the continuous flow of charges.
• Water Analogy:   - A garden hose or kitchen faucet is an "open system" where water exits the system.   - A better analogy for a circuit is a "closed system," such as a typical car cooling loop.
• Cooling Loop Mechanics:   - A pump pulls water in from one side, creating LOW PRESSURE.   - The pump pushes water out the other side, creating HIGH PRESSURE.   - A pump provides a DIFFERENCE IN PRESSURE across its two sides.   - In pushing water through, the pump gives the water energy.   - As water goes through a "load," it loses both pressure and energy.
• Circuits as Recirculation Systems:   - Water represents Electric Charge ($q$).   - Pump represents the Battery.   - Pipes represent the Wires.   - Load represents the Resistor (e.g., light bulb, toaster).

Charge and Current

• Charge ($q$):   - Measured in coulombs (C).   - Example: One proton has a specific unit of charge.
• Current ($I$):   - Refers to the flow of charge through a point in a circuit.   - Unit: ampere (A), where $1\,A = 1\,C/s$.   - Equation: $I = \frac{q}{t}$
• Calculations:   - Example 1: If $12\,C$ of charge flows through a wire in $4\,s$, the current is:     $I = \frac{12\,C}{4\,s} = 3\,A$   - Example 2: A charge of $15\,C$ flows through a wire in $2\,s$.     $I = \frac{15\,C}{2\,s} = 7.5\,A$   - Example 3: If $15\,A$ flows through a load for $5\,\text{minutes}$, how much charge flows?     - $t = 5\,\text{min} \times 60\,s/\text{min} = 300\,s$     - $q = I \times t = 15\,A \times 300\,s = 4500\,C$   - Example 4: If a wire has $50\,mA$ ($0.050\,A$) and delivers $125\,kC$ ($125000\,C$), the time in days is:     - $t = \frac{q}{I} = \frac{125000\,C}{0.050\,A} = 2.5 \times 10^{6}\,s$     - In days: $\frac{2.5 \times 10^{6}\,s}{86400\,s/\text{day}} \approx 28.9\,\text{days}$

Energy and the Law of Conservation

• Definition: Energy ($U$) is a quantity related to "getting stuff done."
• Unit: joule (J).
• Forms of Energy:   - Food energy (Chemical potential energy).   - Energy of motion (Kinetic energy).   - Warmth (Thermal energy).   - Electricity (Electric potential energy).
• Conservation of Energy Law:   - Energy cannot be created or destroyed, only transferred (place to place) or transformed (one kind to another).   - The total energy of the universe is fixed.
• The Energy Story (Example):   - Sun stores nuclear potential energy.   - Sun transforms it into thermal and light energy, which transfers to Earth.   - Plants use photosynthesis to transform light into chemical potential energy.   - A cow eats grass, transferring chemical potential energy to its meat.   - A person eats a hamburger, transferring energy to throw a ball (kinetic energy).   - As the ball rises, kinetic energy transforms into gravitational potential energy (GPE).   - As the ball falls, GPE transforms back into kinetic energy.   - At impact, kinetic energy transforms into thermal energy (the ground warms up).

Concepts of Potential Energy and Work

• Measuring Energy:   - Keeping track of energy in a system, comparing before/after, and checking for transformation.   - Larger values mean more energy is stored or transferred (e.g., a higher roller coaster hill or a faster object).
• Roller Coaster Examples:   - At the top of a hill: $GPE = 100\,J$, $KE = 0\,J$.   - Midway down: $GPE = 50\,J$, $KE = 50\,J$.   - If ball has $75\,J$ of potential energy at a height, and total energy is $100\,J$, it must have $25\,J$ of kinetic energy.
• Energy Losses:   - If a system starts with $2500\,J$ of PE and ends with $2000\,J$ of KE at the bottom (PE = 0), the "missing" $500\,J$ is converted into heat and sound (thermal energy).

PhET Energy Simulation Observations

• Mass: Increasing the mass of a skater does not change the height they reach (energy is proportional to mass on both sides, so mass cancels out in terms of height reaching), but it increases the total kinetic and potential energy values.
• Gravity: Increasing gravity makes the skater go faster down the ramp (more kinetic energy) but doesn't change the ability to reach the original height.
• Friction: Adding friction causes energy to be transformed into thermal energy. The skater reaches a lower height each time until they eventually stop at the bottom.
• Thermal Energy: Represents energy that has spread out and become hard to track, though it remains in the universe.

Electric Potential and Voltage

• Electric Potential ($V$):   - Defined as the amount of electric potential energy per unit of charge.   - Equation: $V = \frac{U}{q}$   - Unit: volt (V) ($1\,V = 1\,J/C$).
• Potential Difference (Voltage, $\Delta V$):   - Change in electric potential between two points.   - Equation: $\Delta V = V_b - V_a$   - Voltage only exists across two points; it does not exist "in" a fixed spot.
• Battery Analogy:   - A battery is like a ladder or slide.   - A higher voltage battery gives each coulomb more energy (a taller slide).   - Charges gain potential energy when forced from the negative side to the positive side of the battery ($\Delta V$ is positive).   - Charges lose potential energy as they move through a load (like a light bulb).
• Calculations:   - Example 1: Battery transfers $120\,J$ to $20\,C$. Potential at positive terminal:     $V = \frac{120\,J}{20\,C} = 6\,V$   - Example 2: A $9\,V$ battery is connected to a bulb.     - $5\,C$ at the positive end has $U = V \times q = 9\,V \times 5\,C = 45\,J$.     - $5\,C$ at the negative end has $0\,J$ (reference point).   - Example 3: Current of $2\,A$ for $15\,s$ with a $9\,V$ battery.     - $q = I \times t = 2\,A \times 15\,s = 30\,C$     - Energy transferred to bulb: $U = \Delta V \times q = 9\,V \times 30\,C = 270\,J$

Electric Power

• Definition: Power ($P$) describes the rate at which energy is transferred or transformed per second.
• Unit: watt (W) ($1\,W = 1\,J/s$).
• Equation: $P = \frac{\Delta U}{\Delta t}$
• Relationship with Voltage and Current:   - Power depends on how much energy each charge carries (voltage) and how many charges pass by per second (current).
• Calculations:   - Example 1: Bulb transfers $30\,J$ in $3\,s$.     $P = \frac{30\,J}{3\,s} = 10\,W$   - Example 2: Bulb has $\Delta V = 6\,V$, $4\,C$ pass in $2\,s$.     - Energy transferred: $\Delta U = V \times q = 6\,V \times 4\,C = 24\,J$     - Power: $P = \frac{24\,J}{2\,s} = 12\,W$
• Important Notes:   - Power is not energy.   - Power does not "flow."   - Zero power means no electrical energy is being transferred.

Energy Costs and the Kilowatt-Hour

• The joule is a very small unit of energy. Homes use energy in large quantities measured in kilowatt-hours (kWh).
• Manitoba Electricity Rates (Example January 2026):   - Basic monthly charge (up to $200\,A$): $\$9.84$   - Energy charge: $9.97\ \text{cents}/kWh$ ($\$0.0997/kWh$).
• Cost Calculation Procedure:   1. Find total power in kilowatts (kW): $\text{Watts} / 1000$.   2. Find total time in hours (h).   3. Calculate energy in kWh: $E = P \times t$.   4. Calculate cost: $\text{Cost} = E \times \text{Rate}$.
• Manya's Wasted Energy Example:   - 7 lights: $3(8\,W) + 4(10\,W) = 64\,W = 0.064\,kW$   - Time: $4\,h$   - Energy: $0.064\,kW \times 4\,h = 0.256\,kWh$   - Cost: $0.256\,kWh \times 9.97\ \text{cents}/kWh \approx 2.55\ \text{cents}$.

Resistance in Electric Circuits

• Definition: Resistance ($R$) describes how much a material opposes the movement of charges.
• Unit: Ohm (\Omega).
• Factors Affecting Resistance:   - Material: Conductivity/Resistivity (e.g., silver and copper have low resistance, graphite is higher).   - Length: Longer wires have greater resistance (charges experience more interactions).   - Thickness (Cross-Sectional Area): Thicker wires have less resistance (more pathways available).   - Temperature: In metals, higher temperature causes atoms to vibrate more, increasing resistance.
• Purpose of Resistors:   - To limit current.   - To protect sensitive components.   - To control brightness or speed.   - To convert electrical energy into heat or light (e.g., light bulb filaments).

Ohm's Law

• Describes the relationship between voltage, current, and resistance.
• Equation: $I = \frac{\Delta V}{R}$ or $\Delta V = I R$
• Circuit behavior:   - In a series circuit, total resistance ($R_{\text{total}}$) is the sum of individual resistances.   - The current is the same at every point in a single-loop circuit.
• Calculations:   - Example 1: A toaster plugged into a $120\,V$ socket draws $13\,A$.     $R = \frac{120\,V}{13\,A} \approx 9.2\,\Omega$   - Example 2: Circuit with $54\,V$ battery and three resistors ($R_1$, $R_2$, $R_3$).     - If $I = 3.0\,A$ and $\Delta V_2 = 9\,V$, then $R_2 = \frac{9\,V}{3.0\,A} = 3\,\Omega$.     - If $R_3 = 7\,\Omega$, then $\Delta V_3 = 3.0\,A \times 7\,\Omega = 21\,V$.     - Voltage for resistor 1: $\Delta V_1 = 54\,V - 9\,V - 21\,V = 24\,V$.     - Resistance for resistor 1: $R_1 = \frac{24\,V}{3.0\,A} = 8\,\Omega$.

Comprehensive Problem Set

• Phone Charger: $540\,C$ in $15\,s$. $I = 540/15 = 36\,A$.
• Small Heater: $1800\,J$ in $12\,s$. $P = 1800/12 = 150\,W$.
• Lamp: $120\,V$, $2.4\,A$.   - $R = 120/2.4 = 50.0\,\Omega$.   - Energy in $5\,\text{min}$: $U = V \times I \times t = 120 \times 2.4 \times 300 = 86400\,J = 8.64 \times 10^{4}\,J$.
• Electric Kettle: $R = 14\,\Omega$, $V = 120\,V$, $t = 4\,\text{min}$.   - $I = 120/14 = 8.57\,A$.   - Energy: $120 \times 8.57 \times 240 \approx 2.47 \times 10^{5}\,J$.
• Portable Speaker: $2.70 \times 10^{4}\,J$ over $25\,\text{min}$.   - $P = \frac{27000}{1500} = 18.0\,W$.   - Current ($12\,V$ source): $I = 18/12 = 1.5\,A$.
• Manitoba Cost Comparison (Space Heater):   - $P = 1.50\,kW$, $t = 8.0\,h$. Energy = $12\,kWh$.   - Manitoba: $12 \times 9.97\ \text{cents} = 119.64\ \text{cents} \approx \$1.20\,CAD$.   - California ($\$0.31\,USD/kWh$): $12 \times 0.31 = \$3.72\,USD$.   - Exchange ($1\,USD = 1.36\,CAD$): \3.72 \times 1.36 = \$5.06\,CAD$.   - Difference: $\$5.06 - \$1.20 = \$3.86\,CAD$.