Electrical Energy and Circuits Vocabulary

Fundamental Definitions in Electricity

Conductor: A material that allows electric charges to flow through it easily.
Insulator: A material that does not allow electric charges to flow easily.
Electrostatics: The study of electric charges at rest (covered in Unit 1).
Voltaic Cell: A cell containing two different metals, known as electrodes, placed in an electrolytic solution that produces an electric charge on the electrodes. * Significance: This provided scientists with the first opportunity to work with a flow of electrons rather than static electricity.
Alessandro Volta: The scientist who created the first voltaic cell.
Battery: A combination of two or more voltaic cells.
Electrodes: Electrical conductors through which a current enters or leaves an electrical device. * Anode: The positive pole of a primary cell or battery. * Cathode: The negative electrode of a cell or battery.

Electric Potential Difference and Work

Definition: The electric potential difference between two points is defined as the change in electrical energy divided by the amount of charge that passes between those two points.
Alternative Names: It is also referred to as the voltage or the voltage drop across two points.
Mathematical Formula: $V = \frac{\Delta E_Q}{q}$ * $V$ = electric potential difference (Voltage, measured in Volts ( $V$ )) * $\Delta E_Q$ = change in potential energy (measured in Joules ( $J$ )) * $q$ = quantity of charge (measured in Coulombs ( $C$ ))
Work Relationship: $\Delta E_Q$ is equal to the amount of work done on an object.
Example 2.1: * Given: $V = 12\,V$ , $q = 48\,C$ , Find: $\Delta E_Q$ (Work done). * Calculation: $12 = \frac{E_Q}{48}$ * $E_Q = 576\,J$ * The work done is equal to the amount of energy between the two points, which is $576\,J$ .
Manual Calculation (Page 3): * Given: $t = 30\,min = 1800\,s$ , $V = 120\,V$ , $q = 25000\,C$ . * Work/Energy: $E_Q = V \times q = 120 \times 25000 = 3.0 \times 10^6\,J$ . * Current: $I = \frac{q}{t} = \frac{25000}{1800} = 13.89\,A$ .

Direction of Charge Flow and Resistance

Electron Flow vs. Conventional Flow: * Electron Flow: Actual physical flow of negative charges (electrons) from the cathode ( $-$ ) to the anode ( $+$ ). * Conventional Current: The flow of positive charge from the anode ( $+$ ) to the cathode ( $-$ ). * Historical Context: Early scientists could not see electrons and assumed current flowed from ( $+$ ) to ( $-$ ). Because so much text and theory were developed using this method, it remains the standard today. It does not affect calculations.
Resistance ( $R$ ): * Resistance is the amount of "friction" encountered by a charge as it flows through a circuit. * Factors Governing Resistance: 1. Length ( $L$ ): The longer the length of the circuit, the greater the resistance. 2. Cross-sectional Area ( $A$ ): The smaller the cross-sectional area (diameter), the greater the resistance. 3. Temperature: Generally, for metallic conductors, resistivity increases as temperature increases.
Resistivity Formula: $R = \rho \frac{L}{A}$ * $R$ = resistance (measured in Ohms, $\Omega$ ) * $\rho$ (rho) = resistivity of the material (measured in $\Omega\,m$ ) * $L$ = length of the conductor (measured in $m$ ) * $A$ = cross-sectional area (measured in $m^2$ )
Area Formulas for Conductors: * For circular conductors: $A = \pi r^2$ or $A = \frac{\pi d^2}{4}$ * For rectangular conductors: $A = l \times w$
Resistivity Table ( $\rho$ in $\Omega\,m$ ): * Silver: $1.6 \times 10^{-8}$ * Copper: $1.7 \times 10^{-8}$ * Aluminum: $2.7 \times 10^{-8}$ * Tungsten: $5.6 \times 10^{-8}$ * Nichrome: $100 \times 10^{-8}$ * Carbon: $3500 \times 10^{-8}$ * Germanium: $0.46$ * Glass: $1 \times 10^{10}$ to $1 \times 10^{14}$

Ohm's Law and Power Dynamics

Ohm's Law (1826): Proposed by George Simon Ohm. The voltage across a load is equal to the current multiplied by the resistance of the load. $V = IR$ * $V$ = voltage (Volts, $V$ ) * $I$ = current (Amps, $A$ ) * $R$ = resistance (Ohms, $\Omega$ ) * Constraint: This law only applies to metal conductors at a constant temperature. Once heated, the relationship is no longer linear.
Electric Power ( $P$ ): * Every appliance has a power rating (e.g., Stove: $12000\,W$ , Lamp: $150\,W$ , Clothes Dryer: $5000\,W$ , Clock: $5\,W$ , CFL: $10-13\,W$ ). * Power Formulas: 1. $P = VI$ 2. $P = I^2 R$ 3. $P = \frac{V^2}{R}$
Worked Cost Calculation (Page 9): * Given: $V = 120\,V$ , $I = 0.4\,A$ , rate = $14.5\,c/kWh$ , duration $t = 4\,h$ . * Power: $P = VI = 120 \times 0.4 = 48\,W = 0.048\,kW$ . * Energy: $E_Q = P \times t = 0.048 \times 4 = 0.192\,kWh$ . * Cost: $0.192 \times 14.5 = 2.78\,cents$ . (If run twice, total cost = $5.57\,cents$ ).

Electrical Environment and Safety

Automobile Grounding: * Instead of running separate return wires for every load (Light 1, Light 2, etc.), the metal body of the vehicle acts as one large common ground wire to return charge to the battery.
Residential Wiring: * Unlike the DC (Direct Current) used in cars, homes use AC (Alternating Current). * In the US, AC reverses direction 60 times per second ( $60\,Hz$ ). * Flicker Effect: TVs and computer screens flicker at $60\,Hz$ . Humans don't notice it, but cameras with different frame rates capture this effect.
Safety Devices: * Circuit Breakers: Switches that flip to break the circuit if current levels become dangerous. * Fuses: Devices that contain a metal strip designed to "blow" (melt) and break the circuit when current is too high.

Circuit Analysis

Circuit Components: * Load: Anything that draws current (resistors, bulbs, motors). * Power Source: Cells, batteries, AC/DC generators. * Switch: Opens (breaks) or Closes (completes) the path.
Series Circuits: * Current remains the same through all components: $I_s = I_1 = I_2 = I_3 = ...$ * Total Voltage is the sum of drops: $V_s = V_1 + V_2 + V_3 + ...$ * Equivalent Resistance: $R_{eq} = R_1 + R_2 + R_3 + ...$
Parallel Circuits: * Voltage is the same across all branches: $V_s = V_1 = V_2 = V_3 = ...$ * Total Current is the sum of branch currents: $I_s = I_1 + I_2 + I_3 + ...$ * Equivalent Resistance Inverse: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$ * Node Concept (Nodal Analysis): Current entering a junction must equal current leaving the junction.

Kirchhoff’s Rules for Complex Circuits

Junction Rule (1st Rule): The sum of all currents entering a node equals the sum of currents leaving.
Loop Rule (2nd Rule): The algebraic sum of changes in potential around any closed circuit loop must be zero. * Convention: Sources are positive, voltage drops (across resistors) are negative when moving in the direction of conventional current.

Matrix Solving Method: * For circuits that cannot be simplified via series-parallel reduction, a system of equations ( $3 \times 3$ matrix) is used. * General form: $I_1 R_1 + I_2 R_2 + I_3 R_3 = V$ . * Equation 1 (Loop 1): $-6I_1 - 3I_2 + 0I_3 = -18$ * Equation 2 (Loop 2): $6I_1 + 0I_2 + 2I_3 = 45$ * Equation 3 (Junction): $I_1 - I_2 - I_3 = 0$ * Solving using $A^{-1} \times B$ yields: * $I_1 = 4.75\,A$ * $I_2 = -3.5\,A$ (Negative implies the assumed direction was wrong) * $I_3 = 8.25\,A$

Efficiency Formulas and Practice

Efficiency ( $Eff$ ): $Eff = \frac{\text{Work Output}}{\text{Energy Input}} \times 100\%$
Practice Problem Result (Motor): * Input Energy: $0.24\,kWh$ . * Work Output: $0.18\,kWh$ . * Efficiency: $\frac{0.18}{0.24} \times 100\% = 75\%$ . * Cost to run (2 hours at $10.81\,c/kWh$ ): $0.24 \times 10.81 = 2.59\,cents$ .