ap physics c: e&m unit 11 study guides

Key Concepts and Definitions

• Electric current ($I$) is the flow of electric charge through a conductor, measured in amperes (A)
• Voltage ($V$) represents the potential difference between two points in a circuit, measured in volts (V)
  • Voltage sources (batteries, power supplies) provide the energy needed to move charges through a circuit
• Resistance ($R$) opposes the flow of electric current, measured in ohms ($\Omega$)
  • Conductors (copper wire) have low resistance, while insulators (rubber, plastic) have high resistance
• Power ($P$) is the rate at which energy is transferred or consumed in a circuit, measured in watts (W)
  • $P = IV$ where $I$ is current and $V$ is voltage
• Ideal circuit elements have constant properties and behave according to simple mathematical models
  • Real circuit components exhibit non-ideal behaviors (temperature dependence, nonlinearity)

Circuit Components and Symbols

• Resistors are components that provide a specific amount of resistance in a circuit, represented by a zigzag line symbol
  • Fixed resistors have a constant resistance value, while variable resistors (potentiometers) allow for adjustable resistance
• Capacitors store electric charge and energy in an electric field, represented by two parallel lines symbol
  • Capacitance ($C$) is measured in farads (F) and depends on the size, shape, and material of the capacitor plates
• Inductors store energy in a magnetic field when current flows through them, represented by a coiled wire symbol
• Switches control the flow of current by opening or closing a circuit path, represented by a line with a break symbol
• Voltage and current sources provide the energy to drive a circuit, represented by circles with plus and minus signs
  • Ideal sources maintain constant voltage or current regardless of the load

Ohm's Law and Resistance

• Ohm's law states that the current through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance
  • $V = IR$ where $V$ is voltage, $I$ is current, and $R$ is resistance
• Resistance is a material property that depends on the length ($L$), cross-sectional area ($A$), and resistivity ($\rho$) of a conductor
  • $R = \rho L/A$ where $\rho$ is the material-specific resistivity
• Resistivity varies with temperature, typically increasing for metals and decreasing for semiconductors as temperature rises
  • Temperature coefficient of resistance ($\alpha$) quantifies the change in resistance per degree of temperature change
• Conductance ($G$) is the reciprocal of resistance and measures a material's ability to conduct electric current
  • $G = 1/R$ and is measured in siemens (S)

Series and Parallel Circuits

• In series circuits, components are connected end-to-end, forming a single path for current flow
  • Total resistance is the sum of individual resistances: $R_{total} = R_1 + R_2 + ... + R_n$
  • Current is the same through all components, while voltage divides across each component proportionally to its resistance
• In parallel circuits, components are connected side-by-side, forming multiple paths for current flow
  • Total resistance is the reciprocal of the sum of reciprocal resistances: $1/R_{total} = 1/R_1 + 1/R_2 + ... + 1/R_n$
  • Voltage is the same across all components, while current divides among the parallel branches
• Series and parallel connections can be combined to form complex circuits
  • Equivalent resistance can be found by simplifying the circuit using series and parallel reduction techniques

Kirchhoff's Laws

• Kirchhoff's current law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node
  • $\sum I_{in} = \sum I_{out}$ at any point in a circuit
  • Ensures conservation of charge in a circuit
• Kirchhoff's voltage law (KVL) states that the sum of voltage drops around any closed loop in a circuit equals zero
  • $\sum V = 0$ for any complete loop
  • Voltage drops across resistors have opposite signs to voltage rises from sources
• KCL and KVL form the basis for circuit analysis techniques (node-voltage, mesh-current)
  • Apply KCL at non-reference nodes and KVL around independent loops to generate equations
  • Solve the system of linear equations to determine unknown currents and voltages

Capacitors and RC Circuits

• Capacitors store energy in an electric field between two conducting plates separated by an insulator (dielectric)
  • Capacitance ($C$) is the ratio of charge stored ($Q$) to the voltage applied ($V$): $C = Q/V$
  • Parallel plate capacitance depends on the plate area ($A$), separation distance ($d$), and dielectric constant ($\kappa$): $C = \kappa \varepsilon_0 A/d$
• In series, the total capacitance is the reciprocal of the sum of reciprocal capacitances: $1/C_{total} = 1/C_1 + 1/C_2 + ... + 1/C_n$
• In parallel, the total capacitance is the sum of individual capacitances: $C_{total} = C_1 + C_2 + ... + C_n$
• RC circuits contain resistors and capacitors, exhibiting time-dependent behavior
  • Charging and discharging curves follow exponential functions with time constants $\tau = RC$
  • Voltage across a capacitor cannot change instantaneously, leading to transient responses

Circuit Analysis Techniques

• Node-voltage method applies KCL to non-reference nodes, expressing currents in terms of unknown node voltages
  • Solve the resulting system of equations to determine node voltages, then calculate branch currents
• Mesh-current method applies KVL around independent loops, expressing voltage drops in terms of unknown loop currents
  • Solve the resulting system of equations to determine loop currents, then calculate node voltages
• Superposition principle allows analyzing circuits with multiple sources by considering the effect of each source independently
  • Find the contribution of each source to the desired quantity (current or voltage) while setting other sources to zero
  • Sum the individual contributions to obtain the total response
• Thévenin and Norton equivalent circuits simplify complex networks by representing them as a single voltage or current source with a series or parallel resistance
  • Thévenin equivalent: Voltage source ($V_{Th}$) in series with resistance ($R_{Th}$)
  • Norton equivalent: Current source ($I_{N}$) in parallel with resistance ($R_{N}$)

Real-World Applications

• Electric power distribution systems use transformers to step up voltage for long-distance transmission and step down voltage for local distribution
  • High voltage minimizes power losses due to resistance in transmission lines
• Household wiring uses parallel circuits to distribute power to multiple devices
  • Outlets, switches, and appliances are connected in parallel to maintain constant voltage
• Electronic devices (smartphones, computers) contain complex integrated circuits with millions of transistors, resistors, and capacitors
  • Integrated circuits perform logic operations, store data, and control the flow of information
• Sensors and transducers convert physical quantities (temperature, pressure, light) into electrical signals
  • Wheatstone bridge circuits detect small changes in resistance, used in strain gauges and load cells
• Electrical safety devices (fuses, circuit breakers) protect against overcurrent conditions
  • Fuses contain a thin wire that melts and breaks the circuit when current exceeds a specified limit
  • Circuit breakers use electromagnets to trip a switch and interrupt the current flow