🧲AP Physics 2 (2025) Unit 11 – Electric Circuits

Key Concepts and Definitions

• Electric current $I$ represents the flow of electric charge, measured in amperes (A)
  • Conventionally, current flows from positive to negative terminals in a circuit
• Voltage $V$, also known as electric potential difference, is the energy per unit charge, measured in volts (V)
  • Voltage is the driving force that causes electric current to flow in a circuit
• Resistance $R$ is the opposition to the flow of electric current, measured in ohms ($\Omega$)
  • Materials with high resistance (insulators) impede current flow, while materials with low resistance (conductors) allow current to flow easily
• Conductance $G$ is the reciprocal of resistance, measured in siemens (S)
  • Conductance represents the ease with which electric current flows through a material
• Capacitance $C$ is the ability of a component to store electric charge, measured in farads (F)
  • Capacitors are devices that store energy in an electric field between two conducting plates

Circuit Components and Symbols

• Resistors are components that oppose the flow of electric current, represented by the symbol $\Omega$
  • Resistors are used to control current flow and voltage drops in a circuit
  • The resistance of a resistor is indicated by its color code or numerical value
• Voltage sources, such as batteries or power supplies, provide the energy to drive current through a circuit, represented by the symbols $\boxed{+}$ and $\boxed{-}$
• Switches are used to open or close a circuit, controlling the flow of current, represented by the symbols $\frac{}{o}$ (open) and $\frac{}{-}$ (closed)
• Capacitors store electric charge and energy, represented by the symbol $\boxed{||}$
  • Capacitors are often used to smooth voltage fluctuations or store energy for later use
• Wires are used to connect components in a circuit, represented by straight lines
  • Ideal wires have negligible resistance and allow current to flow freely

Ohm's Law and Basic Relationships

• Ohm's Law states that the voltage $V$ across a resistor is directly proportional to the current $I$ flowing through it, with the constant of proportionality being the resistance $R$
  • Mathematically, Ohm's Law is expressed as $V = IR$
• The relationship between current, voltage, and resistance can be rearranged to solve for any of the three variables
  • $I = \frac{V}{R}$ and $R = \frac{V}{I}$
• Conductance $G$ is related to resistance by $G = \frac{1}{R}$
  • Ohm's Law can also be expressed using conductance as $I = GV$
• The power $P$ dissipated by a resistor is given by $P = IV$, $P = I^2R$, or $P = \frac{V^2}{R}$
  • Power is measured in watts (W) and represents the rate at which energy is converted from electrical to other forms (heat, light, etc.)

Series and Parallel Circuits

• In a series circuit, components are connected end-to-end, forming a single path for current to flow
  • The current is the same through all components in a series circuit
  • The total voltage across a series circuit is the sum of the voltages across each component
  • The equivalent resistance of a series circuit is the sum of the individual resistances: $R_{eq} = R_1 + R_2 + ... + R_n$
• In a parallel circuit, components are connected side-by-side, forming multiple paths for current to flow
  • The voltage is the same across all components in a parallel circuit
  • The total current in a parallel circuit is the sum of the currents through each branch
  • The reciprocal of the equivalent resistance of a parallel circuit is the sum of the reciprocals of the individual resistances: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
• Complex circuits can be analyzed by breaking them down into series and parallel sections
  • Equivalent resistances can be calculated for each section and then combined to find the total resistance of the circuit

Kirchhoff's Laws

• Kirchhoff's Current Law (KCL) states that the sum of currents entering a node (junction) in a circuit must equal the sum of currents leaving the node
  • Mathematically, $\sum I_{in} = \sum I_{out}$
  • KCL is based on the conservation of electric charge, ensuring that charge does not accumulate at any point in a circuit
• Kirchhoff's Voltage Law (KVL) states that the sum of the voltage drops around any closed loop in a circuit must equal zero
  • Mathematically, $\sum V = 0$ around a closed loop
  • KVL is based on the conservation of energy, ensuring that the total energy change around a closed loop is zero
• Kirchhoff's Laws are essential for analyzing complex circuits with multiple loops and nodes
  • By applying KCL and KVL, you can set up a system of equations to solve for unknown currents and voltages in a circuit

Power and Energy in Circuits

• Power $P$ is the rate at which energy is transferred or converted, measured in watts (W)
  • In electrical circuits, power is the product of voltage and current: $P = IV$
• Energy $E$ is the capacity to do work, measured in joules (J)
  • The energy consumed or stored in a circuit is the product of power and time: $E = Pt$
• The power dissipated by a resistor can be calculated using $P = I^2R$ or $P = \frac{V^2}{R}$
  • Resistors convert electrical energy into heat energy
• The energy stored in a capacitor is given by $E = \frac{1}{2}CV^2$
  • Capacitors store energy in an electric field between their plates
• The efficiency of a circuit is the ratio of useful output power to total input power, expressed as a percentage
  • Efficient circuits minimize power losses due to factors such as resistance and heat dissipation

Capacitors and RC Circuits

• Capacitors are components that store electric charge and energy in an electric field between two conducting plates
  • The capacitance $C$ of a capacitor is the ratio of the charge $Q$ stored to the voltage $V$ across its plates: $C = \frac{Q}{V}$
• In a series connection, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
• In a parallel connection, the equivalent capacitance is the sum of the individual capacitances: $C_{eq} = C_1 + C_2 + ... + C_n$
• RC circuits consist of a resistor and a capacitor connected in series
  • The time constant $\tau$ of an RC circuit is the product of the resistance and capacitance: $\tau = RC$
  • The time constant represents the time it takes for the capacitor to charge or discharge to 63.2% of its final value
• The voltage across a capacitor in an RC circuit during charging is given by $V_C(t) = V_S(1 - e^{-t/\tau})$, where $V_S$ is the supply voltage
• The voltage across a capacitor in an RC circuit during discharging is given by $V_C(t) = V_0e^{-t/\tau}$, where $V_0$ is the initial voltage across the capacitor

Practical Applications and Lab Work

• Voltage dividers are circuits that use resistors in series to produce a desired output voltage
  • The output voltage of a voltage divider is a fraction of the input voltage, determined by the ratio of the resistances
• Wheatstone bridges are circuits used to measure an unknown resistance by balancing the voltages across four resistors
  • When the bridge is balanced, the unknown resistance can be calculated using the values of the other three resistors
• Potentiometers are variable resistors used to control voltage or current in a circuit
  • By adjusting the position of a sliding contact along a resistive element, the resistance and thus the voltage or current can be varied
• Oscilloscopes are instruments used to visualize and measure voltage signals over time
  • Oscilloscopes display voltage on the vertical axis and time on the horizontal axis, allowing for the analysis of waveforms and transient behavior in circuits
• In lab experiments, students may:
  • Construct simple circuits using breadboards, resistors, capacitors, and voltage sources
  • Measure voltage, current, and resistance using multimeters
  • Verify Ohm's Law and Kirchhoff's Laws through experimental data
  • Investigate the charging and discharging behavior of RC circuits
  • Analyze the frequency response of RC filters
  • Build and test voltage dividers, Wheatstone bridges, and potentiometers