🧲AP Physics 2 Unit 4 – Electric Circuits

Key Concepts and Definitions

• Electric current ($I$) represents the flow of electric charge through a circuit measured in amperes (A)
• Voltage ($V$) describes the potential difference between two points in a circuit measured in volts (V)
  • Voltage acts as the driving force that pushes electric current through a circuit
• Resistance ($R$) opposes the flow of electric current in a circuit measured in ohms ($\Omega$)
  • Conductors (copper, aluminum) have low resistance allowing current to flow easily
  • Insulators (rubber, plastic) have high resistance limiting current flow
• Power ($P$) quantifies the rate at which electrical energy is converted into other forms (heat, light) measured in watts (W)
• Conventional current assumes positive charges flow from the positive terminal to the negative terminal
  • In reality, electrons flow from the negative terminal to the positive terminal (electron flow)
• An electric circuit consists of a closed loop that allows electric current to flow from a voltage source to a load and back to the source

Circuit Components and Symbols

• Voltage sources provide the energy needed to drive electric current through a circuit
  • Batteries (DC) and generators (AC) are common examples of voltage sources
  • Ideal voltage sources maintain a constant voltage regardless of the current drawn
• Resistors limit the flow of electric current in a circuit
  • Resistor symbols include a zigzag line or rectangular box labeled with resistance value
  • Variable resistors (potentiometers) allow resistance to be adjusted
• Switches control the flow of current by opening or closing a circuit
  • Open switches break the circuit preventing current flow
  • Closed switches complete the circuit allowing current to flow
• Capacitors store electric charge and energy in an electric field between two conductive plates
  • Capacitor symbols consist of two parallel lines representing the plates
• Inductors store energy in a magnetic field generated by the current flowing through a coil of wire
  • Inductor symbols resemble a coil or spiral
• Diodes allow current to flow in only one direction acting as one-way valves
  • Diode symbols include a triangle pointing in the direction of conventional current flow

Ohm's Law and Resistance

• Ohm's law describes the relationship between voltage ($V$), current ($I$), and resistance ($R$) in a circuit
  • Mathematically expressed as $V = IR$
  • Doubling the voltage across a resistor doubles the current flowing through it
  • Doubling the resistance halves the current flowing through the resistor
• Resistance is an intrinsic property of a material that opposes the flow of electric current
  • Resistance depends on the material's resistivity, length, and cross-sectional area
  • Resistivity is a measure of a material's inherent resistance to current flow
• The resistance of a conductor is directly proportional to its length ($L$) and inversely proportional to its cross-sectional area ($A$)
  • Mathematically expressed as $R = \rho \frac{L}{A}$, where $\rho$ is the material's resistivity
• Temperature affects the resistance of materials
  • Most conductors exhibit an increase in resistance with increasing temperature (positive temperature coefficient)
  • Some materials, such as semiconductors, display a decrease in resistance with increasing temperature (negative temperature coefficient)
• Resistors can be combined in series or parallel to achieve desired resistance values
  • Series combinations: $R_{total} = R_1 + R_2 + ... + R_n$
  • Parallel combinations: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$

Series and Parallel Circuits

• In a series circuit, components are connected end-to-end forming a single path for current flow
  • Current is the same through all components in a series circuit
  • Total voltage across the series circuit equals the sum of voltages across individual components
  • Total resistance in a series circuit is the sum of individual resistances: $R_{total} = R_1 + R_2 + ... + R_n$
• In a parallel circuit, components are connected across the same two nodes forming multiple paths for current flow
  • Voltage is the same across all components connected in parallel
  • Total current in a parallel circuit equals the sum of currents through individual branches
  • Total resistance in a parallel circuit is given by $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
    • Parallel resistance is always less than the smallest individual resistance
• Series and parallel circuits can be combined to form complex networks
  • Identify series and parallel sections, then simplify each section using appropriate formulas
  • Redraw the simplified circuit and repeat the process until the entire circuit is reduced to a single equivalent resistance
• Voltage dividers are series circuits used to produce a desired output voltage
  • Output voltage depends on the ratio of resistances: $V_{out} = V_{in} \frac{R_2}{R_1 + R_2}$
• Current dividers are parallel circuits used to split current into desired proportions
  • Branch currents are inversely proportional to the resistances: $I_1 : I_2 = R_2 : R_1$

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
  • Mathematically, $\sum I_{in} = \sum I_{out}$
  • KCL is based on the conservation of electric charge
  • Applying KCL helps determine unknown currents in a circuit
• Kirchhoff's voltage law (KVL) states that the sum of voltage drops around any closed loop in a circuit equals the sum of voltage rises
  • Mathematically, $\sum V_{rises} = \sum V_{drops}$
  • KVL is based on the conservation of energy
  • Applying KVL helps determine unknown voltages in a circuit
• To apply Kirchhoff's laws:
  • Assign labels and reference directions for currents and voltages
  • Apply KCL at each node to obtain current equations
  • Apply KVL around each independent loop to obtain voltage equations
  • Solve the system of equations to determine unknown currents and voltages
• Kirchhoff's laws are essential for analyzing complex circuits that cannot be simplified using series-parallel combinations alone

Capacitors and Capacitance

• Capacitance ($C$) is a measure of a capacitor's ability to store electric charge and energy
  • Capacitance is measured in farads (F), where 1 F = 1 C/V
  • Typical capacitor values range from picofarads (pF) to microfarads (μF)
• The capacitance of a parallel-plate capacitor depends on the plate area ($A$), plate separation ($d$), and the dielectric material between the plates (permittivity, $\varepsilon$)
  • Mathematically, $C = \frac{\varepsilon A}{d}$
  • Increasing plate area or decreasing plate separation increases capacitance
• Capacitors can be connected in series or parallel to achieve desired capacitance values
  • Series combinations: $\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
  • Parallel combinations: $C_{total} = C_1 + C_2 + ... + C_n$
• The voltage across a capacitor depends on the charge stored ($Q$) and its capacitance: $V = \frac{Q}{C}$
  • Capacitors oppose changes in voltage by storing or releasing charge
  • Capacitors act as open circuits to DC and as short circuits to high-frequency AC
• The energy stored in a capacitor is given by $E = \frac{1}{2}CV^2$ or $E = \frac{1}{2}\frac{Q^2}{C}$
  • Energy is stored in the electric field between the capacitor plates

DC vs. AC Circuits

• Direct current (DC) circuits involve current that flows in one direction at a constant level
  • Batteries and solar cells are examples of DC sources
  • In DC circuits, capacitors act as open circuits and inductors act as short circuits (steady-state)
• Alternating current (AC) circuits involve current that periodically reverses direction
  • AC is characterized by its frequency (Hz), amplitude, and phase
  • Wall outlets (120 V, 60 Hz) and generators are examples of AC sources
• In AC circuits, capacitors and inductors exhibit frequency-dependent behavior
  • Capacitive reactance ($X_C$) decreases with increasing frequency: $X_C = \frac{1}{2\pi fC}$
  • Inductive reactance ($X_L$) increases with increasing frequency: $X_L = 2\pi fL$
• Impedance ($Z$) is the total opposition to current flow in an AC circuit, including resistance, capacitive reactance, and inductive reactance
  • Impedance is a complex quantity measured in ohms: $Z = R + j(X_L - X_C)$
• Power in AC circuits consists of real power (P), reactive power (Q), and apparent power (S)
  • Real power is the average power dissipated as heat (watts)
  • Reactive power is the power stored and released by capacitors and inductors (volt-amperes reactive, VAR)
  • Apparent power is the total power supplied by the source (volt-amperes, VA)

Problem-Solving Strategies and Examples

• Identify the type of circuit (DC or AC, series or parallel) and the components involved
• Draw a clear and labeled circuit diagram, including component values and reference directions for currents and voltages
• Determine the unknown quantities (currents, voltages, resistances, power) and the relevant equations (Ohm's law, Kirchhoff's laws, power formulas)
• Example: In a series circuit with a 12 V battery and two resistors (R1 = 4 Ω, R2 = 6 Ω), find the current through each resistor and the voltage across each resistor
  • Solution:
    • Total resistance: $R_{total} = R_1 + R_2 = 4 \Omega + 6 \Omega = 10 \Omega$
    • Current (using Ohm's law): $I = \frac{V}{R_{total}} = \frac{12 V}{10 \Omega} = 1.2 A$
    • Voltage across R1: $V_{R1} = IR_1 = (1.2 A)(4 \Omega) = 4.8 V$
    • Voltage across R2: $V_{R2} = IR_2 = (1.2 A)(6 \Omega) = 7.2 V$
• Example: In a parallel circuit with a 24 V battery and three resistors (R1 = 2 Ω, R2 = 4 Ω, R3 = 6 Ω), find the total current and the power dissipated by each resistor
  • Solution:
    • Current through each resistor (using Ohm's law):
      • $I_1 = \frac{V}{R_1} = \frac{24 V}{2 \Omega} = 12 A$
      • $I_2 = \frac{V}{R_2} = \frac{24 V}{4 \Omega} = 6 A$
      • $I_3 = \frac{V}{R_3} = \frac{24 V}{6 \Omega} = 4 A$
    • Total current: $I_{total} = I_1 + I_2 + I_3 = 12 A + 6 A + 4 A = 22 A$
    • Power dissipated by each resistor (using $P = I^2R$):
      • $P_1 = I_1^2R_1 = (12 A)^2(2 \Omega) = 288 W$
      • $P_2 = I_2^2R_2 = (6 A)^2(4 \Omega) = 144 W$
      • $P_3 = I_3^2R_3 = (4 A)^2(6 \Omega) = 96 W$
• When solving complex circuits, break the problem into manageable steps and apply appropriate laws and formulas to each step
  • Simplify the circuit by identifying series and parallel combinations
  • Apply Kirchhoff's laws to determine unknown currents and voltages
  • Use Ohm's law and power formulas to calculate resistance, current, voltage, and power