AP Physics 2

🧲AP Physics 2 Unit 4 – Electric Circuits

Electric circuits form the backbone of modern electronics, powering our devices and enabling countless technological advancements. This unit explores the fundamental concepts of electric current, voltage, and resistance, as well as their relationships in various circuit configurations. Students will learn about circuit components, Ohm's Law, and Kirchhoff's Laws, which are essential for analyzing both simple and complex circuits. The unit also covers capacitors, AC vs. DC circuits, and problem-solving strategies for tackling circuit-related challenges in physics and engineering applications.

Study Guides for Unit 4

4.0

Unit 4 Overview: Electric Circuits

6 min read

4.1

Definition and Conservation of Electric Charge

6 min read

4.2

Resistivity and Resistance

7 min read

4.3

Resistance and Capacitance

8 min read

4.4

Kirchhoff’s Loop Rule

4 min read

4.5

Kirchhoff’s Junction Rule and the Conservation of Electric Charge

5 min read

Key Concepts and Definitions

Electric current ( $I$ ) represents the flow of electric charge through a circuit measured in amperes (A)
Voltage ( $V$ $V$ ) 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 ( $R$ $R$ ) opposes the flow of electric current in a circuit measured in ohms ( $\Omega$ $Ω$ )
- 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 ( $V$ $V$ ), current ( $I$ $I$ ), and resistance ( $R$ $R$ ) 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 ( $L$ $L$ ) and inversely proportional to its cross-sectional area ( $A$ $A$ )
- 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 $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$ $\frac{1}{R _{t o t a l}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + ... + \frac{1}{R _{n}}$
  - 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 ( $C$ $C$ ) 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 ( $A$ $A$ ), plate separation ( $d$ $d$ ), and the dielectric material between the plates (permittivity, $\varepsilon$ $ε$ )
- 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 ( $Q$ $Q$ ) and its capacitance: $V = \frac{Q}{C}$ $V = \frac{Q}{C}$
- 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 $E = \frac{1}{2}CV^2$ $E = \frac{1}{2} C V^{2}$ or $E = \frac{1}{2}\frac{Q^2}{C}$ $E = \frac{1}{2} \frac{Q ^{2}}{C}$
- 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 ( $Z$ $Z$ ) 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 $P = I^2R$ $P = I^{2} R$ ):
    - $P_1 = I_1^2R_1 = (12 A)^2(2 \Omega) = 288 W$
    - $P_2 = I_2^2R_2 = (6 A)^2(4 \Omega) = 144 W$
    - $P_3 = I_3^2R_3 = (4 A)^2(6 \Omega) = 96 W$
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