Rucete ✏ AP Physics C In a Nutshell
13. Analysis of Circuits Containing Batteries, Resistors and Capacitors
This chapter introduces the definitions of current, drift velocity, resistance, Ohm’s law, resistor networks, Kirchhoff’s laws for complex circuits, and the behavior of capacitors in circuits with resistors.
Current and Conventional Current
• Current is the rate of charge flow through a surface: I = dQ/dt, units of amperes (A = C/s).
• Conventional current assumes positive charges move from high to low potential, even though electrons actually flow from low to high potential.
• Conventional current direction and actual electron flow are opposite but yield correct circuit analysis results.
Current Density and Drift Velocity
• Current density J is a vector pointing in the direction of conventional current, defined as J = I/A for uniform flow.
• Drift velocity vD is the small net motion of electrons under an electric field, related by J = nevD, where n is the mobile electron density.
• Even though electrons move randomly at high speeds, drift velocity causes net current flow.
Resistance and Resistivity
• Ohm’s law: V = IR relates voltage, current, and resistance.
• Resistance depends on material resistivity ρ and geometry: R = ρL/A.
• Resistivity is a material property; conductivity σ is its reciprocal (σ = 1/ρ).
Calculating Resistance from Geometry
• For a uniform cylindrical resistor, E = V/L, J = E/ρ, and thus I = VA/ρL.
• Final resistance formula: R = ρL/A.
Resistors in Parallel
• Voltage across each resistor is the same.
• Total current is the sum of currents through each resistor: I = I₁ + I₂ + …
• Reciprocal of equivalent resistance: 1/Req = 1/R₁ + 1/R₂ + …
• Adding resistors in parallel decreases overall resistance.
Resistors in Series
• Current through each resistor is the same.
• Total voltage is the sum of voltages across each resistor: V = V₁ + V₂ + …
• Equivalent resistance: Req = R₁ + R₂ + …
• Adding resistors in series increases total resistance.
Complex Resistor Networks
• Reduce complex networks by identifying simple series or parallel groups.
• Replace groups with their equivalent resistance step-by-step.
• Repeat until only one equivalent resistor remains.
Single-Battery Circuit Analysis
• In a circuit with a single battery, wires are assumed to have negligible resistance, so the potential is constant along them.
• At any junction (node), the total current entering equals the total current leaving (Kirchhoff’s node rule).
• To analyze, use Ohm’s law across the total equivalent resistance to find total current, then subdivide as needed.
Voltmeters and Ammeters
• Voltmeters are connected in parallel to measure potential difference; designed with very high resistance to minimize current draw.
• Ammeters are connected in series to measure current; designed with very low resistance to minimize voltage drop.
Kirchhoff’s Laws for Multi-Battery Circuits
• Node rule: Total current entering a node equals total current leaving.
• Loop rule: The sum of potential differences around any closed loop is zero.
• Apply Kirchhoff’s laws systematically by assigning current directions and creating independent equations for each loop and node.
Applying Kirchhoff’s Laws: Key Steps
• Assign current directions (guesses are okay; wrong guesses result in negative currents).
• Create loop equations and node equations until the number of independent equations matches the number of unknowns.
• Solve the system of equations by substitution or elimination.
Power Dissipation in Circuits
• Power dissipated by a resistor: P = IV = I²R = V²/R.
• As current flows through resistors, electrical energy is converted into heat or other forms.
Internal Resistance of Batteries
• Real batteries have internal resistance; terminal voltage is less than EMF when current flows: Vterminal = EMF − Ir.
• Maximum current occurs when external resistance is zero: Imax = EMF/Rinternal.
• Internal resistance explains why high-current circuits reduce terminal voltage.
RC Circuits: Discharging Capacitors
• When a charged capacitor discharges through a resistor, charge and current decrease exponentially.
• Charge as a function of time: Q(t) = Q₀e^(−t/RC).
• Current as a function of time: I(t) = (Q₀/RC)e^(−t/RC).
RC Circuits: Charging Capacitors
• When charging, current starts at maximum and decreases exponentially.
• Charge builds up over time: Q(t) = Qfinal(1 − e^(−t/RC)).
• Current during charging: I(t) = (Qfinal/RC)e^(−t/RC).
Time Constant (τ)
• Defined as τ = RC.
• Time constant represents the characteristic time for charge, current, or voltage to decrease by a factor of e (≈2.718).
• After time t = τ, discharging or charging processes are approximately 63% complete.
In a Nutshell
Understanding circuits involving batteries, resistors, and capacitors requires mastering current flow, voltage differences, and resistance. Ohm’s law and Kirchhoff’s rules provide powerful tools for solving circuit problems. Capacitors introduce time-dependent behaviors, characterized by exponential changes governed by the time constant RC. Analyzing both steady-state and transient circuit behaviors is crucial for deeper insight into real-world electrical systems.