Unit 3: Electric Circuits

📚Study Guide: Electric Circuits

Unit 3: Electric Circuits

Electric circuits in AP Physics C: E&M extend the algebraic treatment of Physics 2 to include current density, microscopic forms of Ohm's law, and the full calculus of RC circuits. Current I is defined as the rate of charge flow, I = dQ/dt. At a microscopic level, current density J = I/A = nqv_d, where n is the charge carrier density, q is the charge per carrier, and v_d is the drift velocity. The microscopic form of Ohm's Law relates current density to electric field: J = σE, where σ is the conductivity (or E = ρJ, where ρ is resistivity). Resistance of a conductor depends on its geometry and material: R = ρL/A. You must be able to derive this from the microscopic relations. Kirchhoff's Loop Rule (ΣΔV = 0) and Junction Rule (ΣI_in = ΣI_out) remain the foundations of circuit analysis, but in Physics C you will apply them to more complex multi-loop circuits and be expected to set up systems of equations. The unit emphasizes RC circuits analyzed through differential equations. For a charging capacitor in series with a resistor and battery, Kirchhoff's Voltage Law gives ε − IR − Q/C = 0. Since I = dQ/dt, this becomes a first-order linear differential equation: dQ/dt + Q/(RC) = ε/R. The solution is Q(t) = Q_max(1 − e^(−t/τ)), where τ = RC is the time constant and Q_max = Cε. For discharging, Q(t) = Q₀e^(−t/τ). The current during charging is I(t) = (ε/R)e^(−t/τ); during discharging, I(t) = −(Q₀/RC)e^(−t/τ). You should be able to derive these solutions by separation of variables. The voltage across the capacitor is V_C = Q/C, and across the resistor is V_R = IR. On the AP Exam, circuit problems require setting up differential equations, solving them, and interpreting the time-dependent behavior of charge, current, and voltage.

Key Concepts

• Current and Current Density: I = dQ/dt. J = I/A = nqv_d. Current density is a vector parallel to the flow of positive charge.
• Microscopic Ohm's Law: J = σE or E = ρJ. The electric field inside a conductor drives the current, and resistivity opposes it.
• Resistance from Geometry: R = ρL/A. Longer and thinner wires have higher resistance.
• EMF and Terminal Voltage: ε is the ideal source voltage. Terminal voltage V_term = ε − Ir, where r is internal resistance.
• Kirchhoff's Rules: Junction Rule (conservation of charge) and Loop Rule (conservation of energy). Essential for multi-loop circuit analysis.
• RC Charging Differential Equation: dQ/dt + Q/(RC) = ε/R. Solution: Q(t) = Cε(1 − e^(−t/τ)).
• RC Discharging: dQ/dt = −Q/(RC). Solution: Q(t) = Q₀e^(−t/τ).

Vocabulary

• Current Density (J): Current per unit area. Vector. Unit: A/m².
• Drift Velocity (v_d): The average velocity of charge carriers in a conductor under an electric field. Very slow (~mm/s) compared to signal speed.
• Resistivity (ρ): An intrinsic property of a material opposing current flow. Unit: Ω·m.
• Conductivity (σ): The reciprocal of resistivity; σ = 1/ρ. Unit: (Ω·m)⁻¹.
• EMF (ε): The energy supplied by a source per unit charge. Ideal voltage.
• Terminal Voltage: Actual voltage across a source's terminals under load.
• Internal Resistance (r): The resistance within a real battery that causes terminal voltage to drop with current.
• Time Constant (τ): For RC circuits, τ = RC. Characterizes the charging/discharging rate.

Essential Formulas

• I = dQ / dt
• J = n*q*v_d = I / A
• E = ρ * J (microscopic Ohm's law)
• R = ρ * L / A
• V = I * R
• ΣI_in = ΣI_out (junction rule)
• ΣΔV = 0 (loop rule)
• τ = R * C
• Q(t) = Q_max * (1 - e^(-t/τ)) (charging)
• Q(t) = Q0 * e^(-t/τ) (discharging)
• I(t) = (ε/R) * e^(-t/τ) (charging current)

Common Mistakes

• Confusing EMF with Terminal Voltage: Under load, terminal voltage is less than emf due to internal resistance: V = ε − Ir.
• Forgetting Internal Resistance Drops Voltage: Real batteries have r > 0. As current increases, terminal voltage decreases.
• Sign Errors in KVL: Walking around a loop, potential rises across batteries (from − to +) and drops across resistors (in direction of current). Be consistent.
• Solving RC Without Separation of Variables: You must separate dQ/dt = (ε − Q/C)/R into dQ/(εC − Q) = dt/(RC) and integrate. Do not guess the form.

AP Exam Strategies

• Assign Current Directions Arbitrarily: If your result is negative, the actual direction is opposite. State this clearly in free-response.
• Walk Loops Consistently: Choose a direction (clockwise/counterclockwise) and stick to it. Add emfs as rises and IR drops as falls.
• For RC, Write the Differential Equation via KVL: Start with ε − IR − Q/C = 0, substitute I = dQ/dt, and separate variables. Show all steps.
• Remember Charging Starts at Q = 0, Discharging Starts at Q₀: Apply the correct initial condition to your integrated equation to find the constant.

Real-World Applications

• Batteries and Internal Resistance: All real batteries have internal resistance, which limits short-circuit current and reduces terminal voltage under heavy loads.
• Electronic Timing Circuits: RC circuits are used in timers, oscillators, and pulse generators because of their predictable exponential charging curves.
• Pacemakers: Implanted medical devices use RC timing circuits to deliver precisely timed electrical pulses to regulate heart rhythm.