Unit 11: Electric Circuits

Current, Resistance, and Ohm’s Law

• Electric Current: Electric current (\(I\)) is the rate at which electric charge flows through a conductor, measured in amperes (A), where \(1\,\text{A} = 1\,\text{C/s}\). In a metal conductor, this movement is due to drifting electrons responding to an electric field applied by a voltage source. While individual electrons move relatively slowly (drift velocity), the electric field propagates nearly at the speed of light, allowing the circuit to respond almost instantly when switched on.
• Resistance: Resistance (\(R\)) quantifies how much a material opposes the flow of electric current, measured in ohms (\(\Omega\)). It depends on the material’s resistivity (\(\rho\)), length (\(L\)), and cross-sectional area (\(A\)) according to \( R = \rho \frac{L}{A} \). Longer conductors and smaller cross-sectional areas lead to greater resistance, while materials with low resistivity like copper or silver are excellent conductors.
• Ohm’s Law: Ohm’s Law states that \( V = IR \), relating voltage (\(V\)), current (\(I\)), and resistance (\(R\)) in an ohmic device. This relationship assumes the resistance is constant and independent of voltage or current, which holds true for many conductors but not for all devices (e.g., diodes, filament bulbs). In calculus-based analysis, instantaneous voltage and current can be considered for time-dependent situations, leading to differential equation modeling in complex circuits.
• Power in Electrical Circuits: Electrical power (\(P\)) describes the rate at which energy is converted or used in a circuit element, given by \( P = IV \), or equivalently \( P = I^2R \) or \( P = \frac{V^2}{R} \). This concept is essential for determining how much energy a resistor dissipates as heat and for ensuring that components operate within safe power limits. Power considerations are also critical in designing circuits for efficiency and safety.
• Microscopic View of Conduction: From a physics standpoint, conduction in metals can be modeled by the Drude model, where electrons behave like a gas of charged particles undergoing collisions with lattice ions. The average time between collisions (relaxation time) influences conductivity, and temperature changes can significantly alter resistivity. This connection bridges macroscopic circuit laws with the microscopic properties of materials.

Resistors in Series and Parallel

• Series Resistors: When resistors are connected end-to-end in a single path, they are in series, and the total resistance is the sum of the individual resistances: \( R_{\text{eq}} = R_1 + R_2 + R_3 + \dots \). This happens because current must pass through each resistor sequentially, and the voltage drop across the combination is the sum of individual voltage drops. In series, the same current flows through all components, but the voltages can differ depending on resistance values.
• Parallel Resistors: In a parallel arrangement, resistors are connected so both ends share the same two nodes, and the reciprocal of the total resistance is the sum of the reciprocals of individual resistances: \( \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots \). Here, the voltage across each resistor is the same, but currents through each branch may differ depending on the resistance. Parallel configurations reduce the equivalent resistance, allowing more current flow for the same applied voltage.
• Mixed Arrangements: Many circuits feature a combination of series and parallel resistors. In such cases, simplification is done step-by-step by first identifying simple series or parallel groups and reducing them to equivalent resistances. This process is crucial before applying Kirchhoff’s Laws to complex circuits.
• Current and Voltage Division: In series circuits, voltage divides proportionally to resistance values (\( V_i = IR_i \)), whereas in parallel circuits, current divides inversely proportional to resistance (\( I_i = \frac{V}{R_i} \)). These rules allow quick determination of how voltage and current are distributed without solving full systems of equations, especially in symmetric circuits.
• Energy Dissipation Differences: In series, all resistors carry the same current, so power dissipation varies depending on each resistance value (\( P = I^2 R \)). In parallel, all resistors share the same voltage, and power dissipation depends on both voltage and resistance. Understanding these differences is essential for designing safe and efficient electrical systems.

Kirchhoff’s Laws

• Kirchhoff’s Junction Rule: The junction rule states that the total current entering a junction must equal the total current leaving it. This is a direct application of the conservation of electric charge — charges cannot accumulate at a point in a steady-state circuit. For example, if 4 A flows into a junction and splits into two branches, one carrying 1.5 A and the other 2.5 A, the sum leaving (1.5 A + 2.5 A) equals the incoming 4 A.
• Kirchhoff’s Loop Rule: The loop rule states that the algebraic sum of all potential differences around any closed loop in a circuit must equal zero. This is a direct consequence of the conservation of energy, as a charge completing a loop returns to its starting point with the same total energy. When applying the loop rule, you must account for both EMF sources (batteries) and voltage drops across resistors, paying attention to sign conventions depending on the direction of traversal.
• Applying the Laws Together: Complex circuits often require simultaneous application of both the junction and loop rules to determine unknown currents and voltages. Typically, you begin with the junction rule to set up relationships between currents, then use the loop rule to form equations involving resistances and EMFs. Solving these systems can involve multiple equations and is especially important for multi-loop problems in AP Physics C.
• Sign Conventions: In the loop rule, moving from the negative to positive terminal of a battery counts as a positive EMF, while moving through a resistor in the direction of current results in a voltage drop (negative value). Reversing direction in the analysis changes these signs accordingly. Consistent sign conventions prevent algebraic mistakes when solving circuit equations.
• Limitations: Kirchhoff’s laws assume steady-state DC conditions where capacitors are fully charged and inductors (if present) have steady current. In time-dependent situations (like charging a capacitor), these rules still apply, but the voltage-current relationships for the components must also reflect their changing states.

RC Circuits and Time-Dependent Behavior

• Introduction to RC Circuits: An RC circuit contains at least one resistor (R) and one capacitor (C), often connected to a DC voltage source. The capacitor stores energy in the electric field between its plates, while the resistor controls the rate of current flow. Unlike purely resistive circuits, RC circuits exhibit time-dependent behavior — the voltage and current change continuously as the capacitor charges or discharges.
• Charging a Capacitor: When a capacitor begins charging through a resistor from an initial uncharged state, the voltage across the capacitor increases exponentially toward the battery voltage \(V_0\). The equation for the capacitor voltage is \( V_C(t) = V_0 \left( 1 - e^{-t/RC} \right) \), where \(RC\) is the time constant. The current decreases over time according to \( I(t) = \frac{V_0}{R} e^{-t/RC} \), meaning the charging happens quickly at first and then slows.
• Discharging a Capacitor: When the circuit is disconnected from the battery and the capacitor is allowed to discharge through the resistor, its voltage decreases exponentially: \( V_C(t) = V_0 e^{-t/RC} \). The current also decays exponentially in the same fashion, following \( I(t) = -\frac{V_0}{R} e^{-t/RC} \). The negative sign indicates the current flows in the opposite direction compared to charging.
• Time Constant (\(\tau\)): The product \( \tau = RC \) is known as the time constant of the circuit, and it determines how quickly the capacitor charges or discharges. After one time constant, the voltage change reaches about 63% of its final value (charging) or decreases to about 37% of its initial value (discharging). Larger resistance or capacitance values increase the time constant, making the process slower.
• Energy Considerations: The energy stored in a fully charged capacitor is given by \( U = \frac{1}{2} C V^2 \). During charging, this energy comes from the battery, but not all of it ends up stored in the capacitor — some is dissipated as thermal energy in the resistor. During discharging, all of the capacitor’s stored energy is released, again partly as heat in the resistor.

Power in Electrical Circuits (In-Depth)

• Definition of Electrical Power: Electrical power is the rate at which energy is transferred or converted within a circuit. In a resistor, power represents the rate at which electrical energy is transformed into thermal energy due to collisions between electrons and the atomic lattice. The basic expression for power is \( P = IV \), where \( I \) is the current through the component and \( V \) is the potential difference across it.
• Alternative Power Formulas: Using Ohm’s Law (\( V = IR \)), we can rewrite the power formula in two other forms: \( P = I^2 R \) and \( P = \frac{V^2}{R} \). The form you choose depends on which quantities are known in the problem. For example, \( P = I^2 R \) is useful when current and resistance are known, while \( P = \frac{V^2}{R} \) is useful when voltage and resistance are given.
• Power in Series vs. Parallel Circuits: In a series circuit, the same current flows through all resistors, so power dissipation in each resistor depends only on its resistance. In parallel circuits, each branch has the same voltage but different currents, so power distribution depends on the branch resistances. Understanding this helps in analyzing which components might overheat or consume more energy.
• Efficiency Considerations: Not all electrical power is converted to useful work. In real systems, resistive losses occur in wires and components, reducing efficiency. Power transmission lines, for example, use high voltages to reduce current and therefore minimize \( I^2 R \) losses over long distances.
• Energy Calculations Over Time: Since energy is the product of power and time (\( E = P \cdot t \)), we can determine how much total energy is consumed or converted in a given interval. This concept is directly applied in household energy billing, where energy use is measured in kilowatt-hours (kWh). In physics problems, using joules instead of kWh ensures consistency with SI units.

Energy Considerations in Circuits

• Conservation of Energy in Circuits: Just as in mechanics, energy in an electrical circuit is conserved. The energy supplied by the power source (such as a battery) is exactly equal to the total energy dissipated or stored in the circuit’s components. In resistors, energy is converted into heat; in capacitors, it is stored in an electric field; and in inductors (covered later), it is stored in a magnetic field.
• Energy Supplied by a Battery: A battery provides electrical energy by converting chemical potential energy into electric potential energy of charges. If a battery delivers a current \( I \) for a time \( t \) at voltage \( V \), the total energy supplied is \( E = IVt \). This energy is then distributed across the components according to their individual voltage drops and resistances.
• Energy Dissipation in Resistors: The primary mechanism of energy loss in most circuits is through resistive heating, described by Joule’s law \( E = I^2 R t \). This heat is an irreversible form of energy transformation, and in practical systems, minimizing these losses is key to improving efficiency — for example, by reducing wire resistance or increasing transmission voltage.
• Energy Storage in Capacitors: Capacitors store energy in the form of an electric field between their plates, given by \( U = \frac{1}{2} C V^2 \). When connected in a circuit, they can release this stored energy to provide a temporary power source, smooth voltage fluctuations, or filter signals, depending on the application.
• Energy Flow in Multi-Component Circuits: In more complex circuits, Kirchhoff’s Loop Rule ensures that the algebraic sum of potential changes is zero, meaning energy gains from sources exactly match energy losses and storage in the loop. This principle allows us to verify solutions and confirm that no energy is lost or gained mysteriously in an idealized system.

Capacitors in Circuits (with Resistors)

• Capacitor Basics in Circuits: A capacitor stores energy in an electric field between its plates and can be combined with resistors in a variety of configurations. When connected to a DC voltage source, a capacitor charges up until the voltage across it matches the source voltage, at which point current in the branch stops. The rate at which it charges or discharges depends on both the resistance and the capacitance in the circuit.
• Charging a Capacitor Through a Resistor: In an RC charging circuit, the capacitor voltage increases over time following \( V_C(t) = V_0 (1 - e^{-t/RC}) \), where \( R \) is the resistance and \( C \) is the capacitance. Initially, the current is maximum and decreases exponentially as the capacitor approaches full charge. This time-dependent behavior is governed by the time constant \( \tau = RC \), which represents the time it takes for the voltage to reach about 63% of its final value.
• Discharging a Capacitor Through a Resistor: When a charged capacitor is connected across a resistor without a power source, it releases its stored energy, and the voltage decreases as \( V_C(t) = V_0 e^{-t/RC} \). The current also decreases exponentially, and after about \( 5\tau \), the capacitor is effectively fully discharged. This principle is used in timing circuits and camera flash mechanisms.
• Series and Parallel Capacitor Combinations: Capacitors combine in the opposite way to resistors: in series, \( \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots \), and in parallel, \( C_{\text{eq}} = C_1 + C_2 + \dots \). This allows precise tuning of total capacitance in a circuit to achieve desired voltage storage or filtering behavior.
• Energy Stored in Capacitors: The energy stored in a capacitor is given by \( U = \frac{1}{2} C V^2 \). In RC circuits, part of this energy is dissipated as heat in the resistor during charging and discharging, showing a direct link between circuit analysis and energy conservation principles. Understanding this helps engineers design circuits to minimize energy loss in sensitive electronics.

Multi-Loop Circuit Analysis (with Calculus Where Needed)

• Purpose of Multi-Loop Analysis: Many real-world circuits are more complex than a single loop and include multiple branches with shared components. Analyzing these circuits requires applying Kirchhoff’s Laws systematically to each loop and junction. This ensures that current and voltage relationships are consistent throughout the network and allows for solving unknown values even when the circuit has multiple sources and resistive paths.
• Applying Kirchhoff’s Loop Rule: For each independent loop, sum the potential rises and drops and set the total equal to zero: \( \sum \Delta V = 0 \). Voltage gains occur across sources in the direction of current, and voltage drops occur across resistors according to Ohm’s Law \( V = IR \). By writing equations for each loop, you generate a system of equations that can be solved simultaneously for unknown currents and voltages.
• Using the Junction Rule for Currents: At any junction (node) where wires split or merge, the sum of currents entering must equal the sum of currents leaving. This reflects charge conservation and allows you to relate currents in different branches before applying loop equations. Combining the junction rule with loop equations often provides just enough information to solve for all unknowns in the circuit.
• Inclusion of Capacitors and Time Dependence: When capacitors are part of a multi-loop circuit, their voltage and current relationships require calculus. The current through a capacitor is given by \( I = C \frac{dV}{dt} \), so a loop equation may lead to a differential equation that describes the charging or discharging process. Solving these equations reveals exponential behavior similar to single-loop RC circuits but now influenced by interactions between multiple loops.
• Strategies for Complex Systems: For especially complex circuits, matrix methods or computational tools can be used to solve simultaneous equations. However, the underlying physics still comes from Kirchhoff’s Laws, Ohm’s Law, and the capacitor current-voltage relationship. By mastering these fundamentals, you can confidently approach any circuit, no matter how many loops or branches it contains.

Steady-State Direct-Current Circuits with Batteries and Resistors Only

• Definition of Steady State: In a direct-current (DC) circuit containing only resistors and batteries, the steady state refers to the condition where currents and voltages are constant in time. This occurs after all transient effects (such as capacitor charging) have died out, meaning the circuit has reached equilibrium and behaves predictably according to Ohm’s Law and Kirchhoff’s Laws.
• Application of Ohm’s Law: In steady state, the voltage across each resistor is directly proportional to the current through it, following \( V = IR \). Since the current is constant, there is no accumulation of charge anywhere in the circuit, and energy is continuously supplied by the battery to balance the energy dissipated as heat in the resistors.
• Single-Loop and Multi-Loop Cases: For a single loop with one battery and one resistor, the steady-state current is simply \( I = \frac{V_{\text{battery}}}{R} \). In multi-loop resistor networks, steady-state analysis requires Kirchhoff’s Loop Rule to account for multiple voltage sources and resistors, ensuring the total potential change in each closed loop is zero.
• Power Considerations: In steady state, the power delivered by the battery is \( P = IV \), which is exactly equal to the sum of the power dissipated in each resistor, \( P_{\text{resistor}} = I^2 R \). This energy balance is a direct application of the conservation of energy in electrical systems.
• Importance in Circuit Design: Understanding steady-state behavior is essential for designing systems that rely on constant operating conditions, such as household wiring or DC power supplies. It also serves as the baseline for analyzing more complex transient behaviors when capacitors or inductors are introduced.

Charging Capacitors in a Circuit

• Physical Picture of Charging: When a switch connects a resistor–capacitor (RC) branch to a battery, electrons are pushed off one plate and drawn onto the other, building equal and opposite charges. The growing charge creates an increasing electric field between the plates and a rising capacitor voltage \( V_C(t) \). Because the resistor limits current, the charge and voltage do not jump instantly to their final values but evolve smoothly in time toward steady state.
• Differential Equation and Time Constant: Applying Kirchhoff’s Loop Rule to a simple series RC charging loop gives \( V_0 - iR - V_C = 0 \) with \( i = \frac{dq}{dt} \) and \( V_C = \frac{q}{C} \). Rearranging yields \( \frac{dq}{dt} = \frac{1}{RC}(C V_0 - q) \), a first-order linear ODE whose solution is \( q(t) = C V_0 \big(1 - e^{-t/RC}\big) \). The parameter \( \tau = RC \) is the **time constant**, setting the characteristic scale for how quickly the capacitor approaches its final charge.
• Key Charging Relations: From the charge solution follow the standard results \( V_C(t) = V_0 \big(1 - e^{-t/\tau}\big) \) and \( i(t) = \frac{V_0}{R} e^{-t/\tau} \). The current starts at a maximum \( i(0)=V_0/R \) because the initially uncharged capacitor behaves like a short, then decays exponentially as the growing \( V_C \) opposes the battery. After about \( 5\tau \), \( V_C \) is within a fraction of a percent of \( V_0 \), which is effectively “fully charged” for practical purposes.
• Energy Flow During Charging: The battery does positive work on charges; part becomes stored as electric potential energy \( U=\tfrac12 C V_C^2 \) in the capacitor and part is dissipated as thermal energy \( I^2 R \) in the resistor. Integrating power shows that, for a full charge from 0 to \( V_0 \), exactly half the energy delivered by the battery is stored in the capacitor and half is lost as heat. This split is independent of \( R \) because a larger \( R \) reduces current but extends the charging time proportionally.
• Multiple RC Branches and Effective \(\tau\): In networks, the effective resistance seen by the capacitor determines the time constant \( \tau = R_{\text{th}}C \) where \( R_{\text{th}} \) is the Thevenin resistance “looking into” the charging path. Replacing the surrounding network by its Thevenin equivalent clarifies dynamics and preserves the same \( V_0 \) and \( \tau \). This method generalizes the single-loop formulas to complex circuits without re-deriving new differential equations each time.

Discharging Capacitors in a Circuit

• Physical Picture of Discharge: If a charged capacitor is disconnected from its source and connected across a resistor, the stored electric field drives charges to flow, reducing the plate charge magnitude. The capacitor’s voltage drops as energy leaves the field and is converted to heat in the resistor. The process is spontaneous and continues until the plates become neutral and no potential difference remains.
• Differential Equation and Solution: For a simple RC discharge loop, Kirchhoff’s Rule gives \( V_C - iR = 0 \) with \( V_C = q/C \) and \( i = -\frac{dq}{dt} \) (since charge on the initially positive plate decreases). This yields \( \frac{dq}{dt} = -\frac{1}{RC} q \), whose solution is \( q(t) = q_0 e^{-t/RC} \). Consequently \( V_C(t) = V_0 e^{-t/\tau} \) and \( i(t) = \frac{V_0}{R} e^{-t/\tau} \) in magnitude, showing the same time constant \( \tau = RC \) as in charging.
• Energy Dissipation: The initial energy in the capacitor is \( U_0 = \tfrac12 C V_0^2 \), and during discharge all of this energy is dissipated as thermal energy in the resistor. The instantaneous power in the resistor is \( P_R(t)=i^2(t)R=\frac{V_0^2}{R} e^{-2t/\tau} \), which integrates to \( U_0 \) over \( t\in[0,\infty) \). No external work is required because the field itself supplies the energy that is lost as heat.
• Initial Conditions and Polarity: The exponential form preserves the sign of \( V_C \): the voltage decays toward zero from its initial value without overshoot in ideal RC circuits. If a loop contains multiple sources or branches, polarity conventions must be consistent so that \( q(0) \) and \( V_0 \) represent the actual initial capacitor voltage. Thevenin equivalents again simplify analysis by reducing the surrounding network seen by the capacitor.
• Practical “End” of Discharge: Mathematically the decay continues indefinitely, but practically the capacitor is considered discharged after about \( 5\tau \). Beyond this time the residual voltage is typically <1% of its initial value and is negligible for most applications. Safety procedures often use this rule of thumb when bleeding off high-voltage capacitors through resistors.

Gauss’s Law (Concise Review in Circuit Context)

• Statement and Meaning: Gauss’s Law states that the total electric flux through a closed surface equals the enclosed charge divided by permittivity, \( \displaystyle \oint \vec{E}\cdot d\vec{A}=\frac{q_{\text{enc}}}{\varepsilon_0} \). In circuit settings this law explains why excess charge on conductors resides on their outer surfaces and why fields inside ideal conductors are zero at electrostatic equilibrium. It also justifies the perpendicular field at a conductor’s surface and relates it to surface charge density via \( E_{\perp}=\sigma/\varepsilon_0 \).
• Steady DC and Local Charge Neutrality: In steady-state DC circuits, wires are nearly neutral everywhere, with only tiny surface charge redistributions establishing the internal electric field that drives current. Gauss’s Law, combined with continuity of current, shows that these surface charges adjust so that \( \vec{E} \) inside the metal points along the wire and maintains constant current without significant bulk charge accumulation. This resolves the common misconception that batteries “push” charges without any role for surface charge patterns.
• Capacitors and Field Confinement: For parallel-plate capacitors, a pillbox Gaussian surface straddling one plate gives \( E=\sigma/\varepsilon_0 \) (vacuum), predicting a nearly uniform field between wide plates. Because equal and opposite charges live on facing surfaces, the flux through a Gaussian surface outside both plates cancels, yielding negligible external field in the ideal limit. This field confinement underlies why capacitors store energy locally and why fringing is minimal when plate spacing is small relative to dimensions.
• Cavities and Shielding: A closed conducting shell with no enclosed charge has zero net flux through any Gaussian surface entirely inside the metal and therefore zero field within the cavity. In circuit environments this is the basis of electrostatic shielding (Faraday cages), protecting sensitive components from external static fields. If a charge is placed inside the cavity, induced surface charges on the inner wall adjust so that field lines begin and end on those surfaces consistent with Gauss’s Law.
• Symmetry vs. Usefulness: While Gauss’s Law is universally true, it is only computationally convenient when sufficient symmetry makes \( \vec{E} \) uniform over parts of the Gaussian surface. In circuits, it serves less for calculation of wire fields and more as a conceptual tool explaining conductor behavior, boundary conditions, and the relation between surface charge and the fields that drive current. Recognizing when symmetry applies prevents wasted effort and guides correct physical intuition.

Practice Problem 1: RC Charging Time to 90%

Question: A capacitor (\(C = 10\,\mu\text{F}\)) is charged through a resistor (\(R = 2.0\,\text{k}\Omega\)) by a battery of emf \(V_0\). How long after closing the switch does the capacitor voltage first reach \(0.90\,V_0\)?

• Write the charging law: For a series RC charging circuit, the capacitor voltage is \( V_C(t) = V_0\!\left(1 - e^{-t/RC}\right) \). Here the time constant is \( \tau = RC \).
• Set the target level: We need \( V_C(t) = 0.90\,V_0 \). Substitute into the charging law: \( 0.90\,V_0 = V_0\!\left(1 - e^{-t/\tau}\right) \Rightarrow 0.90 = 1 - e^{-t/\tau} \).
• Solve for the exponential: Rearranging gives \( e^{-t/\tau} = 0.10 \). Taking natural logs: \( -\frac{t}{\tau} = \ln(0.10) = -\ln(10) \Rightarrow \frac{t}{\tau} = \ln(10) \approx 2.3026 \).
• Compute the time constant: \( \tau = RC = (2.0\times10^3\,\Omega)(10\times10^{-6}\,\text{F}) = 2.0\times10^{-2}\,\text{s} = 0.020\,\text{s} \).
• Evaluate the time: \( t = \tau \ln(10) \approx (0.020\,\text{s})(2.3026) \approx \boxed{4.6\times10^{-2}\,\text{s} \; (46\,\text{ms})} \).

Practice Problem 2: Steady-State Multi-Loop (Batteries + Resistors Only)

Question: Two rectangular loops share a central resistor \(R_3=3\,\Omega\). The left loop has a battery \(12\,\text{V}\) in series with \(R_1=6\,\Omega\); the right loop has a battery \(9\,\text{V}\) in series with \(R_2=3\,\Omega\). The middle \(R_3\) connects the midpoint between \(R_1\) and the 12 V battery to the midpoint between \(R_2\) and the 9 V battery. In steady state (DC), find the magnitude and direction of the current through \(R_3\).

• Define mesh currents: Let \(i_1\) be clockwise current in the left loop and \(i_2\) be clockwise in the right loop. The current through the shared resistor is \(i_{3}=i_1 - i_2\) (left-to-right is positive).
• Write KVL for the left loop: Going clockwise across the 12 V source, \(R_1\), and \(R_3\): \(12 - 6i_1 - 3(i_1 - i_2)=0 \Rightarrow 12 - 9i_1 + 3i_2 = 0\).
• Write KVL for the right loop: Going clockwise across the 9 V source, \(R_2\), and \(R_3\): \(9 - 3i_2 - 3(i_2 - i_1)=0 \Rightarrow 9 + 3i_1 - 6i_2 = 0\).
• Solve the linear system: From the first: \(3i_2 = 9i_1 - 12 \Rightarrow i_2 = 3i_1 - 4\). Substitute into the second: \(9 + 3i_1 - 6(3i_1 - 4)=0 \Rightarrow 9 + 3i_1 - 18i_1 + 24 = 0 \Rightarrow -15i_1 + 33 = 0 \Rightarrow i_1 = 2.2\,\text{A}\). Then \( i_2 = 3(2.2) - 4 = 2.6\,\text{A}\).
• Find \(i_3\) and interpret direction: \( i_3 = i_1 - i_2 = 2.2 - 2.6 = -0.4\,\text{A} \). The negative sign means the actual current is \(0.40\,\text{A}\) from right to left through \(R_3\). Thus the answer is \(\boxed{0.40\,\text{A}, \text{ right-to-left through } R_3}\).

Common Misconceptions in RC Circuits and Steady-State DC Analysis

• Thinking a capacitor always passes current: Many students mistakenly believe that a capacitor continuously allows current to flow in a DC circuit. In reality, a capacitor only allows current while it is charging or discharging. Once fully charged in a steady-state DC circuit, it behaves like an open circuit, and no current flows through it. This misunderstanding often leads to incorrect predictions about long-term circuit behavior.
• Assuming resistors in parallel always share equal current: In parallel connections, currents split according to resistance values, not equally by default. A lower resistance branch will carry more current, while a higher resistance branch carries less. Forgetting this principle can cause serious errors when applying Kirchhoff’s rules or calculating power dissipation.
• Confusing time constant with total charging time: The RC time constant \(\tau = RC\) is the time it takes for the capacitor to reach about 63% of its final voltage (or discharge to 37%). Some students incorrectly assume the capacitor is fully charged or discharged after one \(\tau\). In practice, reaching over 99% takes about \(5\tau\), which is critical when designing or analyzing timing circuits.
• Forgetting direction conventions in Kirchhoff’s Laws: When applying loop and junction rules, students sometimes mix up current directions, especially in shared resistors between loops. While an incorrect initial guess for direction will still lead to a correct magnitude (but negative sign), mixing conventions within the same equation can lead to impossible results and confusion.
• Neglecting the role of internal resistance in “ideal” batteries: Although we often treat batteries as ideal sources in basic problems, real batteries have small internal resistances that affect current distribution. Ignoring this in higher-level analysis (such as AP Physics C free-response) can lead to overestimating current and underestimating voltage drops.