Comprehensive Guide to DC Circuits, Capacitance, and Electrical Measurements

Sources of Electromotive Force (emf)

• Definition of emf (\epsilon):
      * The source that maintains the current in a closed circuit is called a source of emf, represented by $\epsilon$.
      * Any devices that increase the potential energy of charges circulating in circuits are considered sources of emf. Examples include batteries and generators.
      * The SI unit for emf is the volt ($V$).
      * Physically, the emf is the work done per unit charge: $\text{emf} = \frac{\text{Work}}{\text{Charge}}$.

• Emf and Internal Resistance (r):
      * A real battery possesses some internal resistance, denoted as $r$. Because of this, the terminal voltage is not equal to the emf.
      * As charge passes from the negative to the positive terminal of the battery, the potential of the charge increases by the amount $\epsilon$.
      * As the charge moves through the internal resistance $r$, the potential decreases by the amount $Ir$.
      * Terminal Voltage (\Delta V) Formula:
          $\Delta V = V_b - V_a$
          $\Delta V = \epsilon - Ir$
      * Open-Circuit Voltage: $\epsilon$ is equal to the terminal voltage when the current $I = 0$.

• Circuit Relationships:
      * For an entire circuit where a source of emf with internal resistance $r$ is connected to an external resistor $R$ (often called the load resistance):
          $\epsilon = IR + Ir$
      * The current depends on both the external load resistance and the internal resistance of the battery.
      * Power Relationship:
          $I\epsilon = I^2 R + I^2 r$
      * When the load resistance is significantly larger than the internal resistance ($R \gg r$), the internal resistance $r$ can be ignored. In this case, most of the power delivered by the battery is transferred to the load resistor.
      * Battery Variables:
          * The battery is a source of constant emf $\epsilon$.
          * The battery cannot be considered a source of constant current because it depends on circuit resistance.
          * The terminal voltage of a battery is not considered constant because internal resistance may change over time.

Resistors in Series

• Definition: Two or more resistors are connected end-to-end so that they provide a single path for the current.

• Current Relationship: The current is the same in all resistors because any charge flowing through one resistor must flow through the others.
      $I = I_1 = I_2$

• Potential Difference Relationship: The sum of the potential differences across the individual resistors is equal to the total potential difference provided by the source.
      $\Delta V = \Delta V_1 + \Delta V_2$
      $\Delta V = IR_1 + IR_2 = I(R_1 + R_2)$

• Equivalent Resistance (R_{eq}):
      * The equivalent resistance has the same effect on the circuit as the original combination.
      * For three or more resistors in series:
          $R_{eq} = R_1 + R_2 + R_3 + \dots$
      * The equivalent resistance of a series combination is the algebraic sum of individual resistances and is always greater than any of the individual resistances.

• Circuit Failure Case: If one single element in a series circuit fails (e.g., a broken wire or blown bulb), the circuit is no longer complete, and none of the other elements will work.

• Example 18-1 Breakdown:
      * Four resistors are arranged in a series circuit with a closed-circuit terminal voltage of $6.0\,V$.
      * To find $R_{eq}$, sum the values: $R_{eq} = R_1 + R_2 + R_3 + R_4$.
      * To find current ($I$): $I = \frac{\Delta V}{R_{eq}}$.
      * The electric potential at point A is determined based on a positive terminal potential of $6.0\,V$.
      * If the open circuit voltage (emf) $\epsilon$ is $6.2\,V$, internal resistance $r$ can be calculated as:
          $r = \frac{\epsilon - \Delta V}{I}$
      * The fraction $f$ of power delivered to the load is $f = \frac{I^2 R_{eq}}{I\epsilon}$.

Resistors in Parallel

• Definition: Resistors are in parallel when they are connected across the same potential difference.

• Potential Difference Relationship: The potential difference across each resistor is the same because each is connected directly across the battery terminals.
      $\Delta V = \Delta V_1 = \Delta V_2$

• Current Relationship: The total current ($I$) entering a junction must equal the sum of the currents leaving that point.
      $I = I_1 + I_2$
      * Currents through parallel branches are generally not the same unless the resistances are identical.

• Equivalent Resistance (R_{eq}):
      * The inverse of the equivalent resistance is the algebraic sum of the inverses of the individual resistances.
      $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$
      * Key Logic: The equivalent resistance of a parallel combination is always less than the smallest individual resistor in the group.

• Applications: Household circuits are wired in parallel so that individual electrical devices can be turned on or off independently without breaking the entire circuit.

• Example 18-2 Breakdown:
      * Three resistors in parallel with $\Delta V = 18\,V$ between points a and b.
      * Current in each resistor: $I_n = \frac{\Delta V}{R_n}$.
      * Power delivered to each: $P_n = I_n^2 R_n = \frac{(\Delta V)^2}{R_n}$.
      * Total power is the sum of individual powers or $P_{total} = I_{total} \times \Delta V$.

Kirchhoff’s Rules and Complex DC Circuits

• Overview: Many circuits cannot be reduced to simple series or parallel combinations. These complex circuits require Kirchhoff’s Rules.

• Kirchhoff's First Rule (Junction Rule):
      * Based on the Law of Conservation of Charge.
      * Statement: The sum of the currents entering any junction must equal the sum of the currents leaving that junction.
      $\sum I_{in} = \sum I_{out}$
      $I_1 - I_2 - I_3 = 0$

• Kirchhoff's Second Rule (Loop Rule):
      * Based on the Law of Conservation of Energy.
      * Statement: The sum of the potential differences across all elements around any closed circuit loop must be zero.
      $\sum \Delta V = 0$

• Sign Conventions for the Loop Rule:
      * Resistors:
          * If the resistor is traversed in the direction of the assigned current: $\Delta V = -IR$.
          * If the resistor is traversed opposite to the direction of the assigned current: $\Delta V = +IR$.
      * Batteries (Sources of emf):
          * If the source is traversed from the negative terminal to the positive terminal (direction of the emf): $\Delta V = +\epsilon$.
          * If the source is traversed from the positive terminal to the negative terminal (opposite to the emf): $\Delta V = -\epsilon$.

• Problem-Solving Strategy:
      1. Assign symbols and directions to currents in all branches. If your guess is wrong, the final value will be negative but have the correct magnitude.
      2. Choose a direction (clockwise or counterclockwise) to traverse the loops.
      3. Apply the junction rule to junctions and the loop rule to closed loops to generate independent equations.
      4. Solve the simultaneous equations for the unknown currents.

• Exercise Example (15.0V battery, 100.0 ohm resistors):
      * Equation 1 (Junction): $I_1 - I_2 - I_3 = 0$
      * Equation 2 (Loop 1): $\epsilon - I_1 R - I_2 R = 0$
      * Equation 3 (Loop 2): $I_2 R - I_3 R = 0$
      * Results: $I_3 = 0.05\,A$, $I_2 = 0.05\,A$, $I_1 = 0.10\,A$.

• Example 18-5 (Multi-Battery Circuit):
      * Equation (Junction): $I_3 = I_1 + I_2$
      * Loop abcda: $10\,V - (6.0)I_1 - (2.0)I_3 = 0$
      * Loop befch: $-(4.0)I_2 - 14\,V + (6.0)I_1 - 10\,V = 0$
      * Calculating reveals: $I_1 = 2.0\,A$, $I_2 = -3.0\,A$, $I_3 = -1.0\,A$.

RC Circuits (Resistor-Capacitor)

• Behavior Over Time: In circuits containing both capacitors and resistors, the current varies with time.

• Charging a Capacitor:
      * When the circuit is completed, the capacitor starts to charge until it reaches maximum charge ($Q = C\epsilon$).
      * Once fully charged, the current in the circuit drops to zero.
      * Charge Equation: $q = Q(1 - e^{-t/RC})$
      * The constant $e$ is Euler's constant ($e \approx 2.718$).

• Time Constant (\tau):
      * $\tau = RC$
      * Represent the time required for the charge to increase from zero to $63.2\%$ of its maximum value ($Q_{max}$).
      * Large $\tau$: Capacitor charges slowly.
      * Small $\tau$: Capacitor charges quickly.
      * After $t = 10\tau$, the capacitor is effectively over $99.99\%$ charged.

• Discharging a Capacitor:
      * A charged capacitor can be discharged through a resistor.
      * Charge Equation: $q = Q e^{-t/RC}$
      * The charge decreases exponentially. At $t = \tau$, the charge decreases to $0.368 Q_{max}$ (meaning it has lost $63.2\%$ of its initial charge).

Combinations of Capacitors

• Capacitors in Parallel:
      * Potential difference ($\Delta V$) across each capacitor is the same: $\Delta V_1 = \Delta V_2 = \Delta V$.
      * The total charge ($Q$) is the sum of charges on each capacitor: $Q = Q_1 + Q_2$.
      * Equivalent capacitance ($C_{eq}$) is the sum of individual capacitances:
          $C_{eq} = C_1 + C_2 + C_3 + \dots$
      * The equivalent capacitance is always greater than any of the individual capacitors.
      * Mechanism: Electrons transfer until the voltage across the capacitors equals the battery voltage.

• Capacitors in Series:
      * The magnitude of the charge ($Q$) on all plates is the same: $Q = Q_1 = Q_2$.
      * Potential differences across capacitors add up to the battery voltage: $\Delta V = \Delta V_1 + \Delta V_2$.
      * Equivalent capacitance ($C_{eq}$) is found via the sum of the inverses:
          $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$
      * The equivalent capacitance is always less than any individual capacitor in the combination.
      * Inverse Relationship Note: Capacitors in series combine logically like resistors in parallel, and capacitors in parallel combine like resistors in series.

Current and Voltage Measurements in Circuits

• Circuit Fundamentals: A circuit is a closed path around which current circulates. Key quantities are current ($I$) and potential difference ($\Delta V$).

• Ammeter:
      * Function: Measures electrical current.
      * Connection: Must be connected in series so that all charge passing through the component also passes through the meter.
      * Ideal Resistance: Resistance should be as small as possible to avoid affecting the current it is measuring.

• Voltmeter:
      * Function: Measures voltage (potential difference).
      * Connection: Must be connected in parallel across the two contacts of the component being measured.
      * Ideal Resistance: Resistance should be as large as possible to prevent current from being diverted through the meter.