Physics

Chapter 10 Direct-Current Circuits

Imron Rosyadi

Direct-Current Circuits

Learning Objectives

By the end of this chapter, you will be able to:

• Describe the electromotive force (emf) and the internal resistance of a battery.
• Explain the basic operation of a battery.
• Define the term equivalent resistance.
• Calculate the equivalent resistance of resistors connected in series.
• Calculate the equivalent resistance of resistors connected in parallel.
• State Kirchhoff’s junction rule.
• State Kirchhoff’s loop rule.
• Analyze complex circuits using Kirchhoff’s rules.
• Describe how to connect a voltmeter in a circuit to measure voltage.
• Describe how to connect an ammeter in a circuit to measure current.
• Describe the use of an ohmmeter.
• Describe the charging process of a capacitor.
• Describe the discharging process of a capacitor.
• List some applications of RC circuits.
• List the basic concepts involved in house wiring.
• Define the terms thermal hazard and shock hazard.
• Describe the effects of electrical shock on human physiology and their relationship to the amount of current through the body.
• Explain the function of fuses and circuit breakers.

10.1 Electromotive Force (EMF)

• Voltage sources, like batteries, create a potential difference and can supply current.
• The electromotive force (emf) is a special type of potential difference.
• Coined by Alessandro Volta, emf is not a force but a measure of energy per unit charge.
• Often referred to simply as “sources of emf.”

Historically, the term “electromotive force” was used, but it’s important to clarify that it’s not a force in the Newtonian sense. Instead, it’s the energy provided per unit charge by a source, which drives current in a circuit. Volta invented the first battery, the voltaic pile, which was a significant advancement in electricity.

EMF and Batteries

• Batteries are two-terminal devices that maintain a potential difference.
• Positive terminal (higher potential) and negative terminal (lower potential).
• The emf source acts as a charge pump.
• Moves negative charges from the positive terminal to the negative terminal.
• This process requires energy, often from chemical reactions in a battery.

Inside a battery, chemical reactions do work to move charges. This work increases the potential energy of the charges, creating the potential difference between the terminals. When connected to a circuit, these charges then flow from higher to lower potential, delivering energy to components like a lamp.

Visualizing Voltage Sources

• Various sources create voltage:
  • Wind farms
  • Dams (hydroelectric)
  • Solar farms
  • Batteries
• Each device’s voltage output depends on its construction and load.
• Voltage output equals emf only when there is no load.

Caption: A variety of voltage sources. (a) The Brazos Wind Farm in Fluvanna, Texas; (b) the Krasnoyarsk Dam in Russia; (c) a solar farm; (d) a group of nickel metal hydride batteries.

The image illustrates diverse methods of generating electrical potential difference. It’s crucial for students to understand that “voltage source” is a broad term encompassing many energy conversion technologies, not just batteries. The concept of “no load” is important for defining emf precisely.

EMF in a Circuit

• Current Flow:
  • Positive charges (conventional current) leave the positive terminal, travel through the load, and return to the negative terminal.
  • Electrons leave the negative terminal, travel through the load, and return to the positive terminal.
• Charge Pump Action:
  • EMF source moves negative charges from the positive to the negative terminal inside the source.
  • This maintains the potential difference.
  • Work is done on these charges, increasing their potential energy.

Explain that conventional current flow is a historical convention, while electron flow is the physical reality in metallic conductors. Emphasize that the emf source performs work internally to maintain the potential difference, analogous to a pump.

Defining EMF

• EMF (ε) is the work done on charge per unit charge.
• \(\epsilon = \frac{dW}{dq}\)
• Units of EMF are Volts (V), where \(1 \text{ V} = 1 \text{ J/C}\).

Note

An ideal battery has no internal resistance and its terminal voltage equals its emf.

• Terminal voltage (\(V_{\text{terminal}}\)) is the voltage measured across the battery’s terminals.

The definition of emf as work per unit charge reinforces its nature as an energy-transfer mechanism. Distinguish between ideal and real batteries, setting up the concept of internal resistance.

Origin of Battery Potential

• Chemical reactions within the battery create the potential difference.
• Lead-acid battery example:
  • Cathode (positive): Lead oxide plate
  • Anode (negative): Lead plate
  • Electrolyte: Sulfuric acid

• Chemical reactions separate charges:
  • Electrons are placed on the anode (negative).
  • Electrons are removed from the cathode (positive).
• This charge separation drives the potential difference.
• Requires a complete circuit for reactions to proceed.

Caption: Chemical reactions in a lead-acid cell.

Elaborate on the role of the anode, cathode, and electrolyte in the chemical reactions. Emphasize that these reactions are electrochemical, converting chemical energy into electrical energy. Also, point out that internal resistance is inherent because the chemicals themselves have resistance.

Internal Resistance

• Internal resistance (r) is the resistance to current flow within the voltage source.
• It’s a property of real batteries, not ideal ones.
• Causes a drop in terminal voltage when current flows.
• Generally increases as the battery depletes due to chemical changes.

• Battery Model:
  • An idealized emf source (\(\epsilon\)) in series with an internal resistance (r).
• Terminal Voltage Formula: \[V_{\text{terminal}} = \epsilon - Ir\]
  • Where I is the current drawn from the battery.

Caption: A battery modeled as an idealized emf with internal resistance.

Explain that the Ir term represents the voltage drop inside the battery due to its internal resistance. This means the voltage available to the external circuit is always less than the ideal emf when current is flowing. This concept is crucial for understanding real-world battery performance.

Circuit with Load and Internal Resistance

• When connected to a load resistor R:
  • Positive charges leave the positive terminal, flow through R, and return to the negative terminal.
  • The terminal voltage is also the voltage across the load: \(V_{\text{terminal}} = IR\).
• Therefore, \(IR = \epsilon - Ir\).
• The current through the load is: \[I = \frac{\epsilon}{R + r}\]
• A smaller internal resistance (\(r\)) allows the battery to supply more current to the load.

Caption: Schematic of a voltage source and its load resistor R.

This slide connects the internal resistance concept to a practical circuit. Emphasize that the internal resistance effectively acts in series with the external load. This equation highlights why a battery’s performance degrades as its internal resistance increases.

Voltage Graph in a Circuit

• The graph shows voltage changes around the circuit.
• Voltage increases by \(\epsilon\) inside the battery (chemical work).
• Voltage decreases by Ir inside the battery (due to internal resistance).
• Resulting terminal voltage is \(V_{\text{terminal}} = \epsilon - Ir\).
• Voltage further decreases by IR across the load resistor.
• The total voltage change around a closed loop is zero.

This graph is a powerful visual aid for understanding Kirchhoff’s loop rule, even before it’s formally introduced. Walk through each segment of the graph, explaining the potential changes and how they correspond to the components.

Example: Analyzing a Circuit with a Battery and a Load

A 12.00-V battery has an internal resistance of 0.100 \(\Omega\).

1. Terminal voltage with 10.00-\(\Omega\) load:

   \(I = \frac{\epsilon}{R+r} = \frac{12.00 \text{ V}}{10.00 \Omega + 0.100 \Omega} = 1.188 \text{ A}\)

   \(V_{\text{terminal}} = \epsilon - Ir = 12.00 \text{ V} - (1.188 \text{ A})(0.100 \Omega) = 11.90 \text{ V}\)

2. Terminal voltage with 0.500-\(\Omega\) load:

   \(I = \frac{\epsilon}{R+r} = \frac{12.00 \text{ V}}{0.500 \Omega + 0.100 \Omega} = 20.00 \text{ A}\)

   \(V_{\text{terminal}} = \epsilon - Ir = 12.00 \text{ V} - (20.00 \text{ A})(0.100 \Omega) = 10.00 \text{ V}\)

   Note: A significant reduction, implying a “heavy load.”

3. Power dissipated by 0.500-\(\Omega\) load:

   \(P = I^2R = (20.0 \text{ A})^2 (0.500 \Omega) = 200 \text{ W}\)

4. If internal resistance grows to 0.500 \(\Omega\) with 0.500-\(\Omega\) load:

   \(I = \frac{\epsilon}{R+r} = \frac{12.00 \text{ V}}{0.500 \Omega + 0.500 \Omega} = 12.00 \text{ A}\)

   \(V_{\text{terminal}} = \epsilon - Ir = 12.00 \text{ V} - (12.00 \text{ A})(0.500 \Omega) = 6.00 \text{ V}\)

   \(P = I^2R = (12.00 \text{ A})^2 (0.500 \Omega) = 72.00 \text{ W}\)

   Significance: Increased internal resistance drastically reduces terminal voltage, current, and power delivered.

Walk through each step of the calculation, explaining the formula used and the physical meaning of the results. Emphasize how the terminal voltage drops more significantly with a heavier load (smaller R) and how increased internal resistance negatively impacts performance.

Battery Testers & Rechargeable Batteries

Battery Testers:

• Use small load resistors to draw current.
• Measure if terminal potential drops below an acceptable level.
• High internal resistance indicates a weak battery.

Rechargeable Batteries:

• Current is passed through the battery in the opposite direction.
• Charger voltage must be greater than the battery’s emf.
• Terminal voltage becomes \(V = \epsilon - I_r\), but since \(I\) is now negative (charging), \(V > \epsilon\).

Caption: Battery testers.

Caption: A car battery charger.

Explain how battery testers work by intentionally creating a load to observe the terminal voltage drop, which is indicative of internal resistance. For recharging, clarify why the charger voltage must exceed the battery’s emf to force current in the reverse direction, replenishing its chemical potential.

10.2 Resistors in Series and Parallel

• Equivalent resistance is the single resistance that can replace a combination of resistors without changing the circuit’s total current or voltage.
• Circuits often contain multiple resistors connected in different ways.
• The two simplest configurations are series and parallel.

Series Connection:

• Current flows sequentially through each resistor.
• Current is the same through each resistor.
• Total potential drop is the sum of individual potential drops.

Parallel Connection:

• All resistors share the same two connection points.
• Potential drop (voltage) is the same across each resistor.
• Total current is the sum of individual currents through each branch.

Introduce the fundamental concept of equivalent resistance. Clearly distinguish between series and parallel connections by focusing on the behavior of current and voltage in each configuration. Use simple analogies like a single-lane road (series) versus multiple lanes (parallel) for current flow.

Resistors in Series

• Current is the same through each resistor: \(I_{\text{total}} = I_1 = I_2 = I_3\).
• Total voltage drop across the series combination: \(V = V_1 + V_2 + V_3\)
• Using Ohm’s Law (\(V=IR\)): \(IR_S = IR_1 + IR_2 + IR_3\) \(IR_S = I(R_1 + R_2 + R_3)\)
• Equivalent Resistance (\(R_S\)): \[R_S = R_1 + R_2 + R_3 + \dots + R_N = \sum_{i=1}^{N} R_i\]
• If one resistor in a series circuit breaks, the entire circuit is open, and no current flows.

Caption: (a) Three resistors in series. (b) Equivalent circuit.

Derive the series resistance formula step-by-step from Kirchhoff’s loop rule and Ohm’s law. Emphasize the implication of a single point of failure in a series circuit (e.g., old Christmas lights).

Example: Series Circuit Analysis

A 9 V battery (negligible internal resistance) is connected to four 20 \(\Omega\) resistors and one 10 \(\Omega\) resistor in series.

Caption: A simple series circuit with five resistors.

1. Equivalent resistance:

   \(R_S = 20\Omega + 20\Omega + 20\Omega + 20\Omega + 10\Omega = 90\Omega\)

2. Current through each resistor:

   \(I = \frac{V}{R_S} = \frac{9 \text{ V}}{90 \Omega} = 0.1 \text{ A}\)

   (Same current for all resistors in series)

3. Potential drop across each resistor:

   \(V_{20\Omega} = (0.1 \text{ A})(20 \Omega) = 2 \text{ V}\) (for each of the four 20 \(\Omega\) resistors)

   \(V_{10\Omega} = (0.1 \text{ A})(10 \Omega) = 1 \text{ V}\)

   Check: \(4 \times 2 \text{ V} + 1 \text{ V} = 9 \text{ V}\) (equals battery voltage)

4. Total power dissipated by resistors and power supplied by battery:

   \(P_{20\Omega} = I^2R = (0.1 \text{ A})^2(20 \Omega) = 0.2 \text{ W}\) (for each)

   \(P_{10\Omega} = I^2R = (0.1 \text{ A})^2(10 \Omega) = 0.1 \text{ W}\)

   \(P_{\text{dissipated}} = 4 \times 0.2 \text{ W} + 0.1 \text{ W} = 0.9 \text{ W}\)

   \(P_{\text{source}} = IV = (0.1 \text{ A})(9 \text{ V}) = 0.9 \text{ W}\)

Work through the calculations clearly, emphasizing the consistency of the results, especially the sum of potential drops equaling the source voltage and total power dissipated equaling power supplied. Highlight the practical significance, like using multiple resistors when a specific value is unavailable or for heat dissipation.

Resistors in Parallel

• Potential drop is the same across each resistor: \(V = V_1 = V_2 = V_3\).
• Total current from the source divides among branches (Kirchhoff’s Junction Rule): \(I = I_1 + I_2 + I_3\)
• Using Ohm’s Law (\(I=V/R\)): \(\frac{V}{R_P} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}\)
• Equivalent Resistance (\(R_P\)): \[\frac{1}{R_P} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_N}\] \[R_P = \left(\sum_{i=1}^{N} \frac{1}{R_i}\right)^{-1}\]
• The equivalent resistance in parallel is less than the smallest individual resistance.
• Each resistor receives the full voltage of the source.

Caption: (a) Two resistors in parallel. (b) Equivalent circuit.

Derive the parallel resistance formula, again relating it to Kirchhoff’s rules (junction rule and loop rule). Explain why the equivalent resistance decreases when more resistors are added in parallel (more paths for current, thus less overall opposition to flow). Use examples like household wiring to illustrate the benefits of parallel connections.

Example: Parallel Circuit Analysis

Three resistors \(R_1=1.00\Omega\), \(R_2=2.00\Omega\), and \(R_3=2.00\Omega\) are in parallel, connected to a 3.00 V source.

1. Equivalent resistance:

   \(\frac{1}{R_P} = \frac{1}{1.00\Omega} + \frac{1}{2.00\Omega} + \frac{1}{2.00\Omega} = 1.00 + 0.50 + 0.50 = 2.00 \Omega^{-1}\)

   \(R_P = (2.00 \Omega^{-1})^{-1} = 0.50 \Omega\)

   Note: \(R_P\) is less than the smallest individual resistance (1.00 \(\Omega\)).

2. Current supplied by the source:

   \(I = \frac{V}{R_P} = \frac{3.00 \text{ V}}{0.50 \Omega} = 6.00 \text{ A}\)

3. Currents in each resistor:

   \(I_1 = \frac{V}{R_1} = \frac{3.00 \text{ V}}{1.00 \Omega} = 3.00 \text{ A}\)

   \(I_2 = \frac{V}{R_2} = \frac{3.00 \text{ V}}{2.00 \Omega} = 1.50 \text{ A}\)

   \(I_3 = \frac{V}{R_3} = \frac{3.00 \text{ V}}{2.00 \Omega} = 1.50 \text{ A}\)

   Check: \(I_1 + I_2 + I_3 = 3.00 \text{ A} + 1.50 \text{ A} + 1.50 \text{ A} = 6.00 \text{ A}\) (equals total current).

4. Power dissipated by each resistor:

   \(P_1 = \frac{V^2}{R_1} = \frac{(3.00 \text{ V})^2}{1.00 \Omega} = 9.00 \text{ W}\)

   \(P_2 = \frac{V^2}{R_2} = \frac{(3.00 \text{ V})^2}{2.00 \Omega} = 4.50 \text{ W}\)

   \(P_3 = \frac{V^2}{R_3} = \frac{(3.00 \text{ V})^2}{2.00 \Omega} = 4.50 \text{ W}\)

5. Total power output of the source:

   \(P_{\text{source}} = IV = (6.00 \text{ A})(3.00 \text{ V}) = 18.00 \text{ W}\)

   Check: \(P_1 + P_2 + P_3 = 9.00 \text{ W} + 4.50 \text{ W} + 4.50 \text{ W} = 18.00 \text{ W}\).

Go through these calculations. Reinforce the key properties: constant voltage across parallel components, current division, and the lower equivalent resistance compared to individual resistors.

Series and Parallel Summary

	Series Combination	Parallel Combination
Equivalent Capacitance	\(\frac{1}{C_S} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots\)	\(C_P = C_1 + C_2 + C_3 + \dots\)
Equivalent Resistance	\(R_S = R_1 + R_2 + R_3 + \dots\)	\(\frac{1}{R_P} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots\)

This table is a great summary for students to quickly reference. Point out the inverse relationship between how resistance and capacitance combine in series and parallel, which can sometimes be confusing.

Combinations of Series and Parallel

• Many complex circuits are combinations of series and parallel connections.
• Strategy: Reduce the circuit step-by-step.
  1. Identify simple series or parallel sub-circuits.
  2. Replace them with their equivalent resistances.
  3. Repeat until a single equivalent resistance (\(R_{eq}\)) is found.

Caption: Steps to simplify a combination circuit.

Walk through the example shown in the figure, explaining each reduction step. For instance, Rs and R4 are in series, then their equivalent is in parallel with R2, and that combination is in series with R1. Emphasize how this systematic approach simplifies complex circuits.

Practical Implications

• Wire Resistance:
  • Wires themselves have resistance (\(R_W\)).
  • This resistance is in series with other components.
  • Reduces current and power delivered to the load (\(P = I^2R_W\)).
  • Can cause wires to overheat if current is too large.
• Example: Dimming Lights
  • When a large appliance (like a refrigerator motor) turns on, it draws a large current.
  • This large current causes a significant \(IR_W\) voltage drop across the supply wires.
  • The voltage available to other devices (e.g., a light bulb) decreases, causing them to dim.

Caption: Why lights dim when a large appliance is switched on.

Connect the theoretical concepts to everyday observations like dimming lights. Explain the energy loss in the wires due to their resistance and how this affects the performance of other devices on the same circuit. This provides a tangible application for the learned concepts.

Problem-Solving Strategy: Series and Parallel Resistors

1. Draw a clear circuit diagram: Label all resistors, voltage sources, and known values.
2. Identify unknowns: Clearly state what needs to be determined.
3. Determine configurations: Identify which resistors are in series, parallel, or combinations.
4. Apply appropriate rules: Use series or parallel equivalent resistance formulas.
5. Check reasonableness: Verify that answers are consistent and physically reasonable (e.g., \(R_S\) is greater, \(R_P\) is smaller).

This structured approach helps students tackle circuit problems effectively. Encourage them to follow these steps systematically to avoid errors.

10.3 Kirchhoff’s Rules

• For complex circuits that cannot be simplified by series-parallel reduction.
• Two fundamental rules:
  1. Kirchhoff’s First Rule (Junction Rule): Based on conservation of charge.
     • The sum of all currents entering a junction must equal the sum of all currents leaving the junction. \[ \sum I_{\text{in}} = \sum I_{\text{out}} \]
  2. Kirchhoff’s Second Rule (Loop Rule): Based on conservation of energy.
     • The algebraic sum of changes in potential around any closed circuit path (loop) must be zero. \[ \sum V = 0 \]

Emphasize the underlying physics principles for each rule: charge conservation for the junction rule and energy conservation for the loop rule. Introduce the multi-loop circuit example (Figure 10.19) to demonstrate why these rules are necessary.

Kirchhoff’s First Rule (Junction Rule)

• A junction (node) is a point where three or more wires connect.
• Conservation of Charge: Charge cannot accumulate or be lost at a junction.
• Analogy: Water pipes – what flows in must flow out.

Caption: The sum of currents into a junction equals the sum of currents out.

Use the water pipe analogy to make the junction rule intuitive. Highlight that the junction rule applies directly to current, which is the flow of charge.

Kirchhoff’s Second Rule (Loop Rule)

• Applies to potential differences (voltage).
• In a closed loop, the total energy supplied by sources must be dissipated by components.
• Starting from any point in a closed loop, if you traverse the loop and return to the starting point, the algebraic sum of potential changes must be zero.

Caption: A simple loop with no junctions.

Caption: Voltage graph around the circuit.

Revisit the voltage graph (Figure 10.22) from the EMF section to illustrate the loop rule. Show how potential increases across a source and decreases across a resistor. Connect this to the sum of potential differences being zero.

Applying Kirchhoff’s Rules: Sign Conventions

When traversing a loop:

Caption: Sign conventions for potential changes across components.

These sign conventions are critical for correctly applying the loop rule. Go through each scenario: resistors with and against current flow, and voltage sources from negative to positive and positive to negative terminals. Practice with simple examples.

Problem-Solving Strategy: Kirchhoff’s Rules

1. Label points in the circuit diagram (a, b, c, …).
2. Locate junctions: Identify points with three or more wires.
   • Label currents and their assumed directions at each junction.
   • Ensure at least one current flows in and one flows out.
   • Only use linearly independent junction equations.
3. Choose loops: Select enough loops so every component is included in at least one loop.
   • Avoid redundant loops.
4. Apply the junction rule: Write equations for selected junctions.
5. Apply the loop rule: Use the sign conventions to write equations for selected loops.
   • You will have a system of linear equations to solve for unknowns.

Emphasize the importance of choosing independent equations. Explain that if you have N unknowns (e.g., currents), you need N independent equations (junction and loop equations combined) to solve the system.

Multiple Voltage Sources

• Voltage sources (batteries, solar cells) can be connected in series, parallel, or combinations.

Series Connection: - Positive terminal of one to negative of another. - Total EMF = Sum of individual EMFS. - Total internal resistance = Sum of individual internal resistances. - Used to increase total voltage. \(V_{\text{terminal}} = (\sum_{i=1}^{N} \epsilon_i) - I(\sum_{i=1}^{N} r_i)\)

Caption: Two batteries in series.

Illustrate series connection with flashlights or toys. Explain how the emfs add up, but so do the internal resistances, which can be a disadvantage.

Multiple Voltage Sources (cont.)

Parallel Connection: - Positive terminals connected together, negative terminals connected together. - Usually, batteries have identical EMFS in parallel. - Total EMF = Individual EMF (if identical). - Equivalent internal resistance is reduced: \(\frac{1}{r_{eq}} = \sum_{i=1}^{N} \frac{1}{r_i}\) \(V_{\text{terminal}} = \epsilon - I r_{eq}\) - Used to increase total current capacity (e.g., starting a diesel engine).

Caption: Two batteries in parallel.

Explain why identical emfs are preferred for parallel connections to avoid circulating currents between batteries. Highlight that the main advantage of parallel connection is the reduction of equivalent internal resistance, allowing for higher total current delivery.

10.4 Electrical Measuring Instruments

• Voltmeters: Measure voltage (potential difference).
• Ammeters: Measure current.
• Ohmmeters: Measure resistance.

Note

Always ensure proper connection and safety when using measuring instruments.

Caption: Fuel and temperature gauges in a car are voltmeters.

Introduce the three main types of meters and their functions. Mention that many everyday devices incorporate these measurement principles.

Measuring Current with an Ammeter

• Connection: An ammeter must be placed in series with the component whose current is being measured.
  • This ensures the same current flows through both the component and the ammeter.
• Internal Resistance: Ammeters must have very low internal resistance (ideally zero).
  • A significant resistance would change the total circuit resistance and alter the current being measured.
• Safety: Ammeters usually contain a fuse to protect against excessively high currents.

Caption: Ammeters connected in series to measure current.

Explain why series connection is necessary (current is constant in series). Emphasize the requirement for low internal resistance to avoid affecting the measurement. Discuss the importance of fuses for protection.

Measuring Voltage with a Voltmeter

• Connection: A voltmeter must be connected in parallel with the component whose voltage is being measured.
  • This ensures the voltmeter experiences the same potential difference as the component.
• Internal Resistance: Voltmeters must have very high internal resistance (ideally infinite).
  • A low resistance would draw significant current from the circuit, altering the voltage being measured.
• Terminal Voltage: A voltmeter measures the terminal voltage of a battery, not its ideal emf, as it includes the internal resistance.

Caption: Voltmeters connected in parallel to measure potential differences.

Explain why parallel connection is necessary (voltage is constant in parallel). Emphasize the requirement for high internal resistance to avoid shunting current away from the component being measured. Clarify what a voltmeter measures in the context of a real battery.

Ohmmeters

• Function: Measures the resistance of a component or device.
• Principle: Based on Ohm’s Law (\(R = V/I\)).
  • Traditional ohmmeters use an internal voltage source and measure current.
  • Modern digital ohmmeters use a constant current source and measure voltage.
• Important Rules:
  • The component must be isolated from the circuit.
  • Never connect an ohmmeter to a “live” circuit (one with power). Doing so can damage the meter.

Highlight the operational principle of ohmmeters. Stress the critical safety precautions: isolating the component and never connecting to a live circuit. Explain why these precautions are necessary (to avoid measuring equivalent resistance or damaging the meter).

10.5 RC Circuits

• An RC circuit contains a resistor (R) and a capacitor (C).
• Capacitors store electric charge and energy in an electric field.
• Key phenomena: charging and discharging of a capacitor.

Caption: An RC circuit with a two-pole switch for charging and discharging.

Introduce the basic components and function of an RC circuit. Explain that it is fundamental for timing and filtering in electronics.

Charging a Capacitor

• When the switch is moved to position A, the capacitor charges through the resistor.
• Applying Kirchhoff’s loop rule: \(\epsilon - IR - V_C = 0\)
  • Since \(I = \frac{dq}{dt}\) and \(V_C = \frac{q}{C}\), this leads to a differential equation.
• Charge on capacitor as a function of time: \[q(t) = C\epsilon (1 - e^{-t/RC}) = Q(1 - e^{-t/\tau})\]
  • \(Q = C\epsilon\) is the maximum charge.
  • \(\tau = RC\) is the time constant.
• Current through resistor as a function of time: \[I(t) = \frac{dq}{dt} = \frac{\epsilon}{R} e^{-t/RC} = I_0 e^{-t/\tau}\]
  • \(I_0 = \frac{\epsilon}{R}\) is the initial current.

Tip

The time constant (\(\tau\)) is the time it takes for the charge to reach approximately 63.2% of its maximum value during charging, or to fall to 36.8% of its initial value during discharging.

Derive the charging equations qualitatively, focusing on the exponential behavior. Explain the significance of the time constant \(\tau\). Discuss the initial and final states of charge and current during charging.

Charging Curves

• Charge (\(q(t)\)): Increases exponentially from 0 to \(Q=C\epsilon\).
• Current (\(I(t)\)): Decreases exponentially from \(I_0=\epsilon/R\) to 0.
• Capacitor Voltage (\(V_C(t)\)): Increases exponentially from 0 to \(\epsilon\).
• Resistor Voltage (\(V_R(t)\)): Decreases exponentially from \(\epsilon\) to 0.

Walk through each graph, explaining how charge, current, and voltages change over time during the charging process. Point out the inverse relationship between capacitor voltage and resistor voltage, and how they sum to the source voltage at all times.

Discharging a Capacitor

• When the switch is moved to position B, the capacitor discharges through the resistor.
• Applying Kirchhoff’s loop rule: \(-IR - V_C = 0\)
  • This leads to a differential equation for discharge.
• Charge on capacitor as a function of time: \[q(t) = Q e^{-t/RC} = Q e^{-t/\tau}\]
  • \(Q\) is the initial charge on the capacitor.
• Current through resistor as a function of time: \[I(t) = \frac{dq}{dt} = -\frac{Q}{RC} e^{-t/RC} = -I_0 e^{-t/\tau}\]
  • The negative sign indicates current flows in the opposite direction.

Note

Charging typically takes longer than discharging in devices like flash cameras because the charging resistance (including battery internal resistance) is often much greater than the discharging resistance.

Explain the qualitative derivation of discharge equations. Discuss the exponential decay of charge and current. Connect the “flash camera” example back to the difference in resistance during charging and discharging.

Discharging Curves

• Charge (\(q(t)\)): Decreases exponentially from \(Q\) to 0.
• Current (\(I(t)\)): Decreases exponentially from \(I_0\) to 0 (in magnitude).
• Capacitor Voltage (\(V_C(t)\)): Decreases exponentially from \(V_0\) to 0.
• Resistor Voltage (\(V_R(t)\)): Decreases exponentially from \(V_0\) to 0.

Walk through each graph, explaining the exponential decay during discharge. Compare and contrast with the charging curves.

Applications of RC Circuits

• Timers:
  • Intermittent windshield wipers (variable resistor adjusts interval).
  • Pacemakers (control time between voltage signals to the heart).
  • Strobe lights.
  • 555 timers (timed voltage pulses).
• Filters:
  • Used in AC circuits to filter out unwanted frequencies (“noise”).
• Power Supplies:
  • Convert AC voltage to DC voltage (e.g., in computers).

Relaxation Oscillator: An RC circuit with a neon lamp (or transistor) that acts as a voltage-controlled switch, causing periodic flashes or pulses as the capacitor charges and discharges.

Discuss various practical applications, emphasizing how the time constant \(\tau = RC\) is used to control timing. Use the example of windshield wipers to explain how adjusting resistance changes the time interval.

10.6 Household Wiring and Electrical Safety

• Two main electrical hazards:
  1. Thermal Hazard: Excessive current causing undesired heating (e.g., fire).
  2. Shock Hazard: Current passing through a person.

Thermal Hazards

• Short Circuit: A low-resistance path between terminals of a voltage source.
  • Leads to very large current (\(I = V/r_{low}\)).
  • Generates rapid heating (\(P = I^2R\)) which can melt insulation and cause fires.
• Overloaded Wires: Current exceeds rated maximum for wires.
  • Wires overheat (\(P = I^2R_{wire}\)) and can cause damage or fire.
• Protective Devices: Fuses and circuit breakers are designed to open the circuit if current exceeds safe limits.

Caption: A short circuit in a toaster.

Explain short circuits and overloading as causes of thermal hazards. Emphasize the role of fuses and circuit breakers as essential safety mechanisms. Briefly introduce the AC voltage source symbol.

Shock Hazards

• Electric Shock: Injury caused by current passing through the body.
• Severity factors:
  1. Amount of current (I): Most critical factor.
     • 1 mA: Sensation threshold.
     • 5 mA: Maximum “harmless” shock.
     • 5-30 mA: Muscular contractions, “can’t let go.”
     • 300 mA: Heart/diaphragm contraction, potentially fatal.
  2. Path of current: Current across the heart is most dangerous.
  3. Duration of shock.
  4. Frequency of current: AC is generally more dangerous than DC for similar current levels.

Warning

The human body is a relatively good conductor due to its water content. Dry skin has high resistance (\(\approx 200 \text{ k}\Omega\)), wet skin has lower resistance (\(\approx 10.0 \text{ k}\Omega\)).

Detail the physiological effects of different current levels. Explain how body resistance varies (dry vs. wet skin) and how this influences the actual current for a given voltage. Stress safety precautions like insulated shoes and working with one hand.

Electrical Safety: Systems and Devices

• Three-Wire System:
  • Live/Hot Wire: Supplies voltage and current (usually black or brown).
  • Neutral Wire: Return path for current; grounded at source and user location (usually white or blue).
  • Ground Wire: Connects appliance case to ground (usually green or green/yellow).
• Purpose of Grounding:
  • Forces neutral wire to zero volts relative to ground (safe to touch).
  • Provides an alternative path for current if live/hot wire touches the appliance case, tripping the circuit breaker.

Caption: Three-wire system schematic.

Caption: Three-prong plug.

Explain the function of each wire in the three-wire system. Emphasize how the ground wire protects against shocks by ensuring the appliance case is at zero volts. Use the example of worn insulation and how grounding prevents a shock.

Electrical Safety: Ground Fault Circuit Interrupters (GFCIs)

• Function: Detects differences in current between the live/hot and neutral wires.
• Principle: Based on electromagnetic induction.
• Operation:
  • If live/hot current \(\neq\) neutral current, a leakage current is present.
  • This means current is returning via an unintended path (e.g., through a person).
  • GFCIs trip the circuit if leakage current > 5 mA (the “harmless” limit).
• Location: Typically found in kitchens, bathrooms, and outdoor outlets where water is present, increasing shock risk.

Caption: GFCI protection against shock.

Explain the ingenious mechanism of a GFCI by comparing currents in the live and neutral wires. Emphasize that GFCIs protect against leakage currents that might be too small to trip a standard circuit breaker but are still hazardous to humans.

Key Takeaways

• EMF is the work done per unit charge by a source, not a force. Real batteries have internal resistance (\(r\)), causing terminal voltage to be \(V_{\text{terminal}} = \epsilon - Ir\).
• Resistors in Series: \(R_S = \sum R_i\). Current is the same; voltage adds.
• Resistors in Parallel: \(\frac{1}{R_P} = \sum \frac{1}{R_i}\). Voltage is the same; current adds.
• Kirchhoff’s Rules:
  • Junction Rule: \(\sum I_{\text{in}} = \sum I_{\text{out}}\) (Conservation of Charge).
  • Loop Rule: \(\sum V = 0\) (Conservation of Energy).
• Electrical Measuring Instruments:
  • Ammeters are in series, have low resistance.
  • Voltmeters are in parallel, have high resistance.
  • Ohmmeters measure resistance, require isolated, unpowered components.
• RC Circuits: Capacitors charge/discharge exponentially with a time constant \(\tau = RC\). Used for timers and filters.
• Electrical Safety: Protect against thermal hazards (short circuits, overloads) with fuses/circuit breakers, and shock hazards with grounding systems and GFCIs. Current is the primary factor in shock severity.

Review the most important concepts from the chapter to ensure students have a clear understanding of the core material.

Key Equations

Equation	Description
\(\epsilon = \frac{dW}{dq}\)	Electromotive force (emf)
\(V_{\text{terminal}} = \epsilon - Ir\)	Terminal voltage of a battery with internal resistance
\(I = \frac{\epsilon}{R + r}\)	Current through a load connected to a real battery
\(R_S = R_1 + R_2 + R_3 + \dots\)	Equivalent resistance of resistors in series
\(\frac{1}{R_P} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots\)	Equivalent resistance of resistors in parallel
\(\sum I_{\text{in}} = \sum I_{\text{out}}\)	Kirchhoff’s Junction Rule
\(\sum V = 0\)	Kirchhoff’s Loop Rule
\(q(t) = C\epsilon (1 - e^{-t/RC})\)	Charge on a charging capacitor
\(I(t) = \frac{\epsilon}{R} e^{-t/RC}\)	Current in a charging RC circuit
\(q(t) = Q e^{-t/RC}\)	Charge on a discharging capacitor
\(I(t) = -\frac{Q}{RC} e^{-t/RC}\)	Current in a discharging RC circuit
\(\tau = RC\)	RC time constant

This table provides a concise summary of all essential equations, useful for quick reference and review.

Key Terms

Term	Definition
Ammeter	An instrument that measures current in a circuit. It must be connected in series and have very low resistance.
Circuit breaker	An automatic switch that opens a circuit when the current exceeds a preset safe value, preventing thermal hazards.
Electromotive Force (emf)	The potential difference of a source when no current is flowing; a measure of the energy supplied by the source per unit charge. Not a force.
Equivalent Resistance	The single resistance that can replace a combination of resistors without changing the overall current or voltage in the circuit.
Fuse	A wire that melts and opens a circuit if the current exceeds a certain safe value, protecting against thermal hazards.
Ground fault circuit interrupter (GFCI)	A safety device that detects leakage currents to ground and quickly interrupts the circuit to prevent shocks.
Internal Resistance	The inherent resistance within a voltage source (like a battery) that causes a drop in its terminal voltage when current is drawn.
Junction (Node)	A point in an electrical circuit where three or more wires or components are connected, where currents can split or merge.
Kirchhoff’s Junction Rule	States that the sum of currents entering a junction must equal the sum of currents leaving it, reflecting charge conservation.
Kirchhoff’s Loop Rule	States that the algebraic sum of potential differences around any closed loop in a circuit must be zero, reflecting energy conservation.
Ohmmeter	An instrument used to measure the electrical resistance of a component or circuit, usually requiring the component to be isolated and unpowered.
RC Circuit	An electrical circuit comprising a resistor and a capacitor, exhibiting exponential charge and discharge behavior.
Resistors in Parallel	Components connected across the same two points, experiencing the same voltage drop, and allowing current to divide among them.
Resistors in Series	Components connected end-to-end so that the same current flows sequentially through each of them.
Shock Hazard	A danger arising when an electric current passes through a person, causing physiological effects ranging from pain to death.
Terminal Voltage	The actual voltage measured across the terminals of a voltage source when current is being supplied to an external circuit.
Thermal Hazard	A danger arising when excessive electric current causes undesired heating effects, potentially leading to fires or damage.
**Time Constant (\(\tau\))	A characteristic time for a charging or discharging RC circuit, defined as \(RC\), representing the time for a significant fraction of change.
Voltmeter	An instrument that measures potential difference (voltage) across two points in a circuit. It must be connected in parallel and have very high resistance.

Provide definitions for all important terms introduced in the chapter, serving as a glossary for students.