#AP Physics 2: Compound DC Circuits - The Ultimate Study Guide ⚡

Hey future physicist! Let's dive into the world of compound DC circuits. This guide is designed to make sure you're not just prepared, but confident for your AP Physics 2 exam. We'll tackle equivalent resistance, non-ideal components, and measurement techniques, all while keeping it engaging and easy to remember.

#1. Compound Direct Current (DC) Circuits

# 1.1 Equivalent Resistance of Multiple Resistors in a Circuit

Let's start with the basics. Circuits aren't always simple; they often combine resistors in series and parallel. Here's how to break them down:

Series Connection: Think of it like a single lane road 🚗. All charges must travel through each resistor one after the other. The current is the same through each resistor.
Parallel Connection: Now imagine a multi-lane highway 🛣️. Charges have multiple paths they can take. The voltage is the same across each parallel path.
Equivalent Resistance ( $R_{eq}$ ): This is the total resistance of a group of resistors, treated as a single resistor. It simplifies circuit analysis.
- Series Resistors: Just add them up!
  
  $R_{\text{eq}, s} = \sum_{i} R_{i}$
- Parallel Resistors: Use the reciprocal sum formula:
  
  $\frac{1}{R_{\text{eq}, p}} = \sum_{i} \frac{1}{R_{i}}$

Key Concept

Remember: Parallel resistors decrease the overall resistance because they provide more paths for current to flow. Think of it like adding more lanes to a highway; traffic flows more easily.

Memory Aid

Series is Simple, Parallel is Peculiar!

Series: Just add them up, no fuss.
Parallel: Use the reciprocal sum formula. It's a bit more unusual, hence 'peculiar.'

Practice Question

Multiple Choice Questions

Three resistors with resistances of 10 Ω, 20 Ω, and 30 Ω are connected in series. What is the equivalent resistance of the combination? (A) 5 Ω (B) 10 Ω (C) 20 Ω (D) 60 Ω
Two resistors with resistances of 12 Ω and 6 Ω are connected in parallel. What is the equivalent resistance of the combination? (A) 2 Ω (B) 4 Ω (C) 9 Ω (D) 18 Ω

Free Response Question

A circuit consists of a 12 V battery, a 4 Ω resistor, and an 8 Ω resistor. The resistors are connected in parallel. Calculate the following: (a) The equivalent resistance of the parallel combination. (2 points) (b) The total current supplied by the battery. (2 points) (c) The current through each resistor. (2 points) (d) The power dissipated by each resistor. (2 points)

Answer Key

Multiple Choice Questions

Free Response Question

(a) $\frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{8} = \frac{3}{8}$ , so $R_{eq} = \frac{8}{3} \Omega$ (2 points) (b) $I = \frac{V}{R_{eq}} = \frac{12}{\frac{8}{3}} = 4.5 A$ (2 points) (c) Since the resistors are in parallel, the voltage across each is 12 V. $I_4 = \frac{12}{4} = 3 A$ and $I_8 = \frac{12}{8} = 1.5 A$ (2 points) (d) $P_4 = I_4 V = 3 * 12 = 36 W$ and $P_8 = I_8 V = 1.5 * 12 = 18 W$ (2 points)

# 1.2 Circuits with Resistive Wires and Batteries with Internal Resistance

Now, let's get real. Ideal components don't exist in the real world. Here's how to account for non-idealities:

Ideal vs. Non-Ideal:
- Ideal Batteries: They provide a constant voltage, no matter the current. They have zero internal resistance.
- Ideal Wires: They have zero resistance and don't affect the circuit.
Resistive Wires:
- In reality, wires do have some resistance. It's usually negligible, but it can matter in some cases.
- Wire resistance can be ignored when other resistive elements are present in the circuit.
Internal Resistance of Batteries:
- Real batteries have internal resistance ( $r$ ) that impacts the voltage they deliver.
- The potential difference across the terminals of a real battery is less than its electromotive force (emf, $\mathcal{E}$ ) when current flows.
- $\Delta V_{\text{terminal}} = \mathcal{E} - Ir$

Common Mistake

Don't forget to subtract the voltage drop across the internal resistance when calculating the terminal voltage of a battery. It's a common error!

Quick Fact

The emf ( $\mathcal{E}$ ) of a battery is the potential difference across its terminals when no current is flowing. It's the ideal voltage of the battery.

Practice Question

Multiple Choice Questions

A battery with an emf of 12 V and an internal resistance of 1 Ω is connected to a 5 Ω resistor. What is the terminal voltage of the battery? (A) 10 V (B) 11 V (C) 12 V (D) 13 V
Which of the following statements is true regarding the internal resistance of a battery? (A) It increases the terminal voltage of the battery. (B) It decreases the terminal voltage of the battery. (C) It does not affect the terminal voltage of the battery. (D) It is always negligible.

Free Response Question

A 9 V battery with an internal resistance of 0.5 Ω is connected to a circuit containing a 3 Ω resistor. Calculate the following: (a) The current in the circuit. (2 points) (b) The terminal voltage of the battery. (2 points) (c) The power dissipated by the 3 Ω resistor. (2 points) (d) The power dissipated by the internal resistance of the battery. (2 points)

Answer Key

Multiple Choice Questions

Free Response Question

(a) The total resistance in the circuit is $R_{total} = 3 + 0.5 = 3.5 \Omega$ . The current is $I = \frac{V}{R_{total}} = \frac{9}{3.5} \approx 2.57 A$ . (2 points) (b) The terminal voltage is $V_{terminal} = \mathcal{E} - Ir = 9 - (2.57 * 0.5) \approx 7.71 V$ . (2 points) (c) The power dissipated by the 3 Ω resistor is $P = I^2 R = (2.57)^2 * 3 \approx 19.76 W$ . (2 points) (d) The power dissipated by the internal resistance is $P = I^2 r = (2.57)^2 * 0.5 \approx 3.30 W$ . (2 points)

# 1.3 Measuring Current and Potential Difference in a Circuit

Time to measure! Here's how we use ammeters and voltmeters, and what to watch out for:

Ammeters:
- Measure current (in amps) at a specific point in a circuit.
- Connected in series with the circuit element you're measuring.
- Ideal ammeters have zero resistance, so they don't affect the current.
Voltmeters:
- Measure potential difference (voltage) between two points in a circuit.
- Connected in parallel with the circuit element you're measuring.
- Ideal voltmeters have infinite resistance, so they don't draw any current.
Non-Ideal Meters:
- Real ammeters have some resistance, which decreases the current they measure.
- Real voltmeters have some finite resistance, which draws some current, thus affecting the voltage they measure.

Exam Tip

Remember: Ammeters are in series, like the 's' in series and ammeter. Voltmeters are in parallel, like the two 'l's in parallel and voltmeter. 💡

Understanding how non-ideal meters affect measurements is a high-value topic. Be prepared to discuss this qualitatively on the exam.

Practice Question

Multiple Choice Questions

How should an ammeter be connected in a circuit to measure the current through a resistor? (A) In parallel with the resistor. (B) In series with the resistor. (C) In parallel with the voltage source. (D) In series with the voltage source.
An ideal voltmeter has: (A) Zero resistance and is connected in series. (B) Zero resistance and is connected in parallel. (C) Infinite resistance and is connected in series. (D) Infinite resistance and is connected in parallel.

Free Response Question

A circuit consists of a 6 V battery and a 10 Ω resistor. An ammeter and a voltmeter are used to measure the current through and the voltage across the resistor. (a) Draw a diagram of the circuit, including the ammeter and voltmeter. (2 points) (b) Calculate the ideal current through the resistor and the ideal voltage across it. (2 points) (c) Explain how the measurements would be affected if the ammeter had a small internal resistance. (2 points) (d) Explain how the measurements would be affected if the voltmeter had a small finite resistance. (2 points)

Answer Key

Multiple Choice Questions

Free Response Question

(a) The ammeter should be in series with the resistor, and the voltmeter should be in parallel with the resistor. (2 points) (b) The ideal current is $I = \frac{V}{R} = \frac{6}{10} = 0.6 A$ , and the ideal voltage is 6 V. (2 points) (c) If the ammeter has internal resistance, the total resistance in the circuit increases, so the measured current will be less than the ideal current. (2 points) (d) If the voltmeter has finite resistance, it will draw some current, thus decreasing the measured voltage across the resistor. (2 points)

#Final Exam Focus

Okay, you've made it through! Here's what to focus on for the exam:

High-Priority Topics:
- Equivalent resistance calculations (series and parallel).
- Impact of internal resistance on battery terminal voltage.
- Proper placement and effects of non-ideal ammeters and voltmeters.
Common Question Types:
- Calculating equivalent resistance in complex circuits.
- Analyzing circuits with non-ideal batteries and internal resistance.
- Explaining how non-ideal meters affect measurements.
Last-Minute Tips:
- Time Management: Start with the questions you know best. Don't get bogged down on one problem.
- Common Pitfalls: Watch out for the units and make sure you're using the correct formulas for series vs. parallel resistors.
- Strategies: Draw clear circuit diagrams. Label all known and unknown values. Show all your work. Explain your reasoning clearly.

You've got this! Remember, you're not just memorizing formulas; you're understanding how the world works. Go in there, be confident, and show them what you've learned! 🚀

#AP Physics 2: Compound DC Circuits - The Ultimate Study Guide ⚡

#1. Compound Direct Current (DC) Circuits

# 1.1 Equivalent Resistance of Multiple Resistors in a Circuit

Let's start with the basics. Circuits aren't always simple; they often combine resistors in series and parallel. Here's how to break them down:

Series Connection: Think of it like a single lane road 🚗. All charges must travel through each resistor one after the other. The current is the same through each resistor.
Parallel Connection: Now imagine a multi-lane highway 🛣️. Charges have multiple paths they can take. The voltage is the same across each parallel path.
Equivalent Resistance ( $R_{eq}$ ): This is the total resistance of a group of resistors, treated as a single resistor. It simplifies circuit analysis.
- Series Resistors: Just add them up!
  
  $R_{\text{eq}, s} = \sum_{i} R_{i}$
- Parallel Resistors: Use the reciprocal sum formula:
  
  $\frac{1}{R_{\text{eq}, p}} = \sum_{i} \frac{1}{R_{i}}$

Key Concept

Remember: Parallel resistors decrease the overall resistance because they provide more paths for current to flow. Think of it like adding more lanes to a highway; traffic flows more easily.

Memory Aid

Series is Simple, Parallel is Peculiar!

Series: Just add them up, no fuss.
Parallel: Use the reciprocal sum formula. It's a bit more unusual, hence 'peculiar.'

Practice Question

Multiple Choice Questions

Three resistors with resistances of 10 Ω, 20 Ω, and 30 Ω are connected in series. What is the equivalent resistance of the combination? (A) 5 Ω (B) 10 Ω (C) 20 Ω (D) 60 Ω
Two resistors with resistances of 12 Ω and 6 Ω are connected in parallel. What is the equivalent resistance of the combination? (A) 2 Ω (B) 4 Ω (C) 9 Ω (D) 18 Ω

Free Response Question

Answer Key

Multiple Choice Questions

Free Response Question

# 1.2 Circuits with Resistive Wires and Batteries with Internal Resistance

Now, let's get real. Ideal components don't exist in the real world. Here's how to account for non-idealities:

Ideal vs. Non-Ideal:
- Ideal Batteries: They provide a constant voltage, no matter the current. They have zero internal resistance.
- Ideal Wires: They have zero resistance and don't affect the circuit.
Resistive Wires:
- In reality, wires do have some resistance. It's usually negligible, but it can matter in some cases.
- Wire resistance can be ignored when other resistive elements are present in the circuit.
Internal Resistance of Batteries:
- Real batteries have internal resistance ( $r$ ) that impacts the voltage they deliver.
- The potential difference across the terminals of a real battery is less than its electromotive force (emf, $\mathcal{E}$ ) when current flows.
- $\Delta V_{\text{terminal}} = \mathcal{E} - Ir$

Common Mistake

Don't forget to subtract the voltage drop across the internal resistance when calculating the terminal voltage of a battery. It's a common error!

Quick Fact

The emf ( $\mathcal{E}$ ) of a battery is the potential difference across its terminals when no current is flowing. It's the ideal voltage of the battery.

Practice Question

Multiple Choice Questions

A battery with an emf of 12 V and an internal resistance of 1 Ω is connected to a 5 Ω resistor. What is the terminal voltage of the battery? (A) 10 V (B) 11 V (C) 12 V (D) 13 V
Which of the following statements is true regarding the internal resistance of a battery? (A) It increases the terminal voltage of the battery. (B) It decreases the terminal voltage of the battery. (C) It does not affect the terminal voltage of the battery. (D) It is always negligible.

Free Response Question

Answer Key

Multiple Choice Questions

Free Response Question

# 1.3 Measuring Current and Potential Difference in a Circuit

Time to measure! Here's how we use ammeters and voltmeters, and what to watch out for:

Ammeters:
- Measure current (in amps) at a specific point in a circuit.
- Connected in series with the circuit element you're measuring.
- Ideal ammeters have zero resistance, so they don't affect the current.
Voltmeters:
- Measure potential difference (voltage) between two points in a circuit.
- Connected in parallel with the circuit element you're measuring.
- Ideal voltmeters have infinite resistance, so they don't draw any current.
Non-Ideal Meters:
- Real ammeters have some resistance, which decreases the current they measure.
- Real voltmeters have some finite resistance, which draws some current, thus affecting the voltage they measure.

Exam Tip

Remember: Ammeters are in series, like the 's' in series and ammeter. Voltmeters are in parallel, like the two 'l's in parallel and voltmeter. 💡

Understanding how non-ideal meters affect measurements is a high-value topic. Be prepared to discuss this qualitatively on the exam.

Practice Question

Multiple Choice Questions

How should an ammeter be connected in a circuit to measure the current through a resistor? (A) In parallel with the resistor. (B) In series with the resistor. (C) In parallel with the voltage source. (D) In series with the voltage source.
An ideal voltmeter has: (A) Zero resistance and is connected in series. (B) Zero resistance and is connected in parallel. (C) Infinite resistance and is connected in series. (D) Infinite resistance and is connected in parallel.

Free Response Question

Answer Key

Multiple Choice Questions

Free Response Question

#Final Exam Focus

Okay, you've made it through! Here's what to focus on for the exam:

High-Priority Topics:
- Equivalent resistance calculations (series and parallel).
- Impact of internal resistance on battery terminal voltage.
- Proper placement and effects of non-ideal ammeters and voltmeters.
Common Question Types:
- Calculating equivalent resistance in complex circuits.
- Analyzing circuits with non-ideal batteries and internal resistance.
- Explaining how non-ideal meters affect measurements.
Last-Minute Tips:
- Time Management: Start with the questions you know best. Don't get bogged down on one problem.
- Common Pitfalls: Watch out for the units and make sure you're using the correct formulas for series vs. parallel resistors.
- Strategies: Draw clear circuit diagrams. Label all known and unknown values. Show all your work. Explain your reasoning clearly.

You've got this! Remember, you're not just memorizing formulas; you're understanding how the world works. Go in there, be confident, and show them what you've learned! 🚀

Compound Direct Current (DC) Circuits

Isabella Lopez

#AP Physics 2: Compound DC Circuits - The Ultimate Study Guide ⚡

#1. Compound Direct Current (DC) Circuits

# 1.1 Equivalent Resistance of Multiple Resistors in a Circuit

# 1.2 Circuits with Resistive Wires and Batteries with Internal Resistance

# 1.3 Measuring Current and Potential Difference in a Circuit

#Final Exam Focus

Explore more resources

How are we doing?

Compound Direct Current (DC) Circuits

Isabella Lopez

#AP Physics 2: Compound DC Circuits - The Ultimate Study Guide ⚡

#1. Compound Direct Current (DC) Circuits

# 1.1 Equivalent Resistance of Multiple Resistors in a Circuit

# 1.2 Circuits with Resistive Wires and Batteries with Internal Resistance

# 1.3 Measuring Current and Potential Difference in a Circuit

#Final Exam Focus

Explore more resources

How are we doing?