#AP Physics C: E&M - Magnetic Fields Study Guide 🧲

Hey there! Let's get you prepped for the exam with a focused review of Magnetic Fields. This guide is designed to be your go-to resource for a quick and effective review. Let's dive in!

#1. Magnetic Fields and Forces 🧲

#1.1 Forces on Moving Charges in Magnetic Fields

When a charged particle moves through a magnetic field, it experiences a magnetic force. This force is always perpendicular to both the velocity of the particle and the magnetic field.
Formula: $F = q(v \times B)$
- $F$ : Magnetic force (N)
- $q$ : Charge of the particle (C)
- $v$ : Velocity of the particle (m/s)
- $B$ : Magnetic field strength (T)
- The $\times$ denotes the cross product, which means the force is perpendicular to both velocity and magnetic field.
Right-Hand Rule: Point your thumb in the direction of the velocity, your fingers in the direction of the magnetic field, and your palm will point in the direction of the force on a positive charge. For a negative charge, the force is in the opposite direction.

Key Concept

The magnetic force does no work on the charged particle because it's always perpendicular to the motion. It changes the direction of motion, not the speed.

Cyclotron Motion: A charged particle in a uniform magnetic field will move in a circle. The radius of this circle is determined by:
- Velocity of the particle
- Strength of the magnetic field
- Mass and charge of the particle

Memory Aid

Remember the right-hand rule: Thumb for velocity, fingers for the magnetic field, and palm for force (for positive charges). For negative charges, use the back of your hand!

Practice Question

Multiple Choice Questions

A proton moves with a velocity v in a uniform magnetic field B. The magnetic force on the proton is: (A) parallel to v (B) parallel to B (C) perpendicular to both v and B (D) zero if v is parallel to B
An electron is moving in a circular path in a uniform magnetic field. If the magnetic field strength is doubled, the radius of the circular path will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A proton (charge +e, mass m) enters a region of uniform magnetic field B with a velocity v perpendicular to the field.

(a) Draw a diagram showing the path of the proton in the magnetic field. (2 points)

(b) Derive an expression for the radius of the circular path in terms of e, m, v, and B. (5 points)

(c) If the magnetic field is 0.5 T, the velocity of the proton is 2 x 10^6 m/s and the mass of the proton is 1.67 x 10^-27 kg, find the radius of the path. (3 points)

Answer Key

Multiple Choice

Free Response

(a) Diagram should show a circular path. (2 points)

(b) Derivation: - Magnetic force = centripetal force: evB = mv^2/r (2 points) - Solve for r: r = mv/eB (3 points)

#1.2 Forces on Current-Carrying Wires in Magnetic Fields

A current-carrying wire in a magnetic field experiences a force. This is the principle behind electric motors!
Formula: $F = ILB\sin\theta$
- $F$ : Force on the wire (N)
- $I$ : Current in the wire (A)
- $L$ : Length of the wire in the magnetic field (m)
- $B$ : Magnetic field strength (T)
- $\theta$ : Angle between the current and the magnetic field
Right-Hand Rule: Again, use the right-hand rule. Thumb points in the direction of the current, fingers point in the direction of the magnetic field, and the palm shows the direction of the force on the wire.

Exam Tip

Remember, the force is maximized when the current is perpendicular to the magnetic field ( $\theta$ = 90°), and zero when they are parallel ( $\theta$ = 0°).

Common Mistake

Students often mix up the direction of current and force. Always use the right-hand rule carefully!

Quick Fact

The force on a current-carrying wire is the basis for how speakers work. The current in the coil interacts with the magnet to produce sound.

Practice Question

Multiple Choice Questions

A straight wire carrying a current is placed in a uniform magnetic field. The force on the wire is maximum when: (A) the wire is parallel to the magnetic field (B) the wire is perpendicular to the magnetic field (C) the wire is at a 45-degree angle to the magnetic field (D) the force is always zero
A wire of length L carries a current I in a magnetic field B. If the length of the wire is doubled and the current is halved, the magnetic force on the wire will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A straight wire of length 0.5 m carries a current of 2 A in a uniform magnetic field of 0.4 T. The wire is oriented at an angle of 30 degrees to the magnetic field.

(a) Calculate the magnitude of the force on the wire. (3 points)

(b) If the angle between the wire and magnetic field is increased to 90 degrees, what is the new force on the wire? (2 points)

Answer Key

Multiple Choice

Free Response

(a) Calculation: - $F = ILB\sin\theta = (2 A)(0.5 m)(0.4 T)\sin(30^\circ) = 0.2 N$ (3 points)

(b) Calculation: - $F = ILB\sin\theta = (2 A)(0.5 m)(0.4 T)\sin(90^\circ) = 0.4 N$ (2 points)

(c) The direction of the force is perpendicular to both the wire and the magnetic field, determined by the right-hand rule. (2 points)

#2. Magnetic Fields from Currents ⚡

#2.1 Fields of Long Current-Carrying Wires

A long, straight wire with a current produces a circular magnetic field around it.
Formula: $B = \frac{\mu_0 I}{2\pi r}$
- $B$ : Magnetic field strength (T)
- $\mu_0$ : Permeability of free space ( $4\pi \times 10^{-7}$ T⋅m/A)
- $I$ : Current in the wire (A)
- $r$ : Distance from the wire (m)
Right-Hand Rule: Wrap your fingers around the wire in the direction of the current; your thumb points in the direction of the magnetic field.

Key Concept

The magnetic field is strongest close to the wire and decreases as the distance from the wire increases.

Practice Question

Multiple Choice Questions

The magnetic field strength due to a long, straight current-carrying wire is: (A) directly proportional to the current and inversely proportional to the distance from the wire (B) inversely proportional to the current and directly proportional to the distance from the wire (C) directly proportional to both the current and the distance from the wire (D) inversely proportional to both the current and the distance from the wire
If the current in a long, straight wire is doubled, and the distance from the wire is also doubled, the magnetic field strength will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A long, straight wire carries a current of 5 A. Calculate the magnetic field strength at a distance of 0.1 m from the wire.

(a) Calculate the magnetic field strength at a distance of 0.1 m from the wire. (4 points)

(b) If the distance from the wire is doubled, what is the new magnetic field strength? (2 points)

(c) What is the direction of the magnetic field at a point above the wire, assuming the current is moving to the right? (2 points)

Answer Key

Multiple Choice

Free Response

(a) Calculation: - $B = \frac{\mu_0 I}{2\pi r} = \frac{(4\pi \times 10^{-7} T⋅m/A)(5 A)}{2\pi (0.1 m)} = 1 \times 10^{-5} T$ (4 points)

(b) Calculation: - $B = \frac{(4\pi \times 10^{-7} T⋅m/A)(5 A)}{2\pi (0.2 m)} = 0.5 \times 10^{-5} T$ (2 points)

#2.2 Biot-Savart Law and Ampère’s Law

Biot-Savart Law: This law calculates the magnetic field created by a small segment of current-carrying wire. It's useful for complex geometries but can be mathematically intensive.
Ampère’s Law: This law relates the magnetic field around a closed loop to the current passing through the loop. It's a powerful tool for calculating magnetic fields in situations with high symmetry (like long wires and solenoids).
Ampere's Law Formula: $\oint B \cdot dl = \mu_0 I_{enc}$
- The line integral of the magnetic field around a closed loop is proportional to the current enclosed by the loop.

Ampère's Law is crucial for solving problems involving magnetic fields, especially for long wires and solenoids. Focus on understanding how to apply it.

Memory Aid

Think of Ampère's Law as a way to find the 'total magnetic field' around a current, similar to how Gauss's Law finds the 'total electric field' around a charge.

Practice Question

Multiple Choice Questions

Which law is most useful for calculating the magnetic field due to a long, straight current-carrying wire? (A) Biot-Savart Law (B) Ampère’s Law (C) Faraday's Law (D) Gauss's Law
Ampère’s Law relates the magnetic field around a closed loop to: (A) the electric field through the loop (B) the current enclosed by the loop (C) the charge enclosed by the loop (D) the magnetic flux through the loop

Free Response Question

Use Ampère's Law to derive the magnetic field strength at a distance r from a long, straight wire carrying a current I.

(a) State Ampère's Law. (2 points)

(b) Draw a diagram showing the Amperian loop you will use for this derivation. (2 points)

Answer Key

Multiple Choice

Free Response

(a) Ampère's Law: The line integral of the magnetic field around a closed loop is proportional to the current enclosed by the loop. (2 points)

(b) Diagram should show a circular Amperian loop of radius r centered on the wire. (2 points)

(c) Derivation: - $\oint B \cdot dl = B(2\pi r)$ (2 points) - $B(2\pi r) = \mu_0 I$ (1 point) - $B = \frac{\mu_0 I}{2\pi r}$ (1 point)

#Final Exam Focus 🎯

High-Priority Topics:
- Forces on moving charges and current-carrying wires in magnetic fields
- Magnetic fields produced by long wires and solenoids
- Application of Ampère's Law
Common Question Types:
- Calculating magnetic forces on particles and wires
- Determining the direction of magnetic forces and fields using right-hand rules
- Using Ampère's Law to find magnetic fields due to current distributions
- Conceptual questions about the interaction of magnetic fields and matter
Last-Minute Tips:
- Time Management: Start with questions you know how to solve quickly.
- Right-Hand Rule: Practice using the right-hand rule correctly; it's crucial for many problems.
- Units: Always include units in your calculations and answers.
- Formulas: Make sure you know the key formulas and when to use them.
- Conceptual Understanding: Don't just memorize formulas; understand the concepts behind them.

Good luck! You've got this! 💪

Magnetic Fields

Samuel Young

10 min read

Next Topic - Forces on Moving Charges in Magnetic Fields

Listen to this study note

Study Guide Overview

This study guide covers magnetic fields and forces, including forces on moving charges and current-carrying wires. It explains key formulas like F = q(v × B) and F = ILBsinθ, emphasizing the right-hand rule for determining force direction. The guide also covers magnetic fields generated by currents, including the field around a long wire using the formula B = (μ₀I)/(2πr). Finally, it explores the Biot-Savart Law and Ampère's Law, focusing on the latter's application for calculating fields in symmetrical situations.

#AP Physics C: E&M - Magnetic Fields Study Guide 🧲

Hey there! Let's get you prepped for the exam with a focused review of Magnetic Fields. This guide is designed to be your go-to resource for a quick and effective review. Let's dive in!

#1. Magnetic Fields and Forces 🧲

#1.1 Forces on Moving Charges in Magnetic Fields

When a charged particle moves through a magnetic field, it experiences a magnetic force. This force is always perpendicular to both the velocity of the particle and the magnetic field.
Formula: $F = q(v \times B)$
- $F$ : Magnetic force (N)
- $q$ : Charge of the particle (C)
- $v$ : Velocity of the particle (m/s)
- $B$ : Magnetic field strength (T)
- The $\times$ denotes the cross product, which means the force is perpendicular to both velocity and magnetic field.
Right-Hand Rule: Point your thumb in the direction of the velocity, your fingers in the direction of the magnetic field, and your palm will point in the direction of the force on a positive charge. For a negative charge, the force is in the opposite direction.

Key Concept

The magnetic force does no work on the charged particle because it's always perpendicular to the motion. It changes the direction of motion, not the speed.

Cyclotron Motion: A charged particle in a uniform magnetic field will move in a circle. The radius of this circle is determined by:
- Velocity of the particle
- Strength of the magnetic field
- Mass and charge of the particle

Memory Aid

Remember the right-hand rule: Thumb for velocity, fingers for the magnetic field, and palm for force (for positive charges). For negative charges, use the back of your hand!

Practice Question

Multiple Choice Questions

A proton moves with a velocity v in a uniform magnetic field B. The magnetic force on the proton is: (A) parallel to v (B) parallel to B (C) perpendicular to both v and B (D) zero if v is parallel to B
An electron is moving in a circular path in a uniform magnetic field. If the magnetic field strength is doubled, the radius of the circular path will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A proton (charge +e, mass m) enters a region of uniform magnetic field B with a velocity v perpendicular to the field.

(a) Draw a diagram showing the path of the proton in the magnetic field. (2 points)

(b) Derive an expression for the radius of the circular path in terms of e, m, v, and B. (5 points)

(c) If the magnetic field is 0.5 T, the velocity of the proton is 2 x 10^6 m/s and the mass of the proton is 1.67 x 10^-27 kg, find the radius of the path. (3 points)

Answer Key

Multiple Choice

Free Response

(a) Diagram should show a circular path. (2 points)

(b) Derivation: - Magnetic force = centripetal force: evB = mv^2/r (2 points) - Solve for r: r = mv/eB (3 points)

#1.2 Forces on Current-Carrying Wires in Magnetic Fields

A current-carrying wire in a magnetic field experiences a force. This is the principle behind electric motors!
Formula: $F = ILB\sin\theta$
- $F$ : Force on the wire (N)
- $I$ : Current in the wire (A)
- $L$ : Length of the wire in the magnetic field (m)
- $B$ : Magnetic field strength (T)
- $\theta$ : Angle between the current and the magnetic field
Right-Hand Rule: Again, use the right-hand rule. Thumb points in the direction of the current, fingers point in the direction of the magnetic field, and the palm shows the direction of the force on the wire.

Exam Tip

Remember, the force is maximized when the current is perpendicular to the magnetic field ( $\theta$ = 90°), and zero when they are parallel ( $\theta$ = 0°).

Common Mistake

Students often mix up the direction of current and force. Always use the right-hand rule carefully!

Quick Fact

The force on a current-carrying wire is the basis for how speakers work. The current in the coil interacts with the magnet to produce sound.

Practice Question

Multiple Choice Questions

A straight wire carrying a current is placed in a uniform magnetic field. The force on the wire is maximum when: (A) the wire is parallel to the magnetic field (B) the wire is perpendicular to the magnetic field (C) the wire is at a 45-degree angle to the magnetic field (D) the force is always zero
A wire of length L carries a current I in a magnetic field B. If the length of the wire is doubled and the current is halved, the magnetic force on the wire will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A straight wire of length 0.5 m carries a current of 2 A in a uniform magnetic field of 0.4 T. The wire is oriented at an angle of 30 degrees to the magnetic field.

(a) Calculate the magnitude of the force on the wire. (3 points)

(b) If the angle between the wire and magnetic field is increased to 90 degrees, what is the new force on the wire? (2 points)

Answer Key

Multiple Choice

Free Response

(a) Calculation: - $F = ILB\sin\theta = (2 A)(0.5 m)(0.4 T)\sin(30^\circ) = 0.2 N$ (3 points)

(b) Calculation: - $F = ILB\sin\theta = (2 A)(0.5 m)(0.4 T)\sin(90^\circ) = 0.4 N$ (2 points)

(c) The direction of the force is perpendicular to both the wire and the magnetic field, determined by the right-hand rule. (2 points)

#2. Magnetic Fields from Currents ⚡

#2.1 Fields of Long Current-Carrying Wires

A long, straight wire with a current produces a circular magnetic field around it.
Formula: $B = \frac{\mu_0 I}{2\pi r}$
- $B$ : Magnetic field strength (T)
- $\mu_0$ : Permeability of free space ( $4\pi \times 10^{-7}$ T⋅m/A)
- $I$ : Current in the wire (A)
- $r$ : Distance from the wire (m)
Right-Hand Rule: Wrap your fingers around the wire in the direction of the current; your thumb points in the direction of the magnetic field.

Key Concept

The magnetic field is strongest close to the wire and decreases as the distance from the wire increases.

Practice Question

Multiple Choice Questions

The magnetic field strength due to a long, straight current-carrying wire is: (A) directly proportional to the current and inversely proportional to the distance from the wire (B) inversely proportional to the current and directly proportional to the distance from the wire (C) directly proportional to both the current and the distance from the wire (D) inversely proportional to both the current and the distance from the wire
If the current in a long, straight wire is doubled, and the distance from the wire is also doubled, the magnetic field strength will: (A) double (B) halve (C) remain the same (D) quadruple

Free Response Question

A long, straight wire carries a current of 5 A. Calculate the magnetic field strength at a distance of 0.1 m from the wire.

(a) Calculate the magnetic field strength at a distance of 0.1 m from the wire. (4 points)

(b) If the distance from the wire is doubled, what is the new magnetic field strength? (2 points)

(c) What is the direction of the magnetic field at a point above the wire, assuming the current is moving to the right? (2 points)

Answer Key

Multiple Choice

Free Response

(a) Calculation: - $B = \frac{\mu_0 I}{2\pi r} = \frac{(4\pi \times 10^{-7} T⋅m/A)(5 A)}{2\pi (0.1 m)} = 1 \times 10^{-5} T$ (4 points)

(b) Calculation: - $B = \frac{(4\pi \times 10^{-7} T⋅m/A)(5 A)}{2\pi (0.2 m)} = 0.5 \times 10^{-5} T$ (2 points)

#2.2 Biot-Savart Law and Ampère’s Law

Biot-Savart Law: This law calculates the magnetic field created by a small segment of current-carrying wire. It's useful for complex geometries but can be mathematically intensive.
Ampère’s Law: This law relates the magnetic field around a closed loop to the current passing through the loop. It's a powerful tool for calculating magnetic fields in situations with high symmetry (like long wires and solenoids).
Ampere's Law Formula: $\oint B \cdot dl = \mu_0 I_{enc}$
- The line integral of the magnetic field around a closed loop is proportional to the current enclosed by the loop.

Ampère's Law is crucial for solving problems involving magnetic fields, especially for long wires and solenoids. Focus on understanding how to apply it.

Memory Aid

Think of Ampère's Law as a way to find the 'total magnetic field' around a current, similar to how Gauss's Law finds the 'total electric field' around a charge.

Practice Question

Multiple Choice Questions

Which law is most useful for calculating the magnetic field due to a long, straight current-carrying wire? (A) Biot-Savart Law (B) Ampère’s Law (C) Faraday's Law (D) Gauss's Law
Ampère’s Law relates the magnetic field around a closed loop to: (A) the electric field through the loop (B) the current enclosed by the loop (C) the charge enclosed by the loop (D) the magnetic flux through the loop

Free Response Question

Use Ampère's Law to derive the magnetic field strength at a distance r from a long, straight wire carrying a current I.

(a) State Ampère's Law. (2 points)

(b) Draw a diagram showing the Amperian loop you will use for this derivation. (2 points)

Answer Key

Multiple Choice

Free Response

(a) Ampère's Law: The line integral of the magnetic field around a closed loop is proportional to the current enclosed by the loop. (2 points)

(b) Diagram should show a circular Amperian loop of radius r centered on the wire. (2 points)

(c) Derivation: - $\oint B \cdot dl = B(2\pi r)$ (2 points) - $B(2\pi r) = \mu_0 I$ (1 point) - $B = \frac{\mu_0 I}{2\pi r}$ (1 point)

#Final Exam Focus 🎯

High-Priority Topics:
- Forces on moving charges and current-carrying wires in magnetic fields
- Magnetic fields produced by long wires and solenoids
- Application of Ampère's Law
Common Question Types:
- Calculating magnetic forces on particles and wires
- Determining the direction of magnetic forces and fields using right-hand rules
- Using Ampère's Law to find magnetic fields due to current distributions
- Conceptual questions about the interaction of magnetic fields and matter
Last-Minute Tips:
- Time Management: Start with questions you know how to solve quickly.
- Right-Hand Rule: Practice using the right-hand rule correctly; it's crucial for many problems.
- Units: Always include units in your calculations and answers.
- Formulas: Make sure you know the key formulas and when to use them.
- Conceptual Understanding: Don't just memorize formulas; understand the concepts behind them.

Good luck! You've got this! 💪

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A positively charged particle is moving to the right in a magnetic field that points into the page. What is the direction of the magnetic force on the particle? 🚀

Upward

Downward

Leftward

Into the page