#AP Physics C: E&M - Magnetism and Moving Charges ⚡

Hey there! Let's dive into the exciting world of magnetism and moving charges. This is a super important topic, so let's make sure you're feeling confident. We'll break it down step-by-step, so you'll be ready to ace this on the exam.

#Magnetic Fields of Moving Charges

#Single Moving Charge Field 🧲

• A single moving charge creates its own magnetic field. Think of it like a tiny magnet that only appears when the charge is in motion.

• Direction: Use the right-hand rule to find the magnetic field direction. Point your thumb in the direction of the charge's velocity, and your curled fingers will show the direction of the magnetic field lines.

• Strength: The magnetic field is strongest when the velocity and position vectors are perpendicular (90° angle) and weakens as the angle deviates from 90°.

  

  Caption: Magnetic field lines around a moving positive charge. Notice how the field is circular and the direction is determined by the right-hand rule.

#Force on Moving Charges in Magnetic Fields

#Cross-Product of Velocity and Field

• A magnetic field exerts a force on a moving charge. This force is what makes motors and other cool tech work!
• Force Calculation: The magnetic force is given by the equation: $$\vec{F}_{B} = q(\vec{v} \times \vec{B})$$
  • $\vec{F}_{B}$ is the magnetic force vector.
  • $q$ is the charge (positive or negative).
  • $\vec{v}$ is the velocity vector of the charge.
  • $\vec{B}$ is the magnetic field vector.
• The cross product means the force is perpendicular to both the velocity and the magnetic field. Use the right-hand rule again to figure out the direction of the force.

#Combined Electric and Magnetic Fields

• When both electric and magnetic fields are present, a charged particle feels both forces.

• Electric Force: $\vec{F}_{E} = q\vec{E}$ (force due to electric field).

• Magnetic Force: $\vec{F}_{B} = q(\vec{v} \times \vec{B})$ (force due to magnetic field).

• Net Force: The total force is the vector sum of these two: $\vec{F}_{net} = \vec{F}_{E} + \vec{F}_{B} = q\vec{E} + q(\vec{v} \times \vec{B})$. This is known as the Lorentz force.

  

  Caption: A visual representation of the Lorentz force on a positive charge in combined electric and magnetic fields.

#Hall Effect in Conductors

• The Hall effect occurs when a magnetic field is applied perpendicular to the current flow in a conductor. It's like a traffic jam for electrons!

• Charge Separation: The magnetic force pushes charge carriers (electrons or holes) to one side of the conductor, creating a charge imbalance.

• Hall Voltage: This charge separation creates a transverse electric field and a measurable voltage across the conductor, known as the Hall voltage ($\Delta V_{H}$). 🔌

• Equilibrium: The process continues until the electric force from the Hall voltage balances the magnetic force.

• Hall Voltage Formula: The Hall voltage is proportional to the magnetic field strength (B) and the current density (J): $\Delta V_{H} \propto BJ$.

  

  Caption: Illustration of the Hall effect, showing how a magnetic field deflects moving charges in a conductor, leading to a voltage difference.

#Final Exam Focus

Alright, let's get down to the nitty-gritty. Here's what you absolutely need to nail for the exam:

• Right-Hand Rule: Master the right-hand rule for finding the direction of magnetic fields and forces. This is crucial for both multiple-choice and free-response questions. Practice with different orientations of velocity and magnetic fields.
• Magnetic Force Equation: Know the magnetic force equation $\vec{F}_{B} = q(\vec{v} \times \vec{B})$ inside and out. Be comfortable using it with different vector orientations.
• Lorentz Force: Understand that the total force on a charge in combined electric and magnetic fields is the vector sum of the electric and magnetic forces. This is a common concept in FRQs.
• Hall Effect: Be ready to explain the Hall effect and its implications. Understand how the Hall voltage is related to the magnetic field and current density.

#Time Management Tips

• Quick Scan: Start with a quick scan of the questions to identify the easy ones. Tackle these first to build confidence and save time.
• FRQ Strategy: For free-response questions, start by writing down the relevant formulas and known values. This will help you organize your thoughts and earn partial credit even if you don't get the final answer.
• Units: Always include units in your answers. Missing units can cost you points. Check your units at the end of the problem.
• Reasonable Answers: If your final answer seems way off, double-check your calculations. It's better to catch a mistake yourself than to lose points on the exam.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. A proton moves with a velocity $\vec{v} = (2 \times 10^6 m/s) \hat{i}$ in a magnetic field $\vec{B} = (0.5 T) \hat{j}$. What is the magnetic force on the proton? (A) $1.6 \times 10^{-13} N \hat{k}$ (B) $-1.6 \times 10^{-13} N \hat{k}$ (C) $3.2 \times 10^{-13} N \hat{k}$ (D) $-3.2 \times 10^{-13} N \hat{k}$

2. A wire carrying a current is placed in a magnetic field. If the current is doubled and the magnetic field is halved, what happens to the magnetic force on the wire? (A) The force doubles (B) The force halves (C) The force remains the same (D) The force is quadrupled

3. In the Hall effect, which of the following is true about the Hall voltage? (A) It is proportional to the magnetic field and inversely proportional to the current. (B) It is inversely proportional to the magnetic field and proportional to the current. (C) It is proportional to both the magnetic field and the current. (D) It is inversely proportional to both the magnetic field and the current.

#Free Response Question

A long, straight wire carries a current $I$ in the +x direction. A particle with charge $q$ and mass $m$ is moving with velocity $\vec{v}$ in the +y direction at a distance $r$ from the wire. The magnetic field due to the wire is given by $B = \frac{\mu_0 I}{2 \pi r}$, where $\mu_0$ is the permeability of free space.

(a) Determine the direction of the magnetic force on the particle. (2 points) (b) Calculate the magnitude of the magnetic force on the particle. (3 points) (c) If the particle is initially at rest and released, describe the subsequent motion of the particle. (3 points) (d) If an electric field is applied in the -z direction, what is the magnitude of the electric field needed to cancel out the magnetic force? (3 points)

FRQ Scoring Breakdown:

(a) 2 points - 1 point for correctly identifying the direction using the right-hand rule (into the page or -z direction). - 1 point for correct reasoning.

(b) 3 points - 1 point for using the correct formula for the magnetic force: $F = qvB$ - 1 point for using the correct magnetic field expression: $B = \frac{\mu_0 I}{2 \pi r}$ - 1 point for the correct magnitude of the force: $F = \frac{q v \mu_0 I}{2 \pi r}$

(c) 3 points - 1 point for stating that the particle will accelerate in the -z direction due to the magnetic force. - 1 point for stating that the particle will start to deflect from its original path. - 1 point for stating that the particle will move in a curved path due to the constant force perpendicular to its velocity.

(d) 3 points - 1 point for recognizing that the electric force must be equal in magnitude and opposite in direction to the magnetic force. - 1 point for using the correct formula for the electric force: $F_E = qE$ - 1 point for the correct magnitude of the electric field: $E = \frac{v \mu_0 I}{2 \pi r}$

That's it! You've got this! Go ace that exam!