#AP Physics C: E&M - Unit 4: Magnetism - The Night Before 🌃

Hey! Let's get you totally prepped for the exam. This study guide is designed to be your go-to resource, especially for a last-minute review. We'll break down everything you need to know about magnetic fields, forces, and their interactions with electric fields. Let's make this stick! 🚀

#4.0: Overview 👀

In this unit, we're diving into magnetic fields and how they interact with moving charges and electric fields. It's all about forces, motion, and some seriously cool applications. This unit makes up a significant portion of the AP exam, so let's make sure you're comfortable with it.

This unit accounts for 17-23% of the AP Physics C: E&M exam. Make sure you're comfortable with all the concepts.

#4.1: Forces on Moving Charges in a Magnetic Field 🧭

#What Does a Magnetic Field Look Like? 🧲

Remember playing with magnets? Magnetic fields are all around us! Here's the lowdown:

• Magnetic Field Lines: These lines show the direction a north pole would be pushed or pulled. Think of it like a map for magnetic forces.
• Likes Repel, Opposites Attract: Just like with electric charges, north poles repel north poles and attract south poles.
• Field Strength: The closer the field lines, the stronger the magnetic field. It's all about density!

Caption: Visualizing magnetic field lines around a bar magnet.

The Earth's magnetic field is super important! It protects us from harmful solar radiation. Charged particles from the sun spiral along these field lines, creating the beautiful auroras near the poles. 🌌

Caption: The aurora borealis, a result of charged particles interacting with Earth's magnetic field.

Caption: Magnetic field lines around a bar magnet.

#Magnetic Force & the Right Hand Rule for Magnetic Fields 🖐️

Why do charged particles curve in a magnetic field? Because they experience a force! Here’s the magic equation:

$$\vec{F} = q \vec{v} \times \vec{B}$$

To experience a magnetic force:

1. Charge: The object must be charged ($q \ne 0$).
2. Motion: The particle must be moving ($v \ne 0$).
3. Magnetic Field: There must be a magnetic field present ($B \ne 0$).
4. Perpendicularity: The particle's velocity and the magnetic field must have a perpendicular component (cross product).

Caption: The Right-Hand Rule (RHR) for magnetic forces.

#Paths of Charged Objects 🗺️

Because the magnetic force is always perpendicular to the velocity, it causes the particle to curve. Most of the time, we'll assume uniform circular motion (UCM) for calculations. 🔄

Caption: A charged particle moving in a circular path due to a magnetic field.

This is used in tons of tech, like old TVs and particle accelerators! 📺

Caption: A cathode ray tube using magnetic fields to steer electrons.

Physicists also use this to identify particles after collisions. Check out how electrons and positrons spiral in opposite directions!

Caption: Particle tracks in a magnetic field.

#Magnetic & Electric Field Interactions 🏗️

Things get even cooler when we combine electric and magnetic fields! They can work together or against each other. For example:

• Cathode Ray Tubes: Use magnetic fields for vertical steering and electric fields for horizontal steering.
• Velocity Selectors: By carefully balancing electric and magnetic forces, you can select particles with a specific velocity. 🎯
• Mass Spectrometers: Use these fields to determine the mass of particles by measuring the radius of their circular path in a magnetic field.

Caption: A mass spectrometer using both electric and magnetic fields.

#Final Exam Focus 🎯

• High Priority Topics:
  • Magnetic force on a moving charge.
  • Right-Hand Rule (RHR) for determining force direction.
  • Circular motion of charged particles in a magnetic field.
  • Velocity selectors and mass spectrometers.
• Common Question Types:
  • Multiple-choice questions testing RHR and force direction.
  • Free-response questions involving calculations of force, radius of curvature, and particle velocity.
  • Conceptual questions about the applications of magnetic fields.
• Last-Minute Tips:
  • Time Management: Quickly identify the key concepts in each question and apply the relevant formulas.
  • Common Pitfalls: Double-check the direction of the force (especially for negative charges) and ensure you're using the correct units.
  • Strategy: Draw diagrams to visualize the forces and directions. Practice using the RHR until it becomes second nature.

#Practice Questions

Practice Question

Question 1: Free Response

Two plates are set up with a potential difference V between them. A small sphere of mass m and charge -e is placed at the left-hand plate, which has a negative charge, and is allowed to accelerate across the space between the plates and pass through a small opening. After passing through the small opening, the sphere enters a region in which there is a uniform magnetic field of magnitude B directed into the page, as shown above. Ignore gravitational effects. Express all algebraic answers in terms of V, m, e, B, and fundamental constants, as appropriate.

(a) i. What is the initial direction of the force on the sphere as it enters the magnetic field? (Check one.)

_ Into the page, _ Out of the page, _ Towards the top of the page, _ Towards the bottom of the page

ii. Describe the path taken by the sphere after it enters the magnetic field.

(b) Derive an expression for the speed of the sphere as it passes through the small opening.

(c) Derive an expression for the radius of the path taken by the sphere as it moves through the magnetic field.

Answer:

a) i. Towards the bottom of the page. Use the RHR, then remember that the RHR is for positive charged objects, so switch the direction of the force because we have an electron.

ii. With a net force pushing towards the bottom of the page, the particle will travel in a circular path curving towards the bottom of the page.

b)

$$KE = qV$$ $$\frac{1}{2}mv^2 = eV$$ $$v = \sqrt{\frac{2eV}{m}}$$

c)

$$F_B = qvB$$ $$F_c = \frac{mv^2}{r}$$ $$qvB = \frac{mv^2}{r}$$ $$r = \frac{mv}{qB}$$

Substitute in v from part b:

$$r = \frac{m\sqrt{\frac{2eV}{m}}}{eB} = \frac{\sqrt{2meV}}{eB}$$

Scoring Breakdown:

• (a) i. 1 point for correct direction, 0 for incorrect/blank.
• (a) ii. 1 point for circular path, 0 for incorrect/blank.
• (b) 1 point for equating KE and work done by electric field, 1 point for correct expression for speed, 0 for incorrect/blank.
• (c) 1 point for equating magnetic force and centripetal force, 1 point for correct expression for radius, 0 for incorrect/blank.

Practice Question

Question 2: Multiple Choice

Image from apclassroom.collegeboard.org

The figure above shows the paths of five particles as they pass through the region inside the box that contains a uniform magnetic field B directed out of the page. Which particle has a positive charge?

(A) A (B) B (C) C (D) D (E) E

Answer:

(B) Use the RHR with your fingers pointing out of the page.

• A - Force is directed upwards, but shows a downward curve. A must be negatively charged
• B - Force is directed downwards, and the curve is downwards. B must be positively charged
• C - Same as A, must be negatively charged
• D - Force is downwards, curve is upwards. D must be negative as well
• E - No curve, so no charge. E is neutral

Practice Question

Question 3: Multiple Choice

A proton moves with a velocity of 2 x 10^6 m/s to the east in a magnetic field of 0.5 T directed vertically upward. What is the magnitude and direction of the magnetic force on the proton?

(A) 1.6 x 10^-13 N, North (B) 1.6 x 10^-13 N, South (C) 3.2 x 10^-13 N, North (D) 3.2 x 10^-13 N, South (E) 0 N

Answer:

(A) Use the formula F = qvBsinθ. The charge of a proton is 1.6 x 10^-19 C, and the angle between velocity and magnetic field is 90 degrees. F = (1.6 x 10^-19 C)(2 x 10^6 m/s)(0.5 T) = 1.6 x 10^-13 N. Using RHR, the force is directed North.

Alright, you've got this! Remember to stay calm, use your resources, and trust your knowledge. You're going to do great! 🎉