#AP Physics C: E&M - Magnetic Forces and Fields Study Guide 🚀

Hey there, future physicist! Let's get you prepped for the exam with a super-focused review of magnetic forces and fields. We'll break down the key concepts, nail down the formulas, and get you ready to tackle any question they throw your way. Let's do this!

#Forces on Wires in Magnetic Fields

#Force on a Current-Carrying Wire

• Remember how magnetic fields affect individual moving charges? Well, a wire with current is just a bunch of moving charges! 💡

• The force on a current-carrying wire in a magnetic field is given by:

  $$\vec{F} = I \vec{L} \times \vec{B}$$

  • $\vec{F}$ is the magnetic force vector
  • $I$ is the current in the wire
  • $\vec{L}$ is the length vector of the wire (direction is the same as the current)
  • $\vec{B}$ is the magnetic field vector

• The magnitude of the force is:

  $$F = ILB \sin(\theta)$$

  • $\theta$ is the angle between the wire and the magnetic field.

#Right-Hand Rule (RHR) for Wires

• Use the RHR to find the direction of the force:
  1. Thumb: Points in the direction of the current ($I$).
  2. Fingers: Point in the direction of the magnetic field ($B$).
  3. Palm: Points in the direction of the force ($F$).

#Example: Force on a Wire Segment

• Consider a wire with straight and curved sections in a magnetic field (pointing into the page).

• Straight Sections:
  • Force magnitude: $F_s = ILB$ (if the angle is 90 degrees)
  • Direction: Use RHR (force will be either up or down, depending on the current direction).
• Curved Sections:
  • Break into tiny dl segments. However, due to symmetry, the net horizontal force cancels out.
  • Focus on the vertical force components.

#Torque on a Current Loop 🚨

#Torque Basics

• Even though the net force on a closed loop in a magnetic field is zero, there can still be a net torque, causing the loop to rotate.

• Torque is a rotational force: $\tau = rF\sin(\theta)$ (where $r$ is the distance from the axis of rotation).

#Torque on a Current Loop in a Magnetic Field

• Consider a rectangular loop of wire in a magnetic field:

• The forces on the sides of the loop create a torque that causes the loop to rotate.

• Use the RHR to determine the direction of the force on each side of the loop.

• The torque on the loop is given by:

  $$\tau = NIAB \sin(\theta)$$

  • $N$ is the number of turns in the loop.
  • $A$ is the area of the loop.
  • $\theta$ is the angle between the normal to the loop and the magnetic field.

#Magnetic Fields from Current-Carrying Wires 🌐

#Current Creates Magnetic Fields

• A current-carrying wire creates its own magnetic field. 🤯

• The magnetic field lines are circular around the wire.

#Right-Hand Curl Rule (RHCR)

• Use the RHCR to determine the direction of the magnetic field:
  1. Thumb: Points in the direction of the current ($I$).
  2. Fingers: Curl in the direction of the magnetic field ($B$).

#Magnetic Field of a Long Straight Wire

• The magnetic field strength at a distance r from a long, straight wire is:

$$B = \frac{\mu_0 I}{2 \pi r}$$

• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} T \cdot m/A$).

#Forces Between Two Wires 🧲

#Parallel Wires

• When two current-carrying wires are placed near each other, they exert magnetic forces on each other.

• Parallel Currents: Wires with currents in the same direction attract each other.
• Anti-Parallel Currents: Wires with currents in opposite directions repel each other.

#Explanation

• Each wire creates a magnetic field that affects the other wire.
• Use the RHCR to find the magnetic field created by one wire at the location of the other wire.
• Use the RHR to find the force on the second wire due to the magnetic field from the first wire.

#Key Points for Drawing Magnetic Fields

• Magnetic field lines form closed loops.
• The direction of the field is the direction a north pole would point.
• Field line density indicates field strength.
• Use the RHR or RHCR to determine the direction of the field.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A long, straight wire carries a current I in the +z direction. At a point in the xy-plane, the magnetic field due to the wire points in the: (A) +x direction (B) -x direction (C) +y direction (D) -y direction (E) radial direction

2. A rectangular loop of wire is placed in a uniform magnetic field. The loop is oriented such that its plane is parallel to the magnetic field. If a current is passed through the loop, the loop will: (A) experience a net force in the direction of the magnetic field (B) experience a net force perpendicular to the magnetic field (C) experience a net torque that will cause it to rotate (D) experience no net force or torque (E) experience a net force and a net torque

#Free Response Question

The circuit shown above consists of a battery of emf ε in series with a rod of length l , mass m, and resistance r. The rod is suspended by vertical connecting wires of length d, and the horizontal wires that connect to the battery are fixed. All these wires have negligible mass and resistance. The rod is a distance r above a conducting cable. The cable is very long and is located directly below and parallel to the rod. Earth’s gravitational pull is toward the bottom of the page. Express all algebraic answers in terms of the given quantities and fundamental constants.

a) What is the magnitude and direction of the current l in the rod? (2 points)

b) In which direction must there be a current in the cable to exert an upward force on the rod? Justify your answer. (3 points)

c) With the proper current in the cable, the rod can be lifted up such that there is no tension in the connecting wires. Determine the minimum current l_c in the cable that satisfies this situation. (5 points)

#Answers

Multiple Choice Answers

1. (D) Use the RHCR. Thumb in the +z direction, fingers curl in the -y direction at a point in the xy-plane.
2. (C) The loop will experience a torque that will cause it to rotate. There is no net force on the loop.

Free Response Answers

a)

$$I = \frac{\varepsilon}{r}$$

Conventional current leaves the + terminal and travels to the - terminal (CCW in this diagram), so the current in the rod is directed to the left.

• 1 point for correct magnitude of the current
• 1 point for correct direction of the current

b) Because the current in the rod is directed to the left, it will create a magnetic field coming out of the page between the rod and the cable (Right Hand Curl Rule). To lift the rod upwards the cable must create a magnetic field that will repel it, which means that the current in the cable must me traveling to the right (RHCR).

• 1 point for correct direction of current in the cable
• 2 points for correct justification using RHCR

c)

$$F_{mag} = I_c l B_{rod}$$

$$B_{rod} = \frac{\mu_0 I}{2\pi r}$$

$$F_{mag} = I_c l \frac{\mu_0 I}{2\pi r}$$

$$F_{mag} = mg$$

$$I_c l \frac{\mu_0 I}{2\pi r} = mg$$

$$I_c = \frac{2\pi r m g}{\mu_0 I l}$$

$$I_c = \frac{2\pi r m g r}{\mu_0 \varepsilon l}$$

• 1 point for using $F_{mag} = I l B$
• 1 point for calculating $B$ field due to the rod
• 1 point for setting $F_{mag} = mg$
• 1 point for correct substitution
• 1 point for correct final answer

#Final Exam Focus 🎯

#High-Value Topics

• Forces on Wires: Understand how to use the RHR to find force direction and how to calculate force magnitude ($F = ILB \sin(\theta)$).

• Torque on Current Loops: Be able to calculate the torque on a loop ($\tau = NIAB \sin(\theta)$) and understand its applications in electric motors.

• Magnetic Fields from Wires: Know how to use the RHCR to find the direction of the magnetic field around a wire, and how to calculate the magnetic field strength ($B = \frac{\mu_0 I}{2 \pi r}$).

• Forces Between Wires: Understand that parallel currents attract and anti-parallel currents repel, and be able to explain why.

• Connections: AP questions often combine concepts from different units. Be ready to use your knowledge of circuits, forces, and fields together!

#Common Question Types

• Conceptual Questions: Expect questions that ask about the direction of forces and fields using the RHR or RHCR.
• Calculation Questions: Be ready to calculate forces, torques, and magnetic fields using the appropriate formulas.
• Free Response Questions: FRQs often involve a combination of conceptual and calculation-based questions. Practice breaking down complex problems into smaller steps.

#Last-Minute Tips

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back to it later.
• Common Mistakes: Be careful with units, and make sure you're using the correct RHR or RHCR.
• Strategy: Read each question carefully, draw diagrams if necessary, and show all your work. Even if you don't get the final answer, you can still get partial credit for showing your process.

Alright, you've got this! Take a deep breath, trust your preparation, and go ace that exam! You're a physics rockstar! 🌟