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

Hey there! Let's get you totally prepped for the E&M exam. This guide is designed to be your go-to resource, especially the night before the test. We'll break down the concepts, highlight the key points, and make sure you're feeling confident and ready to rock!

#Magnetic Fields of Current-Carrying Wires

#Biot-Savart Law

• The Biot-Savart law is your starting point for understanding how currents create magnetic fields. It's all about calculating the magnetic field generated by a tiny segment of a wire carrying current. 💡
• Formula: $$d \vec{B}=\frac{\mu_{0}}{4 \pi} \frac{I(d\vec{\ell} \times \hat{r})}{r^{2}}$$
• Where:
  • $\mu_0$ is the permeability of free space (a constant).
  • $I$ is the current in the wire.
  • $d\vec{\ell}$ is a small length vector of the wire.
  • $\hat{r}$ is the unit vector from the wire to the point where you're calculating the field.
  • $r$ is the distance from the wire segment to the point.

- **Circular Loop:** At the center of a circular loop, the magnetic field is: - $$B\_{\text {center of loop }}=\frac{\mu\_{0} I}{2 R}$$ - $R$ is the radius of the loop. -

#Magnetic Field Vectors

• Visualize the magnetic field as concentric circles around a wire. 🌀
• The magnetic field vectors are always tangent to these circles.
• Key Point: They never point towards, away from, or parallel to the wire. This is a crucial difference from electric fields.
• The direction of the magnetic field is determined by the Right-Hand Rule:
  • Point your thumb in the direction of the current.
  • Your fingers curl in the direction of the magnetic field. 🖐️

#Magnetic Field Calculations

• To find the total magnetic field, you'll need to integrate the Biot-Savart law over the entire length of the wire. This can sound intimidating, but it's manageable, especially with symmetrical shapes.
• Symmetry is your friend! For straight wires and circular loops, the math simplifies significantly.

#Force on Current-Carrying Wires

#Magnetic Force on Wires

• When a current-carrying wire is in an external magnetic field, it experiences a force. This is the principle behind electric motors!
• Formula: $$\vec{F}_{B}=\int I(d\vec{\ell} \times \vec{B})$$
• Where:
  • $\vec{F}_B$ is the magnetic force on the wire.
  • $I$ is the current.
  • $d\vec{\ell}$ is a small length vector of the wire.
  • $\vec{B}$ is the external magnetic field.

- **Right-Hand Rule for Force:** - Point your fingers in the direction of the current. - Curl your fingers towards the magnetic field. - Your thumb points in the direction of the force. 🖐️ - The force is maximum when the wire is perpendicular to the magnetic field and zero when it's parallel. - - 🪄

#Final Exam Focus

• High-Priority Topics:
  • Biot-Savart Law (especially for straight wires and circular loops).
  • Magnetic force on current-carrying wires.
  • Right-hand rule for both magnetic fields and forces.
• Common Question Types:
  • Calculating magnetic fields at specific points due to current-carrying wires.
  • Determining the force on a wire in a magnetic field.
  • Conceptual questions about the direction of magnetic fields and forces.
• Time Management Tips:
  • Start with the easy questions to build confidence.
  • Don't get stuck on a single problem; move on and come back if you have time.
  • Show all your work; partial credit is your friend.
• Common Pitfalls:
  • Forgetting to use the right-hand rule correctly.
  • Confusing the direction of the magnetic field with the direction of the force.
  • Not paying attention to the geometry of the problem.

#

Practice Question

Practice Questions

#Multiple Choice Questions:

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

2. A circular loop of wire carries a current $I$ in a clockwise direction. The magnetic field at the center of the loop is: (A) directed into the page (B) directed out of the page (C) directed to the right (D) directed to the left (E) zero

3. A wire carrying a current $I$ is placed in a uniform magnetic field $\vec{B}$. If the wire is oriented parallel to the magnetic field, the force on the wire is: (A) maximum (B) minimum but non-zero (C) zero (D) perpendicular to both the current and magnetic field (E) dependent on the length of the wire

#Free Response Question:

A long, straight wire carries a current $I$ along the z-axis. A circular loop of radius $R$ lies in the xy-plane, centered at the origin, and carries a current $I$ in the counterclockwise direction.

(a) Using the Biot-Savart law, derive an expression for the magnetic field at the center of the circular loop due to the current in the loop.

(b) Determine the direction of the magnetic field at the center of the loop due to the current in the loop.

(c) What is the net magnetic field at the center of the loop if the long straight wire also has a current $I$ in the positive z-direction?

(d) Calculate the force on a short segment of the circular loop due to the magnetic field of the wire. Assume the segment is located along the positive x-axis and has a length $d\ell$.

Answer Key and Scoring Rubric:

(a) Derivation of Magnetic Field at Center of Loop (5 points)

• 1 point: Correctly stating Biot-Savart law: $$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{\ell} \times \hat{r}}{r^2}$$
• 1 point: Recognizing that $d\vec{\ell}$ and $\hat{r}$ are perpendicular, so $|d\vec{\ell} \times \hat{r}| = d\ell$
• 1 point: Correctly integrating over the loop: $B = \int dB = \int \frac{\mu_0}{4\pi} \frac{I d\ell}{R^2}$
• 1 point: Recognizing that $\int d\ell = 2\pi R$
• 1 point: Final correct expression: $$B = \frac{\mu_0 I}{2R}$$

(b) Direction of Magnetic Field (2 points)

• 1 point: Applying the right-hand rule correctly.
• 1 point: Correctly stating the direction as out of the page (or positive z-direction).

(c) Net Magnetic Field (3 points)

• 1 point: Recognizing that the magnetic field due to the straight wire is zero at the center of the loop.
• 1 point: Correctly stating that the net magnetic field is equal to the field due to the loop.
• 1 point: Correctly stating the net magnetic field as $\frac{\mu_0 I}{2R}$ out of the page.

(d) Force on a Segment of the Loop (5 points)

• 1 point: Correctly stating the force equation: $d\vec{F} = I d\vec{\ell} \times \vec{B}$
• 1 point: Correctly stating the magnetic field due to the wire at the location of the segment: $B = \frac{\mu_0 I}{2\pi R}$
• 1 point: Recognizing that $d\vec{\ell}$ is in the positive y-direction and the magnetic field is in the negative x-direction.
• 1 point: Correctly using the cross product to find the direction of the force (into the page).
• 1 point: Correctly stating the force magnitude: $dF = \frac{\mu_0 I^2 d\ell}{2\pi R}$

Answers to Multiple Choice Questions:

1. (D)
2. (A)
3. (C)

Remember, you've got this! Go into the exam with confidence, and trust your preparation. You're going to do great! 💪