AP Physics C: Electricity and Magnetism Questions on Ampere's Law

📚 Quick Study Guide

• 🧲 Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop.
• 📝 The formula for Ampere's Law is: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$, where $\vec{B}$ is the magnetic field, $d\vec{l}$ is an infinitesimal length element along the closed loop, $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} T \cdot m/A$), and $I_{enc}$ is the current enclosed by the loop.
• 📏 Choose an Amperian loop that takes advantage of symmetry to simplify the calculation. Common shapes include circles and rectangles.
• 💡 The direction of the magnetic field can be determined using the right-hand rule.
• ➕ When multiple currents are present, $I_{enc}$ is the algebraic sum of the currents passing through the loop, taking direction into account.

✍️ Practice Quiz

1. Question 1: A long, straight wire carries a current of 5A. What is the magnitude of the magnetic field 2 cm from the wire?
   1. A) $2.5 \times 10^{-5} T$
   2. B) $5.0 \times 10^{-5} T$
   3. C) $7.5 \times 10^{-5} T$
   4. D) $1.0 \times 10^{-4} T$
2. Question 2: A solenoid with 2000 turns per meter carries a current of 2A. What is the magnetic field inside the solenoid?
   1. A) $2.5 \times 10^{-3} T$
   2. B) $5.0 \times 10^{-3} T$
   3. C) $7.5 \times 10^{-3} T$
   4. D) $1.0 \times 10^{-2} T$
3. Question 3: A cylindrical conductor of radius R carries a current I uniformly distributed across its cross-section. What is the magnetic field at a distance r < R from the center of the conductor?
   1. A) $\frac{\mu_0 I r}{2 \pi R^2}$
   2. B) $\frac{\mu_0 I}{2 \pi r}$
   3. C) $\frac{\mu_0 I r}{2 \pi R}$
   4. D) $\frac{\mu_0 I}{2 \pi R^2}$
4. Question 4: Two long, parallel wires are separated by a distance of 4 cm and carry currents of 10A in the same direction. What is the magnetic force per unit length between the wires?
   1. A) $2.5 \times 10^{-4} N/m$
   2. B) $5.0 \times 10^{-4} N/m$
   3. C) $7.5 \times 10^{-4} N/m$
   4. D) $1.0 \times 10^{-3} N/m$
5. Question 5: A square loop of side length a carries a current I. What is the magnetic field at the center of the square?
   1. A) $\frac{2\sqrt{2} \mu_0 I}{\pi a}$
   2. B) $\frac{\sqrt{2} \mu_0 I}{\pi a}$
   3. C) $\frac{\mu_0 I}{\pi a}$
   4. D) $\frac{\mu_0 I}{2\pi a}$
6. Question 6: Which of the following is a correct statement of Ampere's Law?
   1. A) $\oint \vec{E} \cdot d\vec{l} = \frac{q_{enc}}{\epsilon_0}$
   2. B) $\oint \vec{B} \cdot d\vec{A} = 0$
   3. C) $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
   4. D) $\oint \vec{E} \cdot d\vec{A} = 0$
7. Question 7: A toroidal solenoid has an inner radius $r_1$ and an outer radius $r_2$, and $N$ total turns. What is the magnetic field inside the toroid at a radius r where $r_1 < r < r_2$?
   1. A) $\frac{\mu_0 N I}{2 \pi r}$
   2. B) $\frac{\mu_0 N I}{2 \pi (r_2 - r_1)}$
   3. C) $\frac{\mu_0 I}{2 \pi r}$
   4. D) 0