Solved Examples: Amperian Loop Calculations for AP Physics C

📚 Quick Study Guide

🔍 The Amperian loop is an imaginary closed loop used to calculate the magnetic field using Ampere's Law. 📏 Choose the loop wisely! Symmetry is your friend; pick a loop where the magnetic field is either constant or zero. 🧭 Ampere's Law: $\oint B \cdot dl = \mu_0 I_{enc}$, where $B$ is the magnetic field, $dl$ is an infinitesimal length element along the 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. 💡 To apply Ampere's Law, determine the direction of the magnetic field and choose the loop such that $B$ is either parallel or perpendicular to $dl$. 𧮄 Remember to only include the current that passes through the Amperian loop. Currents outside the loop do not contribute to the line integral. ➕ When multiple currents are enclosed, consider their directions. Currents flowing in opposite directions have opposite signs. 📐 For wires, a circular Amperian loop centered on the wire is usually the most convenient choice.

🧪 Practice Quiz

1. What is the purpose of an Amperian loop in the context of Ampere's Law?
1. To measure the electric field.
2. To simplify the calculation of the magnetic field.
3. To calculate the gravitational force.
4. To determine the resistance of a wire.
2. Which of the following is the correct mathematical expression for Ampere's Law?
1. $\oint E \cdot dA = \frac{Q_{enc}}{\epsilon_0}$
2. $\oint B \cdot dl = \mu_0 I_{enc}$
3. $\oint B \cdot dA = 0$
4. $\oint E \cdot dl = - \frac{d\Phi_B}{dt}$
3. Consider an infinitely long, straight wire carrying a current $I$. What shape is typically used for the Amperian loop to calculate the magnetic field around the wire?
1. Square
2. Triangle
3. Circle
4. Rectangle
4. A long, straight wire carries a current of 5 A. An Amperian loop with a radius of 2 cm is centered on the wire. What is the value of $\oint B \cdot dl$ around the loop?
1. $4\pi \times 10^{-7} T \cdot m$
2. $10\pi \times 10^{-7} T \cdot m$
3. $2\pi \times 10^{-5} T \cdot m$
4. $5 \times 10^{-7} T \cdot m$
5. Two long, parallel wires carry equal currents in opposite directions. If you choose an Amperian loop that encloses both wires, what is the value of $I_{enc}$?
1. $2I$
2. $I$
3. $0$
4. $-2I$
6. Inside a very long solenoid, the magnetic field is uniform and parallel to the axis. What shape of Amperian loop is often used to determine the magnetic field inside the solenoid?
1. Circle
2. Square/Rectangle
3. Triangle
4. Irregular Shape
7. A cylindrical conductor with radius R carries a uniformly distributed current I. What is the magnetic field at a distance r < R from the axis of the conductor? Assume the current density is uniform.
1. $B = \frac{\mu_0 I r}{2 \pi R^2}$
2. $B = \frac{\mu_0 I}{2 \pi r}$
3. $B = \frac{\mu_0 I}{2 \pi R}$
4. $B = \frac{\mu_0 I r}{2 \pi R}$