Applying Ampere's Law to a Solenoid: Worked Example

📚 Quick Study Guide

• 🧲 Ampere's Law: Relates the integrated magnetic field around a closed loop to the electric current passing through the loop. Mathematically: $\oint B \cdot dl = \mu_0 I_{enc}$, where $B$ is the magnetic field, $dl$ is a differential length element, $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} T \cdot m/A$), and $I_{enc}$ is the enclosed current.
• 🌀 Solenoid: A coil of wire tightly wound into a helix. When current flows through it, it creates a nearly uniform magnetic field inside.
• 📏 Magnetic Field Inside a Solenoid: $B = \mu_0 n I$, where $n = N/L$ is the number of turns per unit length, $N$ is the total number of turns, $L$ is the length of the solenoid, and $I$ is the current. This formula assumes the solenoid is much longer than its diameter.
• 📍 Applying Ampere's Law to a Solenoid: Choose an Amperian loop (typically rectangular) that takes advantage of the symmetry of the solenoid. The magnetic field is approximately parallel to the axis inside the solenoid and nearly zero outside.
• 📐 Amperian Loop Integration: Integrate the magnetic field around the Amperian loop. The only significant contribution to the integral comes from the section inside the solenoid.

Practice Quiz

1. What does Ampere's Law relate?

   1. A) Electric field to charge density
   2. B) Magnetic field to electric current
   3. C) Electric potential to electric field
   4. D) Magnetic force to charge velocity

2. What is the magnetic field outside an ideal, infinitely long solenoid?

   1. A) $\mu_0 n I$
   2. B) $\frac{\mu_0 n I}{2}$
   3. C) Zero
   4. D) Depends on the diameter of the solenoid

3. Which of the following is the correct expression for the magnetic field inside a solenoid?

   1. A) $B = \frac{\mu_0 I}{2\pi r}$
   2. B) $B = \mu_0 n I$
   3. C) $B = \frac{\mu_0 I}{2r}$
   4. D) $B = \frac{\mu_0 n}{I}$

4. What does 'n' represent in the formula $B = \mu_0 n I$ for a solenoid?

   1. A) Number of turns
   2. B) Length of the solenoid
   3. C) Number of turns per unit length
   4. D) Current density

5. In applying Ampere's Law to a solenoid, what shape is typically used for the Amperian loop?

   1. A) Circle
   2. B) Square
   3. C) Rectangle
   4. D) Triangle

6. If the current in a solenoid is doubled, what happens to the magnetic field inside (assuming all other parameters remain constant)?

   1. A) Halves
   2. B) Doubles
   3. C) Quadruples
   4. D) Remains the same

7. What is the permeability of free space ($\mu_0$)?

   1. A) $8.85 \times 10^{-12} F/m$
   2. B) $4\pi \times 10^{-7} T \cdot m/A$
   3. C) $3 \times 10^{8} m/s$
   4. D) $1.6 \times 10^{-19} C$