Solved Examples of Electric Field Calculation Using Gauss's Law

📚 Quick Study Guide

• ⚡ Gauss's Law relates the electric flux through a closed surface to the enclosed electric charge. Mathematically, it's represented as: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\vec{E}$ is the electric field, $d\vec{A}$ is the area vector, $Q_{enc}$ is the enclosed charge, and $\epsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12} C^2/Nm^2$).
• 🧮 To effectively use Gauss's Law:
  • 📐 Choose a Gaussian surface that takes advantage of the symmetry of the charge distribution.
  • ➕ Determine the enclosed charge $Q_{enc}$.
  • ✖️ Calculate the surface integral $\oint \vec{E} \cdot d\vec{A}$.
  • ➗ Solve for the electric field $\vec{E}$.
• 💡 Common Gaussian surfaces include spheres (for spherical symmetry), cylinders (for cylindrical symmetry), and boxes (for planar symmetry).
• 🔑 Symmetry is key! Choose surfaces where the electric field is either constant in magnitude and perpendicular to the surface, or tangent to the surface (so the dot product is zero).

Practice Quiz

1. What does Gauss's Law relate?
   1. (A) The electric field to the magnetic field.
   2. (B) The electric flux through a closed surface to the enclosed charge.
   3. (C) The electric potential to the electric field.
   4. (D) The electric current to the magnetic field.
2. Which of the following is NOT a typical Gaussian surface?
   1. (A) Sphere
   2. (B) Cube
   3. (C) Cylinder
   4. (D) Pyramid
3. If the net charge enclosed within a Gaussian surface is zero, what can be said about the electric flux through the surface?
   1. (A) It must be positive.
   2. (B) It must be negative.
   3. (C) It must be zero.
   4. (D) It could be positive, negative, or zero, depending on the electric field.
4. A point charge *q* is placed at the center of a cube. What is the electric flux through one face of the cube?
   1. (A) $q/\epsilon_0$
   2. (B) $q/4\pi\epsilon_0$
   3. (C) $q/6\epsilon_0$
   4. (D) $q/3\epsilon_0$
5. Gauss's Law is most useful for calculating electric fields in situations with:
   1. (A) Low symmetry
   2. (B) High symmetry
   3. (C) No charges
   4. (D) Moving charges
6. What is the value of $\epsilon_0$ (permittivity of free space) used in Gauss's Law?
   1. (A) $8.854 \times 10^{-12} C^2/Nm^2$
   2. (B) $9 \times 10^{9} Nm^2/C^2$
   3. (C) $4\pi \times 10^{-7} Tm/A$
   4. (D) $1.602 \times 10^{-19} C$
7. Which of the following scenarios would be best suited to solve using Gauss's Law?
   1. (A) Finding the electric field of a single point charge using Coulomb's Law.
   2. (B) Finding the electric field a short distance from a uniformly charged infinite plane.
   3. (C) Finding the electric field of two closely spaced, oppositely charged point charges.
   4. (D) Finding the electric field inside a circuit.