LEC 2: Electric Field and Gauss's Law – Physics 2101A Lecture 24 Study Notes

Electric Field and Gauss's Law

Overview and Learning Goals

This lecture reviews key concepts in electrostatics, focusing on the calculation of electric fields, electric flux through surfaces, and Gauss's law. These topics are foundational for understanding the behavior of electric charges and fields in classical physics.

• Electrostatics involves the study of stationary electric charges and the fields they produce.

• Learning Goals: Calculate electric fields, understand electric flux, and apply Gauss's law in both integral and differential forms.

Recap: Coulomb's Law

Definition and Application

Coulomb's law describes the force between two point charges. It is the basis for calculating electric fields and forces in electrostatics.

• Formula: The force exerted by charge $q_1$ on $q_2$ is given by: $\vec{F}_{12} = \frac{q_1 q_2}{4 \pi \varepsilon_0 r_{12}^2} \hat{r}_{12}$ where $\hat{r}_{12}$ is the unit vector from $q_1$ to $q_2$.

• Superposition Principle: For multiple charges, the net force is the vector sum of individual forces: $\vec{F}_j = \frac{q_j}{4 \pi \varepsilon_0} \sum_i \frac{q_i}{r_{ij}^2} \hat{r}_{ij}$

• Constants: Permittivity of vacuum: $\varepsilon_0 = 8.85 \times 10^{-12} \frac{C^2}{N \cdot m^2}$ Coulomb's constant: $k_e = \frac{1}{4 \pi \varepsilon_0} = 8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$

• Example: Calculating the force between two point charges separated by a distance $r$.

Electric Field

Definition and Calculation

The electric field $\vec{E}$ at a point in space is defined as the force per unit charge experienced by a small test charge placed at that point.

• Formula for N source charges: $\vec{E}(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \sum_{i=1}^N \frac{q_i}{r_i^2} \hat{r}_i$ where $\hat{r}_i$ is the unit vector from each source charge to the point of interest.

• Charge Distributions:

  • Point charge: $\vec{E}(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \sum_i \frac{q_i}{r_i^2} \hat{r}_i$

  • Line charge: $\vec{E}(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\lambda \, d\ell}{r^2} \hat{r}$

  • Surface charge: $\vec{E}(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\sigma \, dA}{r^2} \hat{r}$

  • Volume charge: $\vec{E}(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho \, dV}{r^2} \hat{r}$

• Example: Calculating the field from a continuous line of charge using integration.

Example Problem: Electric Field of an Infinitely Long Line Charge

Direct Calculation

To find the electric field at a distance $r$ from an infinitely long line with uniform charge density $\lambda$, use symmetry and integration.

• Setup: The field is radially symmetric; only the radial component remains after integrating over the line.

• Key Steps:

  • Express $dq = \lambda dz$ for a segment $dz$.

  • Integrate the contributions $d\vec{E}$ from each segment, considering only the radial component.

  • Final result: $E_{\text{total}} = \frac{\lambda}{2 \pi \varepsilon_0 r}$

• Application: Used to model the field near charged wires or rods.

Electric Flux

Definition and Calculation

Electric flux $\phi_e$ through a surface $\vec{A}$ quantifies the number of electric field lines passing through the surface.

• Formula: $\phi_e = \vec{E} \cdot \hat{n} A$ where $\hat{n}$ is the unit vector perpendicular to the surface.

• Interpretation: For closed surfaces, flux measures the net field entering or leaving the volume.

• Example: Calculating flux through a plane or curved surface.

Gauss's Law

Integral Form

Gauss's law relates the net electric flux through a closed surface to the total charge enclosed by that surface.

• Integral Form: $\oint_S \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$

• Physical Meaning: The net flux through a closed surface is proportional to the enclosed charge.

• Shape Independence: The result does not depend on the shape of the surface, only on the total enclosed charge.

• Example: Calculating flux for a point charge inside a sphere.

Solid Angle and Surface Integrals

For arbitrary surfaces, the integral can be expressed in terms of the solid angle $d\Omega$ over the closed surface.

• Solid Angle: $\int d\Omega = 4\pi$ for a sphere.

• Application: Used to simplify calculations for symmetric charge distributions.

Example Problem: Electric Field of a Line Charge via Gauss's Law

Using Gauss's law, the field of an infinitely long line charge is derived by considering a cylindrical Gaussian surface.

• Setup: Cylinder of radius $r$ and length $L$ encloses charge $Q = \lambda L$.

• Calculation:

  • Flux through the side: $\oint \vec{E} \cdot d\vec{A} = E \cdot 2\pi r L$

  • Flux through ends: zero (field is radial)

  • Result: $E = \frac{\lambda}{2\pi \varepsilon_0 r}$

• Comparison: Same result as direct integration, but simpler due to symmetry.

Special Case of Gauss's Law

If no charge is enclosed by the surface, the net electric flux is zero.

• Formula: $\oint_S \vec{E} \cdot d\vec{A} = 0$ when $q$ is outside the volume.

• Application: Used to show that external charges do not contribute to flux through a closed surface.

Differential Form of Gauss's Law

Mathematical Formulation

The differential form of Gauss's law relates the divergence of the electric field to the local charge density.

• Formula: $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$

• Derivation: Uses the divergence theorem to convert the surface integral to a volume integral.

• Physical Meaning: The divergence of $\vec{E}$ at a point equals the charge density at that point divided by $\varepsilon_0$.

• Application: Essential for Maxwell's equations and understanding local field behavior.

Summary Table: Gauss's Law Forms

Form	Equation	Physical Meaning
Integral	$\oint_S \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$	Net flux through closed surface equals enclosed charge divided by $\varepsilon_0$
Differential	$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$	Divergence of $\vec{E}$ equals local charge density divided by $\varepsilon_0$

Summary

• Electric flux quantifies the number of field lines passing through a surface.

• Gauss's law provides a powerful method for calculating electric fields, especially for symmetric charge distributions.

• The integral form relates total flux to enclosed charge; the differential form relates field divergence to local charge density.

• Both forms are mathematically equivalent and foundational for further study in electromagnetism.

Additional info: The differential form of Gauss's law is a precursor to Maxwell's equations, which unify electricity and magnetism.