Gauss's Law Notes

Learning Outcomes

• Determine the amount of charge within a closed surface by examining the electric field on the surface.
• Understand electric flux and how to calculate it.
• Learn how Gauss’s law relates electric flux through a closed surface to the charge enclosed by the surface.
• Use Gauss’s law to calculate the electric field due to a symmetric charge distribution.
• Determine where the charge is located on a charged conductor.

Introduction

• A child acquires an electric charge by touching a charged metal shell.
• The charged hairs on the child’s head repel and stand out.
• Symmetry properties play an important role in physics.
• Gauss’s law allows electric-field calculations using symmetry principles.

What Is Gauss's Law All About?

• Gauss's Law involves surrounding a charge distribution with an imaginary surface that encloses the charge.
• It examines the electric field at various points on this imaginary surface.
• Gauss’s law is a relationship between the field at all points on the surface and the total charge enclosed within the surface.

Charge and Electric Flux

• Positive Charge:
  • A positive charge within a box produces an outward-pointing electric flux through the surface of the box.
  • The field patterns on the surfaces differ based on charge distribution (e.g., one point charge vs. two).
• Negative Charge:
  • Negative charges inside a box result in an inward-pointing electric flux on the surface.

Zero Net Charge Inside a Box

• Zero Charge:
  • If the box is empty and the electric field is zero everywhere, there is no electric flux into or out of the box.
• Electric Field Exists:
  • An electirc field "flows" into the box on one half and "flows" out of the box on the other half.
  • This results in no net electric flux into or out of the box.
• Charge Outside the Box:
  • If charge is near the box but not inside it, the flux points into the box on one end and out of the box on the opposite end.
  • On the sides, the field is parallel to the surface, so the flux is zero.
  • The net electric flux through the box is zero.

What Affects the Flux Through a Box?

• The net electric flux is directly proportional to the net amount of charge enclosed within the surface.
• The net electric flux is independent of the size of the closed surface.

Calculating Electric Flux

• Flat Area Perpendicular to a Uniform Electric Field:
  • Increasing the area means more electric field lines pass through, increasing the flux.
  • A stronger field means more closely spaced lines, therefore more flux.
• Area Not Perpendicular to the Field:
  • If the area is not perpendicular to the field, fewer field lines pass through it.
  • The area that counts is the silhouette area seen when looking in the direction of the field.
• Area Edge-On to the Field:
  • If the area is edge-on to the field, it's perpendicular to the field, and the flux is zero.

Flux of a Nonuniform Electric Field

• The flux through a surface must be computed using a surface integral over the area:
• $\Phi = \int E \cdot dA$
• The SI unit for electric flux is $N \cdot m^2 / C$

Gauss's Law

• Gauss contritubed to several branches of mathematics, including differential geometry, real analysis, and number theory.
• He also investigated the earth’s magnetism and calculated the orbit of the first asteroid to be discovered.
• Gauss’s law provides a different way to express the relationship between electric charge and electric field, equivalent to Coulomb’s law.

Point Charge Centered in a Spherical Surface

• The projection of an area element $dA$ of a sphere of radius $R$ onto a concentric sphere of radius $2R$ is considered
• The area element on the larger sphere is $4 dA$, but the electric field magnitude is 1/4 as great on the sphere of radius $2R$ as on the sphere of radius $R$.
• The electric flux is the same for both areas and is independent of the radius of the sphere.

Point Charge Inside a Nonspherical Surface

• The flux is independent of the surface and depends only on the charge inside.

Gauss's Law in a Vacuum

• For a closed surface enclosing no charge:
  • $\Phi = \int E \cdot dA = 0$
• If an electric field line from an external charge enters the surface at one point, it must leave at another.

General Form of Gauss's Law

• Let $Q_{encl}$ be the total charge enclosed by a surface.
• Gauss’s law states that the total electric flux through a closed surface is equal to the total (net) electric charge inside the surface, divided by $\epsilon_0$:

Positive and Negative Flux

• A surface around a positive charge has a positive (outward) flux, and a surface around a negative charge has a negative (inward) flux.

Applications of Gauss's Law

• Without integration, Gauss’s law can determine electric flux through closed surfaces.
  • $\Phi = \frac{q}{\epsilon_0}$

Gauss's Law Inside a Conductor

• If a Gaussian surface is constructed inside a conductor, $E = 0$ everywhere on this surface.
• Gauss’s Law requires that the net charge inside the surface is zero.
• Under electrostatic conditions (charges not in motion), any excess charge on a solid conductor resides entirely on the conductor’s surface.

Field of a Uniform Line Charge

• Electric charge is distributed uniformly along an infinitely long, thin wire with charge per unit length $\lambda$ (assumed positive).
• Using Gauss’s law, the electric field is found to be:
  • $E = \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r}$

Field of an Infinite Plane Sheet of Charge

• Gauss’s law can be used to find the electric field caused by a thin, flat, infinite sheet with a uniform positive surface charge density $\sigma$:
  • $E = \frac{\sigma}{2 \epsilon_0}$

Charges on Conductors

• Solid Conductor with a Hollow Cavity:
  • If there is no charge within the cavity, a Gaussian surface (A) shows the net charge on the cavity's surface must be zero because $E = 0$ everywhere on the Gaussian surface.
• Charge Inside a Cavity:
  • If a small object with charge $q$ is placed inside a cavity within a conductor, a charge $-q$ is distributed on the cavity's surface, drawn there by the charge $q$ inside the cavity.
  • The total charge on the conductor must remain zero, so a charge $+q$ must appear on its outer surface.

Faraday's Icepail Experiment

• A conducting container is mounted on an insulating stand and is initially uncharged.
• A charged metal ball is hung from an insulating thread and lowered into the container.
• Charges are induced on the walls of the container.
• Let the ball touch the inner wall: the ball loses all its charge.

The Van De Graaff Generator

• Operates on the same principle as in Faraday’s icepail experiment.
• The electron sink at the bottom draws electrons from the belt, giving the belt a positive charge.
• At the top, the belt attracts electrons away from the conducting shell, giving the shell a positive charge.

Electrostatic Shielding

• A conducting box is immersed in a uniform electric field.
• The field of the induced charges on the box combines with the uniform field to give zero total field inside the box.
• To protect an object from electric fields, surround it with a conducting box called a Faraday cage.
• Little to no electric field can penetrate inside the box.

Field at the Surface of a Conductor

• Gauss’s law shows the electric field at the surface of any conductor is always perpendicular to the surface.
• The magnitude of the electric field just outside a charged conductor is proportional to the surface charge density $\sigma$.