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.