Flux & Gauss' Law

• Zero net electric charge inside the oval:
• flux = zero
• number of field lines entering=field lines leaving
• positive charge inside:
• flux=positive if net charge inside of oval is positive
• net flux of field lines out of the oval
• Note: the positive electric flux out gets bigger as the charge inside the oval increases
• negative charge inside:
• flux is negative if the net charge inside the oval is zero
• there is a flux of field lines into the oval
• Note: the electric flux into the oval gets bigger as the charge inside the oval becomes more negative

Below is an image of an electric field with two charges. The net flux is zero because the same number of lines entering are leaving. 

Below is an image of a single charge within an electric field. There are only lines leaving and none entering, therefore there is a non-zero positive flux. 

The net flux is directly proportional to the net charge inside.

Below is a chart made to display the net charges and net flux of an electric we drew up.

Gauss' Law Demo:

A Faraday Cage is used in this demo. Attached to the cage are three pieces of aluminum foil chunks on pieces of string. One end of each string is outside of the cage and the other ends is inside the cage. A van de Graff generator is turned on, putting a large negative charge onto the cylinder cage. 

• when the cylinder is charged both the inner and outer foils move away from the cylinder
• by conduction the foils are negatively charged as well
• since the foil and cylinder cage have the same charge, the aluminum pieces will move away and repel the cage

 Given the fact that as like charges, the excess charges will repel each other; the charges will move as far away from each other as possible. They will go as far as the outer ring and can be on top of it.

 Below are the formulas for the circumference of a circle, area or a circle and volume of a sphere. Also displayed is how the circumference, area and volume are changed if the radius is doubled. 

The following shows the relationship between volume and surface area. The ratio of surface area to volume is 3/r. 

Note that if a charge is spread uniformly throughout the volume of an object, the fraction of a charge at r/2 is Q/8.

Gauss' Law was used to find the equation describing the electric field at a distance r from a point charge. 

Charge density is used to find how much charge is contained within a given radius in a charged sphere.

The orientation of the surface area is usually defined by a perpendicular to the surface and has a magnitude equal to the surface area. The normal vector points away from the outside of the surface. Since the angle between the two vectors makes cos(theta)=1, the dot product can be replaced by simply EdA. The charges exert forces on each other that have the same magnitude and are in opposite directions. This is why the magnitude of the electric field is the same at all points on the spherical shell. 

The following activity used Gauss' Law to find the magnitude of the electric field inside of a uniformly charged sphere of radius R and having a total charge Q. 

The volume of a cylinder is displayed below as well as what fraction of charge resides in a uniformly charged cylinder with r/2. 

 Charge per unit length is lambda. It is used to find the electric field outside and inside the cylinder. The electric field is constant. The ends of the cylinder can be ignored because they are perpendicular to the electric field. 

 Since the charge per unit length and distance are given. We were able to solve for the E field. 

Below is the gravitational version of Gauss' Law. The electric field lines spread out their density ( and hence the strength of the electric field) decreases at the same rate that the area of an enclosing surface increases can ultimately be derived from the 1/r^2 dependence of electrical force on distance. 

 Here the gravitational force of the earth is calculated using the new version of Gauss' Law.

Here are videos of Prof. Mason putting a light bulb, a CD, and a fork into the microwave. Each time the microwave sparked and lit up. 

The CD contained a plasma that reacted in the microwave.

With the fork, the electrons collected at the edges of the pointy teeth and finally sparked.

The filament in the light bulb lit up and got brighter as the time increased. 

Summary: 

• Gauss' Law is used to solve for many things relating to electric fields. It can even be used to solve for the gravitational force on the earth. 
• In regards to Flux
• the net flux is directly proportional to the net charge inside 
• # of field lines entering-#field lines leaving=net flux
• the angle between E and dA determines whether E can be a constant and removed from the Gauss' Law integral formula
• the magnitude of the electric field is dependent on the distance