Gauss's Law

Purpose:  In this lab, we take a look at some of the behaviors of electrical fields. To do this, we use Gauss's Laws to get an idea of the magnitude of the electrical field at a given distance from the charge. We will also take a look at various situations involving microwaves.

Cal Tech's Electric Field App

To begin, we take a look at a program, on Cal Techs website, that shows a visual representation of electric field lines between two charges. In this picture, we see two charges of opposite charge create symmetrical lines in between. This is very typical of dipoles. We can see the red point is surrounded by red circles. This is to show that the electric field shoots out radially. The same occurs with the blue point, but the electric field shoots inward instead of outward. The sum of the two electric fields at a given point is shown by the white lines.

This now shows the behavior of the electric field lines if another charge of the same magnitude is put in. We can see a repelling behavior between same charges and a dipole between opposite charges.

Redrawing the Field Lines

We re-draw one of the visual representations of the electric field lines and put arrows to display the direction of the electric field lines. The electric field lines of a negative charge point inward while the electric field lines of a positive charge point outward. We also draw some Gaussian surfaces and take the sum of these charges within these surfaces. After, we find the flux through the surfaces and calculate the net flux.

The Electric Field of a Conducting Cylinder

We utilize the vandegraff generator to distribute charge along a conducting cylinder. We were given the task to predict what would happen to the pieces of metal that hang inside the cylinder and outside the cylinder.

Before we find out what happens to the pieces of metal, we discuss the relationship between flux and charge. We write that flux and charge are proportional to each other. We also cover a little unit manipulation  and solve for the k value that multiplies charge to account for the proportionality between the two. It turns out the k was equal to one over epsilon knot. We then say that flux equals to the charge enclosed within the surface divided by the permittivity of free space.

Our group believed that nothing would happen to the pieces of metal that lie next to the conducting cylinder.

We find that our prediction of nothing moving is wrong, as the metal on the outside of the cylinder tries to distance itself from the conducting cylinder. The metal on the inside does not move. This is explained using Gauss's Law. There is no electric field inside the conducting cylinder but there is an electric field outside the cylinder. In the equation for flux, the charge enclose inside the cylinder is zero.

What Do Charges Do Inside Conductors?

We draw how eight charges behave inside a conductor. These charges look for the furthest distance apart. This means that they lie on the outer end of a cylinder. Charges are able to move, almost completely freely, within a conductor. We also discuss the the best possible place to be during a lightning storm. We say that it is best to be inside a car because the car acts as a Faraday cage, where a large portion of the charge flows through the car and into the ground. Some charge may enter the car, but if a person does not touch anything metal inside, a person should be safe, as electrical charge tries to flow through the outer ends of the metal.

What Happens If We Double The Radius?

We take a look at some arithmetic. If we double the radius of the circumference, the circumference doubles as well, but if we do the same for the area, the area increases by 4 times. This is because the 2 is also squared. The same goes for volume, which involves a power of 3.

What Happens If The Radius Is Halved?

We found that if we halve the radius, the volume goes to one over eight. We also see that we generally do not have to worry about the angle between the electric field vector and the area vector, as they generally point in the same direction, causing the angle to be zero and cosine of zero is one.

We take a look at the applications of Gauss's Law. We say that the charge density of an object with a small radius is equal to the charge density of an object with a large radius. We solve for the charge of the small volume and this gives us a relationship with a larger charge. Because we are looking at the small charge, we draw an imaginary Gaussian surface at the radius r and this gives us a surface area at radius r. The r's cancel and we get a simple equation for the electric field in terms of r and R. We, then start to look at a cylinder and its total surface area.

Electric Field Inside of an Insulator

We take a look at the electric field inside an insulator. We use the relationship of charge q and Q using charge density. With some substitution, we solve for the electric field inside an insulator in terms of the radius within the insulator.

Gauss's Law In Terms of Gravity

We took a different look at Gauss's Law by looking at it in terms of gravitation. We set the mass of earth over a constant k equal to the integral of Y dA. When we plugged everything in, we found that Y was equal to the acceleration of gravity.

Steel Wool in an Microwave

We start of the "what you should never do" by putting steel wool inside a microwave. Sparks flew and we were able to see that the electric field was strongest on the points as discharge was occurring on the points.

Fork Inside a Microwave

We see the same kind of effect that occurred in the steel wool happening to a fork. The tips of the fork began to light up, as the electric field was strongest there. This is because the charge density is highest at these points. We can draw a circle representing a sphere and place this circle over the points on the fork and we will see that there are more electric field lines passing through the surface area. The are "less" electric field lines passing through the same surface on the other end of the fork.

Compact Disk in a Microwave

Again, sparks occur on the CD because the CD contains some metal. Like the steel wool and the fork, the sparks occur where the electric field is highest.

Light Bulb Inside a Microwave

When we apply an electric field around a light bulb, the light bulb lights up and then changes color. It changes color because plasma is create when the temperature reaches a high temperature.

Conclusion:  We find that Gauss's Law helps us visualize how electric fields behave and this is easy using simple surfaces. We went into the changes of area and volume if we change the radius value. We, also, took a look at the electric field inside a conductor and inside an insulator. Inside a conductor, the electric field is zero. Inside an insulator, the electric field varies with position inside the insulator. To finish things off, we take a look at what happens when we get an electric field build-up on a metal. Build-up, generally, leads to sparks.