Gauss' Law Activities (3/31/2015)

Gauss' Law in Flatland

Two opposite charges were placed with 

low magnitude in order to observe how 

the electric field lines of the charges interacted.

Then, 3 enclosed surfaces were drawn 

(in dotted red/blue, and solid black lines)

 in order to observe and make note of various 

properties affecting the surfaces, such as the

lines of flux in/out and overall charges.

These equations depict what was learned in the 

previous experiment, that the charge enclosed 

by a surface is proportional to the flux of that surface. 

Therefore, flux is proportional to charge.

Gauss' Law Demo

A Van de Graff generator attached to a Faraday cage. The cage has three aluminum foil chunks hanging inside, and three hanging outside.

We were asked what would happen to the 6 foil pieces once the generator was turned on, and a current was running through the cage. Our table guessed incorrectly, because the proper answer was B, that only the outer foils will move away from the cylinder while the inner foils did not move. The reason for this is because any excess charge is outside the cage, so nothing happens inside because the net charge is zero.

Various Gauss' Law Activities

A sketch depicting where excess charge is located on a sphere, which happens to be at the very outermost points, evenly distributed.

When asked what one should do when in a car, which is then struck by lightning, our table concluded that you should stay in your car (which is the correct answer). The car served as the conductor, and every point inside a conductor has an electric field of zero.

Above are answers to various questions involving the review of geometry. The point of this exercise was to analyze how to properly use various geometrical shapes when conducting Gauss' Law calculations.

The calculation showing how Gauss' Law can be used to find the electric field magnitude at any distance, r, from a point charge +q.

Gauss' Law and Spherically Symmetric Charge Distribution

For the calculations pictured above, we were asked to compute the magnitude of the electric field at a distance r from the center of a uniformly charged sphere of radius R wit a total charge of Q throughout its volume, where r<R. This was done using ideas from the previous activity, as you can see how from how the E=kq/r^2 equation was derived into E=kQr/R^3.

Microwave Fun

Shown above is how steel wool reacts in a microwave. What was witnessed is that the electricity discharged at the points or tips of the wool. Considering steel wool has points everywhere, it makes sense that the discharge is seemingly wild.

Next up is a metal fork. Having assessed that an electrical current prefers to discharge at narrow tips, it was expected that the prong of the fork is where sparks would be seen. The result aligned with our prediction.

The third item to be microwaved was a regular CD. In this example, the current actually discharged itself long the surface of the CD. In the last picture you can see the direction the current traveled by the dark imprint left on the CD.

The final item to be placed in the microwave was a light bulb. The result was that the current traveled to the tungsten filament located inside the bulb and lit it up.

Geometry of Cylinders

The objective here was to understand what happens to the electric field when there is cylindrical symmetry. What we realized is that if a charge Q spread uniformly throughout the volume of a cylinder of radius r and length L, the fraction of the charge lying within a radius r/2 is 1/4, not 1/2. The reason for this is that r is squared in this volume equation, so if you divide r by 2, it actually becomes divided by 4.

Gauss' Law and Cylindrical Symmetry

In the pictures above we calculated the electric field at a distance r from a very long, straight, uniformly charged cylinder.

Gauss' Law for Electrical and Gravitational Forces

The gravitational version of Gauss' law.The electric field vector E is converted to Y as the gravitational field vector. When it comes to electrical calculations and forces,  mass, m, is synonymous with charge, q. The e0 value should be denoted as k.

Shown above is the calculation for the magnitude of the gravitational field, Y, at a height h above the surface of Earth. The Earth is treated as a sphere when calculating its area, G=6.67e-11, r=6371km, and m=6e24kg. The concluding equation of Y= G*m/r^2 was computed and the result was a value of 9.865 m/s^2, which is strikingly similar to the gravitational acceleration value of 9.81m/s^2. Our realization is that acceleration due to gravity has been calculated based on the magnitude of Earth's gravitational field and how quickly it draws objects toward itself.

Summary

In summation, we looked at Gauss' Law and how it can be used in analyzing both electrical and gravitational fields, as well as how it is applied to objects of various shape and size. We applied this to real world situations by calculating the magnitude of Earth's gravitational field, and realizing that it is the same value as gravitational acceleration. We also analyzed the geometry of various shapes in order to fully understand how to use them in future equations. As an added bonus, several objects were placed in a microwave so that we could visually see the preferred paths of electrical discharges.