Flux: The Electric Field through an imaginary surface

Today we started our study of flux by first examining the behavior of electric fields. Below is a picture of an electric field that goes straight up and the trajectory of an electron that passes through it.

Then we defined inertia. For charges, inertia is the resistance to move due to electric fields and it's equation is defined below.

Then the we calculated the torque due to an electric field on a dipole. Below is a diagram of the problem we are solving. The dipole consists of a positive charge and a negative charge bonded together. It is at an angle compared to the direction of the electric field.

The torque is the effective force at a distance from the center of mass or rotation of an object. In this case, this equals the force times sine of the angle and the distance from the center of rotation. Also, there is a force for every charge. This expression can be written as the cross product of the distance and the force. This can also be written as the cross product of the dipole and the electric field.

Then we calculated the work done by the torque. This resulted in the expression shown below.

When this is applied to the change in orientation of the dipole, the initial angle is unknown but the final angle is 90. This yields the expression in the right which can be written in vector notation as a dot product and is equal to the potential.

Then we did an experiment involving the apparatus below. The cage is a conductor that is going to be on top and there are a few strips of cloth that are attached to aluminum.

I predicted that the inflow of electrons on the cage would create an electric field that would direct the aluminum that is outside the cage outwards and the aluminum that is inside the cage inwards. Nevertheless, the experiment showed that the aluminum that was hanging inside the cage did not move. This is due to the electric field being zero inside the cage. Also, there is an effect called shielding. 

Below is a picture of what it looked like when the aluminum was moving and the apparatus was turned on.

Then we did a computer exercise where we answered questions about a fictitious electric field due to a charge. With it we understood a technique that allows us to visualize the electric field  to be shown as lines coming outwards or inwards. The number of lines represents the magnitude that an electric field may have due to the charge that causes it. 

Below is a visualization of the electric field between a positive and a negative electric charge. The electric field points towards the negative charge form the positive charge. The number of lines indicates the magnitude of the electric field lines.

Then Prof. Mason did a demonstration on the concept of flux. In the pictures below, he is holding a tablet with lots of sticks pointing outwards. These represent electric field lines. He also holding a rectangle that represents the surface through which electric field lines go. In the picture below, the surface is perpendicular to the lines. This allows for the maximum number of electric field lines to cross the surface.

In the picture below, the surface is at an angle with respect to the lines. This permits fewer lines to cross the surface and illustrates the concept that the surface and the electric field lines need to be perpendicular to each other.

Below Prof. Mason is showing another angle that doesn't allow as many electric field lines in as the perpendicular position.

Below Prof. Mason is demonstrating that when the surface is parallel to the electric field lines, no electric field lines cross the surface.

These examples show that the orientation of the surface and the electric field lines yield a maximum value when the surface is perpendicular to the electric field lines. Also, it shows that no electric field lines cross the surface when they are parallel to each other. Since the surface is going to be defined as a vector, this vector is going to be perpendicular to the surface itself. This means that, when this vector is parallel to the electric field lines, the flux is maximum. Also, when this vector is perpendicular to the electric field, it is zero. This is consistent with the behavior of the cosine function which is included in the formula for electric flux.

The electric flux is defined as the dot product between the electric field and the surface area that it crosses. This is equal to the product of their magnitudes and the cosine of the angle.

Then we answered questions from a computer assignment that asked us to look at graphs of electric fields and surfaces and to derive conclusions about flux. 

Then we did a problem where we predicted the flux through the surfaces of a cube as in the picture below. The conclusion was that the faces that were perpendicular to the electric field in any way had zero flux and the faces facing the electric field had flux.

These faces were labeled 3 and 4. 

Finally we attended a meeting about Raman Spectroscopy. In here, we learned that Raman spectroscopy picks up on the signals given off by compounds when they are hit with photons and spread the energy around. This is a different type of spectroscopy from Raleigh Spectroscopy. The Raman spectrum of a compound seems to be a signature mark and it seems possible that it can be used to identify compounds. 

Summary

In order to understand flux, we started by studying the behavior of electric fields and charges on them. One of the effects of electric fields is that they can change the path of charged particles. Also, they can rotate dipoles. And, the electric field changes with the presence of another charged particles. These provide a trajectory to charged molecules. We also saw experiments were we saw the effect of electric fields on aluminum. Then we studied the concept of flux. It was defined as the dot product of the electric field and the surface that it crosses. Prof. Mason used a demonstration to show how flux changes with orientation and we did exercises to see how other factors change such as the size and shape of the surface. Finally, we learned about Raman spectroscopy. This taught a little of quantum theory and the scattering of photons in compounds. 

Studying Electric Fields

The electric field is a definition of the attraction that a charge feels when another charge is present. It has units of N/C and it is a vector. Also, it is not real. In fact, it is a concept that allows the prediction of how a charged object will react when another sufficiently large charge is sufficiently close by. But, how does this charge know that there is another charge? The truth is it is not known. There are theories that have been developed but have not been proven. Nevertheless, it is very useful. In order to understand it, we compare it to the concept of a gravitational field. 

Below is a picture of 4 statements that originally described the behavior of mass in a gravitational field. In this case, we re-wrote these statements to reflect how electromagnetic fields and charges are defined and work. 

Basically they put the equation for calculating electric fields due to a charge at a certain distance in words. They say that, when a charge is present, an electromagnetic field is also present. Quantitatively, the value of the electric field is dependent on the value of the charge and the distance between the charge that makes the electric field and the charge that is being influenced. A charge that is present within an electrical field will experience a force.

Then, we answered some questions that made us reason with the concept of electric fields and study them. In a computer simulation, there was an electric field and a charge. We moved the charge around and recorded our observations under Question 1. These basically restated the formula for calculating electric fields.

We also practiced with a problem. Given the value of the force a charge feels within an electric field, we found the value of the electric field at that point. Then, we learned to interpret a way of graphing electric fields. This method uses lines coming from the cause of the electric field and concentric circles to represent the strength of the electric field and the points at which the values of the electric field are the same.

Next we described that the magnitudes of the electric field at different distances changes with uniformity and not suddenly.

After answering these questions, Prof. Mason showed us a nice way to visualize the interaction between a charge and an electromagnetic field. In the video he explained that a positive charge will go towards a negative charge in a similar fashion as a ball rolling down the hollow, white surface. If we look closely, we notice that the ball seems to go faster as it approaches the center. Also, if the ball is heavier, it will roll faster as well. This is very similar to how charges act when they are in the presence of an electric field. One of the differences is that charge replaces mass. Also, the charge will approach the center in a parabolic fashion rather than circular. 

Then, Prof. Mason pulled another prop that helped visualize what happens when there are a positive charge and a negative charge that is consistent with the convention of electric fields that positively charged particles will go away from other positively charged particles and towards negatively charged ones. These paths are represented by the electric field. The ball in the photo went from the top of a summit to the bottom of a valley. A positively charged particle will move similarly when it is in the presence of a positive charge and a negative charge. The positive charge is represented by the summit and the negative charge is represented by the valley. These props also helped us understand that when we talk about electric fields and moving particles, they are usually drawn in 2D. Nevertheless, it is something that actually happens in 3D. 

Then we solved a problem involving two charges where we calculated the electric field at a point in 2D.

The charges had the same magnitude but one was positive (right) and the other negative (left). First we calculated the electric field caused by the positive charge in terms of vectors and then we calculated the electric field of the negative charge, also in terms of vectors. The vertical components ended up canceling, and, since the electric fields pointed in the same direction (positive), they added up and yielded a total electric field in the horizontal direction.

Then, we tried something different. 

We calculated the electric field caused by a charge of 2.00 nC at different distances from 0.005 m to 0.1 m in increments of 0.005 m, in excel. This gave us the above table with values for the electric fields at different distances. Then, we used this method and values to calculate the electric field caused by two charges in 2D. The problem is stated in the picture below. 

To solve this problem, we calculated the electric field caused by the charges at different distances. This time the closest distance is 0.005 m and the farthest distance is 0.1 m and the electric field was calculated at every 0.005 m increment. This yielded the table below  which turned out to be exactly the same as the previous one. The reason for this is that both of the charges in the problem are equal in magnitude. Also, we can gather the values calculated from the table to learn the electric field at any point as long as we take in count the direction of the values. For this we drew the diagram below taking in count that the electric field arrows go from positive to negative.

The electric field at points #1, #2, #3, and #4 ended up getting calculated by simply adding up the appropriate values from the table. The electric fields are 50,000 N/C, 22,500 N/C, 22,500 N/C and -43,200 N/C, respectively.

Then, we tried to use the same method to solve another problem.

In this case, we can calculate the electric field caused by a charge at different distances and add them up to find the net electric field at any point. The table looked like the table below.

We were successful but Prof. Mason showed us something very interesting. As we increased the number of partitions and decreased the length of individual pieces (increments), our answer changed a little bit but approximated the value 6 * 10^4 N/C. Therefore, we decided to find a way to calculate the value that was the limit that was reached as the number of partitions increased and the length of individual pieces decreased. The technique involves using differentials as our smallest pieces and added them up appropriately. Therefore, we had to use integration. For this, we developed the concept that there was a very small charge dq that had an electric field dE that belong to the bar. Then, we add up all the dE to find the net electric field. Our work is depicted in the board below.

Then we tested our formula and came up with the same number that was the limit that was approached in the earlier calculations. This gave integration techniques a large importance because they can give accurate values rather than approximate ones that could change.

Finally, we decided to play a game to develop intuition about how charges move in electric fields and have some fun. There were three levels with increased difficulty and we cleared all of them. They were challenging but fun.

This is the first level. In here we ended up using 3 positive charges and 2 negative charges.

This is the second level. In here we used 2 positive charges and 1 negative charge.

This is the third and highest level. In here we used 1 positive charge and 6 negative charges.

Summary

Today we had a lot of fun while learning about electric fields and learning to work with them to find interesting theories about how the universe seems to work. First we developed different ways to state how charges and electric fields interact. We put them in words that were previously used to describe how matter and gravitational fields work and realized that charges and electric fields seem to work similarly. Then we did a few exercises that got us thinking about the behavior of forces caused by electric fields. Next Prof. Mason showed us a demo that taught us that electromagnetic fields aren't 2-dimensional but 3-dimensional. This was very useful and fun for visual thinkers. Then, we dived into calculating electric fields at different distances and angles from the location of the charges. Next, we calculated the electric fields caused not by point charges but by, distributed charges on solids and surfaces. We worked first with tables and learned that individual values can be added up to find our net electric  field. Then we found out that, when dealing with charges spread over surfaces, the number of partitions that is used matters and the highest number of partitions yields the most accurate calculation. Therefore, we used differentials and integrated them. Finally we played some games of electric field hockey that would help us see how certain details affect the movements of charges. These details include location, magnitude and that they affect every particle around them in a radial way.

Introduction to Electromagnetism

Today we investigated electrostatic forces.

We started our journey by predicting what would happen to balloons if we rubbed them with animal fur. My group predicted that this would somehow cause a difference in the charges present in the balloon. And ,as a result it would be electromagnetic. Glass is a material that lets its electrons move very easily. So, when the balloon is left against the glass, it would stick.

We were a little bit nervous as Prof. Mason demonstrated what would happen, but it turned out we were right.

 So, we drew the free-body diagram of the balloon on the side with red ink. The balloon has mass and is subject to Earth's gravitational field, so there is a force pointing down trying to get the balloon to move downwards. Nevertheless, it doesn't, so this means that there is an equal and opposite force present as well. This one is the friction force which is the result of the balloon being pressed against the glass. The balloon is pressed against the glass when there is a force in the direction of it and this is the electromagnetic force due to the charges present on the balloon and the glass.

Then, instead of rubbing the the balloon with fur, it was rubbed with silk, and the question was to think about what would happen now. But the effect is similar to what happened earlier. Basically, there is still a difference in charges present in the balloon but this time it's the opposite way. Before, the side that was rubbed gained a positive charge and the glass gained a negative charge. This time the balloon's side gained negative charge, and the glass reacted by having its surface change to positive charge. The result is the same and the balloon stays on the glass without falling.

Then, we tried to define mass and charge. My group defined mass as the amount of stuff that attracts other stuff via gravity and charge as the amount of stuff that attracts other stuff via electricity. Their equations look very similar, and they are both forces at a distance.

Then we did an experiment with electrostatic forces. We cut two strips of Scotch tape that were 10 cm long and taped them to the table. At the same time we peeled them off and brought them close together with their non-sticky sides facing each other.

We observed that they seemed to repelled each other more and more as we brought them close together. Then we labeled these strips 'bottom' and got a couple more that we labeled 'top.' We taped the bottom ones to the table and then taped the top ones to these. Then, one of us grabbed one of the couples that were taped together and the other one grabbed the other couple that were taped together and simultaneously peeled them off from the table. Then we peeled them off from each other. When we brought the two top strips together, we saw they repelled each other. The same happened with the two bottom strips. Nevertheless, when we brought a top strip and a bottom strip together, they attracted each other as they got closer. In the picture below, we summarized our results.

This experiment is consistent with the hypothesis that there are two types of charge.

Next we decided to quantify this new force law with a pendulum. But first we wanted to calculate the angle the it would move as a functions of known quantities. So, we used the length of the pendulum and the horizontal distance that it moves. This yields the formula shown below.

Then we drew a free-body diagram wrote down the sum of forces in the horizontal and vertical directions to find an equation for the new force which is the one that keeps the body at an angle. This formula uses measurable and known quantities. 

Next we drew what we think is the relationship between this new force and the distance at which interactions happened. We used the graph of the gravitational force vs r as a model, but it works pretty well for the interactions between this new force and r. If we were to write this in equation in the form of F=A*r^B, B would have a value of -2. And, we would solve for the the value of A in the equation. 

Next we used video analysis with Logger Pro to get our data which looked like the picture in the below.

Next we calculated the new force using the formula we derived above and the data we collected from the video analysis. The data we used is mass, acceleration of gravity, the horizontal displacement and the length of the pendulum. Then we plotted the values of this force vs horizontal displacement and matched a couple of best fit equations. First we tried to fit the data with an equation that looked like F=Ar^-1, then we tried to fit the data with an equation that looked like F=Ar^-2, and we tried to fit the data with an equation that looked like F=Ar^B. 

The equation that had r squared inverse had a pretty good RMSE value. The value for B was 1.998 giving a percent difference of 99.9 %. Hence, it seems like having B=-2 is a simple way that works for measuring our values. If the charges would've been the same, then they would've been 0.102 nC. If the charge on the hanging ball is half the charge of the other ball, then the charge of  the hanging ball would've been 0.144 nC and the other would've been 0.722 nC. Nevertheless, it is not possible to tell which is the sign of the charges with the data we have. We can only tell that both are positive or negative, but we can't tell which because we have nothing to test it or compare it to. Also, some of the measurements may be a little off and that is because it when we were marking the points, the marks weren't perfect. This could explain why not all the points fit the curve perfectly or why the points may not be in full agreement with Coulomb's Law. 

Next we starting 'reading' the equation and getting used to interpreting it with some exercises. These were focused on the placement and direction of the unit vector r hat if the forces were repulsive or attractive. If we are talking about the distance between the atoms starting on q2 and ending on q1, we write r hat 12 meaning on 1 from 2. 

In the Coulomb equation, the magnitudes of the charges are proportional to the force. And, the magnitude of the force decreases as the distance increases. The reason is that their relationship is inverse and squared. Coulomb's Law is consistent with Newton's Third Law because it shows that there are two forces with the same magnitude but opposite direction. 

Next, we solved a problem involving Coulomb's Law. 

We found the force between two point charges, 2.0 *10^-9 C and -3.0*10^-9 C, at a distance of 2 cm. We got that to be -1.35*10^-9 N i^. 

Then, calculated the components of a vector in 2D. Since r12 hat is a unit vector, its components are simply the cosine and sine of the angle theta.

This formula is going to help us solve the electromagnetic force problems in 2D like the next one.

In here, we have the same conditions as the previous problem. The previous problem is written to the left and the new problem is solved to the right. The difference is in that q2 is now at (5,6) where the numbers mean distances in centimeters from the origin. Hence, we had to find the sum of the horizontal and vertical forces and then we wrote our answer in vector form. 

Finally, we called it a day by observing some wonderful machines transform mechanical work into electricity. 

Demonstrations of the Van de Graff Generator directing electrons to objects and making them move

Demonstration of the Storm Ball

Summary

Today we saw a lot cool things. We started by observing and studying what happens when we rub a balloon and let it get stuck against a glass by itself. And, we found out that the force keeping it there is the electromagnetic force which is a new kind that we haven't studied yet. Then, we defined charge and mass as the quantities that generate attraction through electricity and gravity, respectively. And, we did an experiment to test our hypothesis that there are only two charges and to see how they interact with each other and other materials. Then, we prepared ourselves to do another experiment by finding how to calculate an angle through distances we can measure. This would allow us to calculate the force that moves a point charge on a swing. Also, it allowed us to find relationships between the electromagnetic force and distance between the charges by applying this formula to a range of data points. And, we played a little with it by calculating the charges on a couple of possible scenarios: one where the charges are the same and the other where one of the charges is half the other. Next we studied Coulomb's Law itself a little bit by looking closer at what it stated about distances, charges, forces and their directions (if they had any). Later, we solved a couple of problems with Coulomb's Law: one with charges on 1D and another one on 2D. Finally, we observed a Van de Graff Generator direct charges on materials and saw them move and expand as they gained charges. We also observed the Storm Ball direct electrons towards the glass and pass charges across the students.