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.
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.
No comments:
Post a Comment