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Superposition principle

2016-04-17T06:42:16Z

Asundaresan6: /* Simple */

'''Claimed by Abhinav Sundaresan: March 15, 2016'''

This topic covers the Superposition Principle. This was the first and original article on this topic (also the best).

==The Main Idea==

The superposition principle essentially states that in a given closed system, the 'reaction' of an object to outside forces is the sum of the outside force acting on it. These forces can act along multiple dimensions, therefore meaning that vector notation will be needed.

===A Mathematical Model===

The net electric field due to two or more charges is the vector sum of each field due to each individual charge. This not only applies to Electric Fields, but Magnetic Fields as well. It is important to note that in the superposition principle, the electric field caused by a charge is not affected by the presence of other charges.

:<math>F(x_1+x_2)=F(x_1)+F(x_2) \,</math>

:<math>F(a x)=a F(x) \,</math> 

===A Computational Model===

Here is a python program that calculates and displays the electric as well as magnetic field at a specific observation location for a '''moving dipole'''. The dipole is a negative and positive charge ''q'' and distance ''s''. Write a porgram that will use the given constants to do this.

# GlowScript 2.0 VPython

magconstant = 1e-7
oofpez = 9e9
q=1.6e-19
s = 1e-9
pluscharge = sphere(pos=vector(-5*s, 0,-s/2), radius=1e-10, color=color.red)
minuscharge = sphere(pos=vector(-5*s, 0, s/2), radius=1e-10, color=color.blue)
velocity = vector(4e4,0,0) # The dipoles cm velocity # The dipoles cm velocity
robs = vector(0,s,0)

# Initializes two arrows (E and B) at the observation location
E = arrow(pos = robs, axis=vector(0,0,0), color = color.cyan)
B = arrow(pos = robs, axis=vector(0,0,0), color = color.magenta)

# Loop that updates dipole position as well as electric and magnetic field
dt = 1e-18

while pluscharge.pos.x < 10*s:

rate(100)

rplus = robs - pluscharge.pos
rplusmag = mag(rplus)
rplushat = norm(rplus)

Eplus = ((oofpez * q) / (rplusmag ** 2))
Bplus = magconstant * q * cross(velocity, rplushat) / rplusmag ** 2

rminus = robs - minuscharge.pos
rminusmag = mag(rminus)
rminushat = norm(rminus)

Eminus = (oofpez * (-q) / (rminus ** 2)) * rminus
Bminus = (magconstant * (-q) * cross(velocity, rminushat)) / (rminushat ** 2)

#Calculates the new E and B net fields
Enet = Eplus + Eminus
Bnet = Bplus + Bminus

# Updates the E and B fields arrows
E.axis = Enet
B.axis = Bnet
from visual import * #import the visual module

rod = cylinder(pos=(0,2,1), axis=(5,0,0), radius=1)

Notice that this uses the principle of superposition. The magnetic and electric fields of the two particles in the dipole are added. This determines the net field at the observation location and is therefore a perfect example of superposition.

==Examples==
<gallery widths="300px">
File:Superposition Principle.JPG| This picture shows the electric field at the location of q3. Note that the Electric Fields of both q1 and q2 were both calculated individually (but do not react because of one another) and summed up to get the net electric field
File:1111.PNG| Even by adding more source charges, the individual electric fields created by each source charge are unaffected by subsequent charges.
</gallery>

===Simple===

[[File:Electric_Forces_Fields_P1.JPG]]

Say that we have two positive particles at the above locations. Ignore the charges and distance provided for now. The electric field at the location of the negative charge is <math>\vec{E_1} = <2\hat{i}+0\hat{j}+0\hat{k}> </math>N/C and <math>\vec{E_2} = <0\hat{i}+2\hat{j}+0\hat{k}> </math>N/C. What is the net electric field at the location of the electron? Use the principle of superposition.

''Answer:''

Since we know the electric fields from both particles at the point, the superposition principle tells us that we must add the fields together to find the net field. Doing this gives us:
<math>\vec{E_1} + \vec{E_2} = <2\hat{i}+0\hat{j}+0\hat{k}> </math>N/C + <math><0\hat{i}+2\hat{j}+0\hat{k}> </math>N/C
<math> = <2\hat{i}+2\hat{j}+0\hat{k}> </math>

Therefore, the net electric field at the electron's location is <math><2\hat{i}+2\hat{j}+0\hat{k}> </math>N/C

=== Difficult ===

[[File:1234556.JPG]]

If <math>Q_1</math> and <math>Q_2</math> are positive and negative charges with charges of 6 nC and -5 nC respectively, what is the net electric field at point A located at (0,0,0)? Charge <math>Q_1</math> is located at (2,-2,0). Charge <math>Q_2</math> is located at (5,0,0).
To begin this problem, the first step is to find <math>r_1</math> <math>r_2</math>, the vectors from the charges to point A as well as their magnitudes:

<math>\vec{r_1} = 0\hat{i}+0\hat{j}-(2\hat{i}+-2\hat{j})\Rightarrow\vec{r_1} = -2\hat{i}+2\hat{j}</math>

<math>||\vec{r_1}|| = \sqrt{2^2 + 2^2} =\sqrt{8}</math>

<math>\vec{r_2} = 0\hat{i}+0\hat{j}-(5\hat{i}+0\hat{j})\Rightarrow\vec{r_2} = -2\hat{i}+0\hat{j}</math>

<math>||\vec{r_2}|| = \sqrt{-5^2 + 0^2} =\sqrt{25}</math>

Using Coloumb's Law, you get:

<math>\vec{E_1} = \frac{1}{4 \pi \epsilon_0}\frac{Q_1}{||r_1||^2}\hat{r_1}=\frac{1}{4 \pi \epsilon_0}\frac{e}{8}<\frac{-2}{\sqrt{8}}\hat{i}+\frac{-2}{\sqrt{8}}\hat{j}> = <-4.77\hat{i}+4.77\hat{j}> </math>N/C

<math>\vec{E_2} = \frac{1}{4 \pi \epsilon_0}\frac{Q_2}{||r_2||^2}\hat{r_2}=\frac{1}{4 \pi \epsilon_0}\frac{e}{5}<\frac{-5}{\sqrt{25}}\hat{i}+\frac{0}{\sqrt{25}}\hat{j}> = <1.8\hat{i}+0\hat{j}> </math>N/C

Next, you need to add the two electric fields together. Because of the superposition principle, the electric field caused by Q1, does not effect the electric field created by Q2, but both can be summed together to create the net electric field at point A.

<math>\vec{E}=\vec{E_1}+\vec{E_2} = <-2.97\hat{i}+0\hat{j}></math>

==Connectedness==
How is this topic connected to something that you are interested in?
* The Superposition Principle is important because it makes your life easier in Physics. You can make assumptions based on the fact that the net field at any location is equal to the sum of all the invidiual fields. You don't know one of the individual fields, but know the net field? Simple subtraction can help you calculate each individual field. Imagine if all Electric Fields influenced each other? It'd be really difficult to calculate net electric fields without a complicated equation that takes into account one field as a function of another. yuck. Thank the Physics Gods for the Superposition Principle.

How is it connected to your major?
* Since the Superposition Principle can be applied to circuits, and since Biomedical Engineering sometimes create medical device that require circuity, knowing how to create a circuit and calculate the electric field with the superposition principle is an important tool to have.

Is there an interesting industrial application?
* In [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1479978 this] study, the Superposition Principle was used to analyze Solar Cells.

==History==
Daniel Bernouilli, in 1753, first proposed the idea of the Superposition Principle. He stated that "The general motion of a vibrating system is given by a superposition of its proper vibrations." His claim was rejected by mathematicians Leonhard Euler, and Joseph Lagrange. However they were eventually proved wrong by Joseph Fourier and it is now the concept you see today.

== See also ==

*[[Coulomb's Law]]
*[[Electric Field]]
*[[Electric Dipole]]
*[[Electric Force]]
*[[Magnetic Force]]
*[[Magnetic Field]]

===Further reading===
*[https://www.boundless.com/physics/textbooks/boundless-physics-textbook/electric-charge-and-field-17/coulomb-s-law-135/superposition-of-forces-483-853/ Additional Textbook Explanation]

*[http://www.sparknotes.com/testprep/books/sat2/physics/chapter17section4.rhtml Sparknotes Explanation]

*[http://www.physicsbook.gatech.edu/Superposition_Principle This guy who made a Superposition principle on this wiki even though I already made one]

===External links===

[https://www.youtube.com/watch?v=S1TXN1M9t18 Instructional video on how to calculate the net electric field using the superposition principle]

==References==

Chabay, Ruth W.; Sherwood, Bruce A. (2014-12-23). Matter and Interactions, 4th Edition: 1-2 (Page 522). Wiley. Kindle Edition.

[[Fields]]

2015-12-05T05:05:54Z

Asundaresan6: /* References */

Created by Abhinav Sundaresan

[[File:FOR.jpg|250px|thumb|right|Frame of Reference]]

The frame of reference is a tool that can be used to change the way in which a system is viewed, thus adding a layer of complexity but in some cases simplifying the problem.

==The Main Idea==

A frame of reference is the point in the way or perspective from which you view the events that take place within a system. In this class we use inertial frames of reference (frames where the velocity is constant). It is easy to think about if you imagine a person and what they would see if they were standing where you want a frame of reference to be. Consider the below example where there are two frames of reference: The one that observes the whole train and the boy and the one from the point of view of the boy on the train.

[[File:Train FOR.png]]

To the first observer (and his frame of reference), the boy is moving to the right along with the train. To the boy on the train (and his different frame of reference), he is standing still on the train. Here is a basic example of how changing the frame of reference can be a very powerful tool.

===A Mathematical Model===

In order to calculate what the velocity is, we will use the following equation so that the frame of reference will not change the overall value of the velocity:

<math>\vec{v} = \vec{v}' + \vec{v}_{frame}</math>

where <math>\vec{v}</math> is the velocity in a stationary frame, <math>\vec{v}_{frame}</math> is the velocity of the moving reference frame, and <math>\vec{v}'</math> is the velocity observed in the moving frame.

===A Computational Model===

There are some great resources that can help you become more acquainted with the topic of frames of reference.

[[File:SimulationFOR.png]]

The following website allows you to change initial conditions and frames of reference to see exactly how your decisions will change what is seen:
http://www.wiley.com/college/halliday/0470469080/simulations/sim45/sim45.html

Also, here are two YouTube videos that do a great job of breaking down frames of reference so that they can be easily understood:

https://www.youtube.com/watch?v=mYH_nODWkqk

https://www.youtube.com/watch?v=jYMU6bn5GHY

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===

Say that you are pulled over on the side of the road and a car passes by you at a speed of 50 m/s. You then start to drive at a speed of 43 m/s and you see the car that just drove by in from of you. What is the speed of the car from your moving frame of reference?

As we know,

<math>\vec{v} = \vec{v}' + \vec{v}_{frame}</math>

In this scenario, <math>\vec{v} = 50 m/s</math> since that is the velocity from our initial stationary reference frame. <math>\vec{v}_{frame} = 43 m/s</math> because that is now the velocity of our frame of reference, meaning that <math>\vec{v}'</math> is what we are solving for.

Plugging in what we know, <math>50 m/s = 43 m/s + \vec{v}'</math>

This means that <math>\vec{v}' = 7 m/s</math>

===Middling===
===Difficult===

==Connectedness==
# I have always been interested in how we perceive the world around us and take measurements. It had never occurred to me that something as simple as speed could change based on the observer and so researching and understanding this topic proved to be very fun and interesting.
# This topic has nothing to do with my current major of Computer Science.
# Analyzing frames of reference can provide many key insights in industrial settings. Whether it be looking at colliding particles in the LHC or why the speed of light is the absolute speed limit of the universe, looking at these things can help scientists understand the world around us and hopefully make great inventions based of these results.

==History==

[[File:SirIsaacNewton.jpg ‎|150px|thumb|right|Isaac Newton]]

Although it is not known who came up with this concept of reference frames, it has been key to interpreting and understanding the world around us. Newton assumed a Euclidean space when looking at frames of reference which he used when he came up with important ideas and theories such as gravity. Einstein came up with transformations between reference frames to come up with his special theory of relativity. These are just some examples of how frames of reference have been used through the history of science and man.

== See also ==

Relevant Materials

===Further reading===

Matter and Interactions Fourth Edition By Ruth W. Chabay, Bruce A. Sherwood - Chapter 10 Section 5

===External links===

:'''1:''' https://www.youtube.com/watch?v=mYH_nODWkqk
:'''2:''' https://www.youtube.com/watch?v=jYMU6bn5GHY
:'''3:''' http://www.wiley.com/college/halliday/0470469080/simulations/sim45/sim45.html
:'''4:''' https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-first-law-of-motion

==References==

:'''1:''' Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

:'''2:''' "Frames of Reference." Isaac Physics. N.p., n.d. Web. <https://isaacphysics.org/concepts/cp_frame_reference>.

:'''3:''' "Reference Frame." Everthing Maths & Science. N.p., n.d. Web. <http://www.everythingmaths.co.za/science/grade-10/21-motion-in-one-dimension/21-motion-in-one-dimension-02.cnxmlplus>.

:'''4:''' Wikipedia. Wikimedia Foundation, n.d. Web. <https://en.wikipedia.org/wiki/Frame_of_reference#Remarks>.

:'''5:''' Wikipedia. Wikimedia Foundation, n.d. Web. <https://en.wikipedia.org/wiki/Inertial_frame_of_reference>.

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