Physics Book - User contributions [en]

Bar Magnet

2016-04-26T18:40:49Z

Mparker73: Typos

claimed by Samah

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

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== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
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[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

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Bohr Model

2016-04-26T18:01:56Z

Mparker73: Typos

by Pearl Ruparel

This page gives basic information about the Bohr Model and Quantization. It also includes examples using Bohr Model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons in orbit similar in structure to the solar system. In this model, the neutrons and protons occupy a dense central region (the nucleus), while the electrons orbit the nucleus, like the planets orbit the Sun. This is why the Bohr Model is commonly referred to as the "planetary model" [2].It is taught as an introduction to quantum physics. In the Bohr Model, electrons can only be at certain, different, distances from the proton to which it is bound. Energy is quantized as explained by the Bohr Model. This means that only orbits with certain radii are allowed, while orbits in between simply don't exist. These levels are knows an quantized energy levels and are labeled with integer N known as quantum number. The lowest energy state is generally termed the ground state. The states with successively more energy than the ground state are called the first excited state, the second excited state, and so on. As the electrons become further away from the nucleus, they become larger and have higher energy. Beyond an energy called the ionization potential the single electron of the hydrogen atom is no longer bound to the atom. The Bohr model works well for very simple atoms such as hydrogen (which has 1 electron) but not for more complex atoms. Although the Bohr model is still used today, especially in elementary textbooks, a more complex model known as the quantum mechanical model is the more accurate version of the Bohr Model and used universally.

===A Mathematical Model===
This model gives us the formula for the radius derived from translational angular momentum.

|Ltrans,nucleus| = Nh* where N = 1,2,3

Note: h* is used when is not the actual notation for it.

h* = h/2π = 1.05 * 10^-34 J*s

1)
|Ltrans,nucleus| = Nh*,rp = Nh*

2)
From curving motion:
<math>|Fperpendicular| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)
Substitute in Bohr's Condition:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)
Solve for the Radius
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math> where N = 1,2,3

Thus Bohr's Model derives equation for the radius.

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

===A Computational Model===

[[File: Screen_Shot_2015-12-02_at_8.27.42_PM.png |left|500x500px|thumb|Energy vs Time Graph, [3]]]

Here is a visualization of the Bohr Model, and its graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels.
Here is the link to visualization to try out different levels and see the energy graphs accordingly. In order to do this visualization, visit the link given below: http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels
[[File: Screen_Shot_2015-12-02_at_8.27.28_PM.png |center|300x300px|thumb|Energy vs Distance Graph, [4]]]

==Examples==

===Simple===
How much energy in electron volts is required to ionize a hydrogen atom, if initially the atom is in the state N = 3?
Here we can use the formula for the hydrogen atom which is

1)
<math>E = {\frac{-13.6 eV}{N^2}}</math>

2)
<math>={\frac{-13.6 eV}{3^2}}</math>
= -1.51 Joules

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?
[[Image:Screen Shot 2015-12-03 at 9.37.48 PM.png|right|250x250px, [6]]]

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

To the right is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|left|350x350px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

==History==

[[Image:bohr.jpg|right|100x100px, [5]]] [5] The first successful model of hydrogen was developed by Bohr in 1913, and incorporated the new ideas of quantum theory. Neils Bohr explained the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr’s model improved the classical atomic models of physicists J. J. Thomson and Ernest Rutherford by incorporating quantum theory. While working on his doctoral dissertation at Copenhagen University, Bohr studied physicist Max Planck’s quantum theory of radiation. Then after graduating, Bohr worked in England with Thomson and subsequently with Rutherford to come up with this model. During this time, Bohr developed his model of atomic structure. Initially, Rutherford's planetary model predicted a continuous spectrum of light from hydrogen. However, Bohr corrected for this by proposing that the translational angular momentum of the electron can be quantized. Although this model is not entirely correct, it has many features that are and is therefore used in physics.

== Shortcomings of the Bohr Model ==
The Bohr Model is an important predecessor to the current quantum mechanical models of the atom. However, there are some characteristics of the Bohr model that are not entirely correct. The actual quantization rules in a hydrogen atom are much more complex than those assumed in the Bohr Model. The translational angular momentum in ground state (N = 1), is zero, not h, and for the next higher state of N = 2, the z component of translational angular momentum can either be zero or h. Other issues with the Bohr Model include that it violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit. It also makes poor predictions regarding spectra of larger atoms, and does not predict the relative intensities of spectral lines.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]

[[Category:Energy]]

Magnetic Field of a Solenoid Using Ampere's Law

2016-04-26T16:53:40Z

Mparker73: Typos

---- CREATED BY JAKE WEBB ----
This page explains how to use Ampere's Law to solve for the magnetic field of a solenoid.

==The Main Idea==

A solenoid is a long coil of wire with a very small diameter, often used to make electromagnets due to their ability to create strong magnetic fields. The magnetic field can be easily calculated along the axis of the solenoid using Ampere's Law, and the magnitude and direction of the field is constant throughout the entirety of the solenoid, excluding the ends.

[[File:solenoidfrombook.png]]

As you can see in this model, at the ends of a solenoid the magnetic field begins to point outward at angles from the axis, with some of the field pointing directly perpendicular. In the middle, however, the field is constant throughout the entirety of the interior and is parallel to the axis.

===A Mathematical Model===

If there are <math>N</math> loops of wire that compose a solenoid of length <math>L</math>, and we know that Ampere's Law for Magnetism gives us the form:

<math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside path}}</math>

[[File:solenoidwithvars.png|frame|Solenoid with variables labeled]]

As stated above, the magnetic field inside a solenoid is constant and parallel to <math>dl⃗</math>, therefore Ampere's Law can be simplified to:

<math>BL=μ0NI</math>

By simply solving for <math>B</math> we can find the equation for the magnetic field of a solenoid:

<math>{B = \frac{μ_{0}NI}{L}}</math>

The direction can be found using the Right Hand Rule with your fingers curling around in the direction of the current and your thumb pointing in the direction of the magnetic field. It will always be along the axis for a solenoid.

[[File:Righthand.png]]

===A Computational Model===
This gif shows how the magnetic field forms in the solenoid when there is a current running through it.
[[File:solenoid-o.gif]]

==Examples==

===Simple===
Find the magnetic field produced by a 40 cm long solenoid if the number of loops is 400 and current passing through it is 5 A.
[[File:Examplesolenoid.jpg|700x900px]]

This example is very simple, as all of the variables of the equation are provided. All you need to do is plug them in and find the direction using the right hand rule.
===Middling===
A solenoid has a magnetic field of 5x10-4 T, 3000 turns, and is 30 cm long. What is the current through the solenoid?
[[File:example2solenoid.jpg|700x900px]]

This example is just a reverse solving of the equation derived above for the magnetic field of a solenoid. It is not munch more difficult than the previous example.
===Difficult===
A long, tightly wound solenoid has a circular cross section of radius 0.02 m. The solenoid is connected to a power supply (not shown), and this current changes with time, so the magnetic field inside the solenoid also changes with time. At time t = 0 s, the magnitude of the magnetic field inside the solenoid is 1.8 T. At time t = 0.3 s, the magnitude of the magnetic field inside the solenoid is 0.5 T. The direction of the magnetic field inside the solenoid is shown in the diagram. The solenoid passes through a coil of wire, which has 45 turns of radius 0.15 m. The coil is connected to a voltmeter (not shown).

[[File:hardexdigram.png]]

a.) At the same instant (t = 0.3 seconds), what will the absolute value of the reading on the
voltmeter be?
[[File:harda.jpg|700x900px]]

b.) The resistance of the coil is 2 ohms. At time t = 0.3 seconds, what is the conventional current
in the coil?
[[File:hardb.jpg|700x900px]]

c.) At time t = 0.3 seconds, what is the magnitude of the non-coulomb electric field ENC in
the coil?
[[File:hardc.jpg|700x900px]]

==Connectedness==
#Solenoids are important because they are the easiest way to create electromagnets and are an important piece in transformers.
#As a Computer Engineering major, solenoids are important, at least for the EE parts of my major, because as stated above they are used to create transformers which are a major piece in large and small scale power grids.

==See Also==
[[Ampere's Law]]

[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]

[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

[[Magnetic Field of a Toroid Using Ampere's Law]]

===Further Reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External Links===
*http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/electromagnetism/electromagnet.html

*http://www.physicstutorials.org/home/magnetism/magnetic-field-around-a-solenoid

*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

==References==
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 887-888.
Physics 2212 Spring 2016 Lab Quiz 8

[[Category:Maxwell's Equations]]