Magnetic torque is induced when a magnetic field causes a current carrying coil of wire to twist.
[[File:torqueexample.png|thumb|Example of Magnetic Torque]]
==The Main Idea==

The idea behind this concept is that as current flows through a wire, a magnetic field is produced. This magnetic field causes a force to act upon the wire causing it to twist. An example of this phenomenon is the movement of a compass needle by the Earth's magnetic field. Another example is a hanging coil that twists in the direction of the magnetic field of a bar magnet.

The magnetic torque acts on the dipole, and it is highly dependent on the magnetic moment and external magnetic field.

Several factors besides the magnetic moment and external magnetic field can affect the magnetic torque. In a loop or other three dimensional object the orientation of the object relative to the magnetic field highly affects the torque.

Through the following general example you can see how this phenomena occurs:
[[File:Magnetic_Torque_Main_Idea_1.png]]

On the sides h, the magnetic force is horizontal pointing outwards causing the loop to stretch; while on the sides of length w the magnetic forces are horizontal and tend to make the loop twist on the axle. This causes the loop to rotate counterclockwise. When the plate of the loop is perpendicular to the magnetic field don't exert any twist.

There are two configurations: Stable and Unstable

[[File:Torque stable unstable.jpeg]]

In the stable configuration, magnetic forces will twist the loop back up to the horizontal plane. In the unstable configuration, small displacement away from the horizontal leads to magnetic forces that rotate it even farther out of the plane.

This relationship can be seen in this video:
[https://www.youtube.com/watch?v=E-3yQqgu8OA]

Here is a video on Asymmetric Magnet Torque
[http://www.youtube.com/watch?v=LD6TX5IH5po Asymmetric Magnet Torque]

===A Mathematical Model===
The overarching equation that encapsulates this physical phenomena is as follows:
: <math> \boldsymbol{\tau} = \boldsymbol{\mu} \times\mathbf{B}</math>

where:

'''τ''' is the variable describing torque

'''μ''' is the magnetic dipole and can be found using many expressions including that of a wire which relates magnetic dipole to the current in the wire multiplied by its cross sectional area. For a magnet, this quanity is not easily derived, and is a little outside the scope of this discussion. This quanitity is usually given in the problem statement. However, for a video that helps describe the magnetic dipole moment of a magnet: [https://www.youtube.com/watch?v=lOSmfcS1Vrg]

'''B''' is the magnetic field

The torque provided by each of the magnetic forces around the axle is equal to the distance from the axle times the component of the force perpendicular to the lever. Twist applied is due to the w - sides of the loop where torque acts out of the page. This causes a clockwise twist.

Fperpendicular = IwBsin(x) where the arm is equal to h/2
each side exerts a force of 2(IwBsin(x))(h/2)

'''τ''' = IwB(sinx) and '''µ''' = Iwh

'''µ''' x '''B''' = µBsin(x) = '''τ'''

The right hand rule for the direction of torque is as follows: the fingers of your right hand curl in the direction the loop will rotate, and your thumb will point the the direction of torque. The direction of the torque vector will be along the axle around which the loop rotates.

===Magnetic Dipole Moment===

The magnetic dipole moment of a current carrying loop of wire, '''µ''', is defined as a vector pointing in the direction of the magnetic field that the loop makes along its axis given by the right hand rule.

µ = IA = Iwh

The coil tends to twist in a direction to make '''µ''' line up with '''B'''.

[[File:Magnetic dipole moment.jpeg]]

===Units Discussion===
'''τ''' has units of N*m

'''µ''' has units of A*m^^2

'''B''' has units of tesla

From this, it must be that one N*m(which interestingly defines work) is equal to one tesla * A*m^^2. From a discussion of units alone, it is important to think about what sorts of questions the professor might ask, meaning questions could include an analyses of the work that must be added to a system to keep it stationary for example.

===A Computational Model===

Click here to view the PHET Interactive Model created by the University of Colorado

[https://phet.colorado.edu/en/simulation/legacy/magnet-and-compass PHET Interactive Magnet and Compass Model]

==Examples==
Essentially, there are only a few categories of questions that can be asked relating to magnetic torque. These questions include a simple computation of magnetic torque given the dipole moment of a magnet, and the magnetic field being applied to the observation location. In this situation, you can either utilize a simple cross product, as in the equation listed above, or if the values are given as scalars, and it is known that they are perpendicular to each other in direction, you can utilize the equation: '''|τ| = µ*Bcos|90|= µ*B'''. This is the essential question involving the equation listed above for magnetic torque. However, the professor can also ask questions relating to material learned from physics 1 involving angular frequencies and other products of angular momentum. The relationship is defined in the following illustration

[[File:IMG_1362.jpg]]

[https://www.youtube.com/watch?v=xER1_SYql44 Torque on Current Carrying Loop]

===Simple===

A bar magnet whose magnetic dipole moment is <3, 0, 1.8> A · m2 is suspended from a thread in a region where external coils apply a magnetic field of <0.6, 0, 0> T. What is the vector torque that acts on the bar magnet?

[[File:SimpleWikiProb.JPG]]

===Middling===
A bar magnet whose magnetic dipole moment is 14 A · m2 is aligned with an applied magnetic field of 5.4 T. How much work must you do to rotate the bar magnet 180° to point in the direction opposite to the magnetic field?

[[File:MiddleWikiProb.JPG]]

===Difficult===

A cylindrical bar magnet whose mass is 0.09 kg, diameter is 1 cm, length is 3 cm, and whose magnetic dipole moment is <4, 0, 0> A · m2
is suspended on a low-friction pivot in a region where external coils apply a magnetic field of <2.0, 0, 0> T. You rotate the bar magnet slightly in the horizontal plane and release it. (For small angles in radians, assume sin(θ) ≈ θ.)

(a) What is the angular frequency of the oscillating magnet?

(b) What would be the angular frequency if the applied magnetic field were <4.0, 0, 0> T?

[[File:DifficultWikiProb.JPG]]

A detailed description and symbolic representation of magnetic torque can be seen here:
[https://www.youtube.com/watch?v=K1FEepXKETM Magnetic Torque and Magnetic Dipole Moment]

==Connectedness==

[[File:Compass.jpg|thumb|A standard compass http://helenotway.edublogs.org/2011/01/02/different-compass-point-same-ultimate-direction/]]

Utilizing a compass is a basic survival need and it just so happens to depend on the torque produced by the Earth's magnetic field. As a Biology major, field work is a large part of what I do, especially studying ecological systems and different habitats. In order to navigate in unfamiliar locations, such as deserts and dense tropical forests, scientists rely heavily on basic survival skills and this includes the use of compasses and maps. Physics, biology, and chemistry make up part of the science family and each heavily depends on the other, this is why it is important to study each one to bridge the relationship.

First paragraph of "Connectedness" written by Demetria Hubbard 2015

The Earth has a complex magnetic field and magnetic dipole moment that creates a magnetic torque. The necessity of all three of these magnetic properties is rarely known; however, all three are essential for life on earth. Earth's magnetic field serves to deflect most of the solar wind, so without the magnetic properties of the earth, the charged solar wind would have stripped the ozone layer from earth which would have exposed everything on earth to dangerous UV radiation.

[[File:Earth's magnetic field, schematic.svg|thumb|right|Earth's magnetic field, schematic]]

One interesting development in the field of magnetic torque is the experimentation, and initial prototyping of magnetic gears for application in a wide variety of industries, but that has a main focus in the wind turbine industry. The issue with strictly mechanical gearing today is in a high stress situation, the “teeth” or connection between gears, will fracture as a result of being over torqued. This results in a very powerful stall out that can gravely damage the broader mechanics of the instrument that the gears are in. Magnetic gears provide an interesting solution to the problem because there is no “physical” interaction between gear faces, only magnetic forces. This mitigates the stalling issue and provides a higher torque range by which machines utilizing this technology can operate. Just to give a specific example of this application, in the oil drilling industry, specifically where mud motors are applied to prospect oil, there is an incredible amount of power that must be applied via torque translation from the power section to the drill bit. An issue often seen is the wearing down of gears along the drill chain as a result of lubrication leaking, and rubbing of two components together, leading to stall outs which can damage the drill overall. To counteract this problem, research has been started to develop magnetic transmission sections to transmit the torque provided by the power section to the drill bit with minimal part damage due to minimal rubbing of components. The introduction of the magnetic gear will also mitigate the cost of lubricants, which is a very high cost especially when expensive lubricants are required.

==History==

Refer to [[Magnetic Force]]

The great importance of magnetic torque that is used in compasses cannot be ignored. The history of the compass and earth's magnetic field are very valuable.
The tendency of a magnet to align itself was discovered by the Chinese about 2000 years ago. The magnetic compass became a valuable commodity to European navigators in the 12th century, and in 1600, William Gilbert published De Magnete, which concluded that the earth behaves as a giant magnet.
Several theories since then have been made to explain how a magnetic field is produced by the earth. The most accepted theory is that the energy from the radioactivity of the earth's core travels outwards as heat. This heat produces a thermal convection core that creates the earth's magnetic field.

== See also ==

* [[Torque]]
* [[Magnetic Field]]
* [[Magnetic Force]]
* [[Bar Magnet]]

===Further reading===

* Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 3rd ed. Hoboken, NJ: Wiley, 2011. Print.
* Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, p. 269, 1985.
* Griffiths, D. J. Introduction to Electrodynamics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, p. 220, 1989.

===External links===

[http://scienceworld.wolfram.com/physics/MagneticTorque.html Magnetic Torque]

==References==

* [http://commons.wikimedia.org/wiki/File:Momento_torcente_magnetico.svg Torque Example]
* Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 3rd ed. Hoboken, NJ: Wiley, 2011. Print.
* "Magnet and Compass PHET Interaction Model." PhET. Ed. Chris Malley. University of Colorado, 2015. Web. 5 Dec. 2015. <https://phet.colorado.edu/en/simulation/legacy/magnet-and-compass>.
* Torque on Current-Carrying Loop in Magnetic Field. Doc Schuster. 23 Jan. 2013. Video. https://www.youtube.com/watch?v=xER1_SYql44
* http://helenotway.edublogs.org/2011/01/02/different-compass-point-same-ultimate-direction/
* Weisstein, Eric. "Magnetic Torque." Eric Weisstein's World of Physics. Wolfram Research, 1996. Web. 5 Dec. 2015. <http://scienceworld.wolfram.com/physics/MagneticTorque.html>.
* "Magnetic Torques and Amp's Law." Rochester Institute of Technology. Web. 5 Dec. 2015. <http://spiff.rit.edu/classes/phys213/lectures/amp/amp_long.html>.
* "Homework 11." WebAssign. Web. 5 Dec. 2015. <http://webassign.net/>.
* Magnetic Torque. Animations for Physics and Astronomy. 15 Feb. 2008. Video. https://www.youtube.com/watch?v=xER1_SYql44
* Digital image. N.p., n.d. Web. 17 Apr. 2016.
* "Discovery of the Earth’s Magnetic Field." GNS Science. N.p., n.d. Web. 17 Apr. 2016. <http://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field>.
* "Magnetic Dipole Moment." Hyperphysics, n.d. Web. 17 Apr. 2016. <http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html>.
* Magnetic Torque and Magnetic Dipole Moment. AK Lectures. 7 Dec. 2013. Video. https://www.youtube.com/watch?v=K1FEepXKETM
* "Magnetism." DISCovering Science. Gale Research, 1996. Reproduced in Discovering Collection. Farmington Hills, Mich.: Gale Group. December, 2000. http://galenet.galegroup.com/servlet/DC/
* Jun 19, 2014 Leland Teschler | Machine Design. "Could Magnetic Gears Make Wind Turbines Say Goodbye to Mechanical Gearboxes?" Machine Design. Penton, 19 June 2014. Web. 27 Nov. 2016.

[[Category:Fields]]

2019-08-23T21:12:38Z

Wpoe: /* Driving Current */

'''Motional emf''' is an electromagnetic field caused by current that is produced through the motion of a conductor in a magnetic field. Polarization of the bar occurs which is similar to the Hall effect except that the Hall effect involves polarization through the force of a magnetic field on charged particles that are already moving inside a motionless conductor. Motional emf is difficult to observe visually using batteries, light bulbs, and compasses because it is so small.

==The Main Idea==

Moving a metal bar (or similar conductive material) will naturally also move the mobile charges within the metal bar. if this is done in a magnetic field it will create a magnetic force, which acts on the charged particles inside the bar polarizing it (charge separation). This makes the bar similar to a battery which means that If the bar is part of a circuit, the magnetic force produced causes a current to run through it.

Additionally, when the metal bar is polarized, because of the charge separation, an electric force is created in the bar opposite to the magnetic force on charged particles.

===Polarization and Steady State===
[[File:Polarizationmotionalemf.png|thumb|alt=Picture showing a conducting bar moving in a magnetic field being polarized as a result of magnetic force|A conducting bar moving with velocity ''v'' is polarized as a result of the upwards magnetic force; see section 1.3 for a right-hand rule applicable to the scenario]]

Polarization occurs due to the shift of the mobile electron sea in one direction caused by the magnetic force. Eventually, the shifting will stop when enough electrons will shift in a particular direction so that the electric force, in the opposite direction, balances out the magnetic force (<math>qvB=qE</math>). Consequently, in the steady state, <math>E=vB</math> and there is no net force on the bar, so the bar does not require any additional force to keep it moving at a constant velocity, assuming it moves in a frictionless environment.
<br />
<br />
In the steady state, a nonzero E-field exists inside the metal bar; however, if the bar is not connected in a circuit, there is no current. This is because the electric force is balanced with the magnetic force, resulting in zero net force on the mobile electrons. The potential difference across the metal bar is then the product of the electric field and the length of the bar.

===Driving Current===
[[File:Circuitmotionalemf.png|thumb|left|alt=Picture demonstrating a possible circuit setup involving a moving conducting bar|A circuit formed with a moving conducting bar on conducting, frictionless rails connected by a resistor]]

[[File:electromagnetforce.png|thumb|alt=sssa|The x and y components of the magnetic force on a mobile electron in the bar]]

If the metal bar is used to form a circuit, where the bar is slid along on two frictionless metal rails that are also connected, then the bar's polarization in the bar mimics a battery and can drive a current through the bar and the rails it slides on, given the bars are connected at some other point.
<br />
<br />
Once the bar, moving within the magnetic field, has a velocity, the direction of the magnetic force can be determined through the right-hand rule. There will be a magnetic force in the direction of the length of the bar on the electrons which will cause the electrons to have a small velocity through the bar. This is in addition to the velocity caused by the moving bar. In the diagram shown on the left the velocity vector of the electrons would be <math> < V_{bar}, V, 0 > </math>. Magnetic force on the electrons is <math> <-e*V*B,-e*V_{bar}*B, 0> </math>. B is the magnetic field being applied. It is clear that there is a magnetic force on the electrons that is opposite to the horizontal force of the moving bar. This means that the horizontal net force for the bar is <math> F_{net} = F_{applied} - N*V*B*e </math> with N being the number of mobile electrons. If the bar moves at higher speeds, the vertical magnetic force on the electrons becomes greater along with the vertical velocity of the electrons. This in turn, increases the magnetic force to the left of the bar. If the bar continues to accelerate, the horizontal net force will get smaller and smaller until it reaches zero at which point the bar will move at a constant velocity with one force pulling it in one direction and the magnetic force pulling it in the opposite direction. A current now runs through the bar. The mobile electrons in the bar move toward the negatively charged end, rather than the positively charged end, because the continuous depletion of charge means that the electric field is always slightly less than what is needed to balance out the opposing magnetic field. Therefore, the electrons move to the negative end to maintain charge separation and the electric field.
<br />
<br />
The potential difference across the bar is still the product of the electric field and the length of the bar. If the bar does have some resistance, then it is treated like a battery with internal resistance. In the circuit, the potential difference round-trip for the entire circuit is zero.

===Ideal Conditions===
Motional emf is difficult to observe with light bulbs and batteries because it is relatively small. In order to obtain a sizeable emf, a larger magnetic field (B) needs to be applied over larger regions (L) all the while the bar is moving at greater velocities (V) through the region. If dealing with a wire, adding multiple turns will also help. Even then if you have a large emf, it only lasts for a little amount of time. Also detectors like lightbulbs and compasses are not very sensitive, so in order to actually detect motional emf more sensitive equipment is needed.

==A Mathematical Model==
[[File:RHRmotionalemf.JPG|thumb|alt=Picture showing a right-hand rule applicable to motional emf|A right-hand rule determining the directions of magnetic force, conventional current, and magnetic field]]

The potential difference across the metal bar is <math>\Delta V=emf=EL=v_{bar}BL</math>. If the bar has resistance, then <math>\Delta V=emf-r_{int}I</math>.
<br />
<br />
When the bar in the circuit, in steady state, moves a distance <math>\Delta x</math> over a time <math>\Delta t</math>, the work done is <math>F \Delta x</math> and the power supplied is <math>{F \Delta x}/{\Delta t}</math>, or <math>Fv_{bar}</math>. If the rails in the circuit are connected by a resistor, then the power dissipated in the resistor is <math>ILBv_{bar}=I*emf={emf/R}*emf={emf^{2}}/R=RI^{2}</math>.
<br />
<br />
Motional emf can be calculated in terms of magnetic flux: <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>.

===Equations to remember===
'''1.''' <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>.
<br />
'''2.''' During steady state, the electric force balances with the magnetic force(<math>qvB=qE</math>), so <math>E=vB</math>.
<br />
<br />
'''3.'''We know <math>\Delta V=emf</math>, <math>\Delta V=EL</math>, and <math>E=vB</math>, so <math>\Delta V=v_{bar}BL</math>.
<br />
<br />
'''4.''' Power <math>P=I\Delta V</math> so <math>P=I(IR)=I^{2}R</math>.
<br />

==A Computational Model==
A simulation of motional emf can be found [http://www.phy.hk/wiki/englishhtm/Induction.htm here].

==Examples==

===Simple===
[[File:Motionalemfsimple.png|center|alt=Diagram for simple example]]

''Taken from the'' Matter & Interactions ''textbook, variation of P60''.

A neutral iron bar is dragged to the left at speed v through a region with a magnetic field B that points out of the page, as shown above. '''(a)''' In which direction is the magnetic force? '''(b)''' Which diagram (1–5) best shows the state of the bar?
<br />
<br />
'''(a)''' The magnetic force points ''upwards''. Use the right hand rule.
<br />
'''(b)''' ''Diagram 3'' shows the correct polarization based on the direction of the magnetic force.
<br />

===Middling===
[[File:21.X.091-moving-bar01.gif|center|alt=Diagram for middle example]]
''Taken from the ''Matter & Interactions ''textbook, variation of P63.''

A neutral metal rod of length '''0.21 m''' slides horizontally at a constant speed of '''7 m/s''' on frictionless insulating rails through a region of uniform magnetic field of magnitude '''0.18 tesla''', directed '''out of the page''' as shown in the diagram. '''(a)''' Is the top of the moving rod positive or negative? '''(b)''' After the initial transient, what is the magnitude of the net force on a mobile electron inside the rod? '''(c)''' What is the magnitude of the electric force on a mobile electron inside the rod? '''(d)''' What is the magnitude of the magnetic force on a mobile electron inside the rod? '''(e)''' What is the magnitude of the potential difference across the rod? '''(f)''' In what direction must you exert a force to keep the rod moving at constant speed?
<br />
<br />
'''(a)''' The top of the moving rod is ''negative''. Use the right-hand rule.
<br />
'''(b)''' <math>\left | \overrightarrow{F}_{net} \right |=0 \: N</math> because after the initial transient the system is in steady state and the electric force balances out the magnetic force.
<br />
'''(c)''' First we recognize that the system is in steady state, so <math>E=vB=7 \times 0.18=1.26</math>. Then <math>\left | \overrightarrow{F}_{E} \right |=\left | qE \right |=\left | -1.6 \times 10^{-19}(1.26) \right |=2.016 \times 10^{-19} \: N</math>.
<br />
'''(d)''' In steady state, <math>\left | \overrightarrow{F}_{E} \right |=\left | \overrightarrow{F}_{B} \right |</math>. So, <math>\left | \overrightarrow{F}_{B} \right |=2.016 \times 10^{-19} \: N</math>.
<br />
'''(e)''' <math>\Delta V=EL=v_{bar}BL=7 \times 0.18 \times 0.21=0.2646 \: V</math>.
<br />
'''(f)''' ''No force is needed''. The system is in steady state, so no additional force needs to be applied to keep the rod at constant speed.
<br />

===Difficult===
[[File:Middlemotionalemf.png|center|alt=Picture for difficult example]]

''Taken from the'' Matter & Interactions ''textbook, P64''.
<br />

A metal bar of mass ''M'' and length ''L'' slides with negligible friction but with good electrical contact down between two vertical metal posts, as shown above. The bar falls at a constant speed ''v''. The falling bar and the vertical metal posts have negligible electrical resistance, but the bottom rod is a resistor with resistance ''R''. Throughout the entire region there is a uniform magnetic field of magnitude ''B'' coming straight out of the page. '''(a)''' Calculate the amount of current ''I'' running through the resistor. '''(b)''' What is the direction of the conventional current ''I''? '''(c)''' Calculate the constant speed ''v'' of the falling bar. '''(d)''' What is the direction of the electric force acting on a positive mobile charge? '''(e)''' What is the direction of the magnetic force acting on the bar?
<br />
<br />
'''(a)''' The bar is already falling at a constant velocity, so <math>\left | \overrightarrow {F}_{net} \right |=0</math> and <math>E=vB</math>. We know that <math>I={emf}/R</math> and <math>{emf}=EL=vBL</math> so it follows that <math>I={vBL}/R</math>.
<br />
'''(b)''' The magnetic force <math>qvB</math> causes positive charges to move left and negative charges to move right. Conventional current flows from the positively charged end of the bar to the negative end, so it runs ''counterclockwise''.
<br />
'''(c)''' The conventional current ''I'' running through the bar causes a magnetic force upwards of magnitude <math>ILB\sin 90^{\circ}=({vBL}/R)LB={vL^{2}B^{2}}/R</math>. This magnitude must balance out the gravitational force <math>Mg</math> because the magnetic and gravitational forces must cancel for <math>\left | \overrightarrow {F}_{net} \right |=0</math> and the bar to have a constant velocity ''v''. So, setting the forces equal to each other, <math>v={MgR}/{L^{2}B^{2}}</math>.
<br />
<br />
Note: We can also find the speed of the falling bar with <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>. Using the loop rule, <math>0=IR - {emf}</math>. So <math>\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |=IR</math>. We can transform that to <math>IR=\frac{\mathrm{d} (BA)}{\mathrm{d} t}</math>. Since B is constant, <math>IR=B(\frac{\mathrm{d} (A)}{\mathrm{d} t})</math>. We can transform that to <math>IR=B(\frac{\mathrm{d} (Lh)}{\mathrm{d} t})</math>, where h is the distance of the bar above the bottom rod. Since L is constant, <math>IR=BL(\frac{\mathrm{d} (h)}{\mathrm{d} t})</math>. <math>\frac{\mathrm{d} (h)}{\mathrm{d} t})</math> is the velocity of the bar, so <math>IR=BLv</math>. Solving for v, <math>v={MgR}/{L^{2}B^{2}}</math>.
<br />
'''(d)''' The electric force acting on a positive mobile charge points towards the ''right'' in the bar. Using the right hand rule, with B pointing out of the page, v pointing downwards, we know that the magnetic force points to the right. This means that the bar is polarized with positive charges on the left side of the bar and negative charges on the right side. The electric force points from positive charges to negative charges, so the electric force points towards the right.
<br />
'''(e)''' The magnetic force acting on the bar points ''upwards''. We know that current flows from left to right in the bar because positive charges are polarized on the left side of the bar and negative charges are on the right side of the bar. Using the right hand rule, with B pointing out of the page and current pointing to the right, we find that the magnetic force points upwards.

==Connectedness==
The concept of power generated from a mechanical input is how electric generators function. The mechanical energy is commonly supplied by falling water at dams or expanding steam in turbines.
<br />
Motional emf is also applied in the DC motors of airplanes, when mechanical energy is transformed into electrical energy. Additionally, motional emf is applied in breaking systems. When an object moves through a magnetic field, it resists the change(movement) by converting the mechanical energy into electrical energy. So if an object moves with a sufficient amount of force to move, through a magnetic field, it can convert enough mechanical energy to stop a system. This concept is applied in, for example, roller coasters.

==History==
[[File:Faraday-Millikan-Gale-1913.jpg|thumb|alt=Picture of Michael Faraday|Michael Faraday]] The phenomenon of electromagnetic induction, the emf produced from the interactions between a magnetic field and an electric circuit, was discovered independently in 1831 by Michael Faraday and 1832 by Joseph Henry (Faraday published his results first). Many of Faraday's ideas were rejected by the scientists of the day because they had no mathematical basis, but James Clerk Maxwell used these ideas to formulate his electromagnetic theory and Maxwell's equations.
<br />
At the time, it was not well understood how Maxwell's equations related to moving charged objects. J.J. Thomson was the first to attempt to derive, from Maxwell's equations, an equation describing effects on moving charged particles, and Oliver Heaviside built upon this work and fixed errors in Thomson's derivation. In 1892, Hendrik Lorentz finally derived the correct equation called the Lorentz force. The magnetic force component of the Lorentz force describes motional emf, as the magnetic force pushing electrons in the moving charged object induces the emf.

== See also ==
*[[Motional Emf using Faraday's Law]]: An expansion of this concept
*[[Lorentz Force]]: Combining electric and magnetic forces
*[[Generator]]: Real-world application
*[[Right-Hand Rule]]: How it works and other RHRs
===Further reading===
*''Matter & Interactions, Vol 2''
*''The Feynman Lectures on Physics, Vol 2'', Ch 16

===External links===
*[http://web.mit.edu/8.02t/www/materials/StudyGuide/guide10.pdf Guide from MIT]

==References==

*''Matter & Interactions Vol 2''
*MIT OpenCourseWare
*[http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf A Brief History of The Development of Classical Electrodynamics]

[[Category:Fields]]

Motional Emf

2019-08-23T20:57:07Z

Wpoe: /* Driving Current */

'''Motional emf''' is an electromagnetic field caused by current that is produced through the motion of a conductor in a magnetic field. Polarization of the bar occurs which is similar to the Hall effect except that the Hall effect involves polarization through the force of a magnetic field on charged particles that are already moving inside a motionless conductor. Motional emf is difficult to observe visually using batteries, light bulbs, and compasses because it is so small.

==The Main Idea==

Moving a metal bar (or similar conductive material) will naturally also move the mobile charges within the metal bar. if this is done in a magnetic field it will create a magnetic force, which acts on the charged particles inside the bar polarizing it (charge separation). This makes the bar similar to a battery which means that If the bar is part of a circuit, the magnetic force produced causes a current to run through it.

Additionally, when the metal bar is polarized, because of the charge separation, an electric force is created in the bar opposite to the magnetic force on charged particles.

===Polarization and Steady State===
[[File:Polarizationmotionalemf.png|thumb|alt=Picture showing a conducting bar moving in a magnetic field being polarized as a result of magnetic force|A conducting bar moving with velocity ''v'' is polarized as a result of the upwards magnetic force; see section 1.3 for a right-hand rule applicable to the scenario]]

Polarization occurs due to the shift of the mobile electron sea in one direction caused by the magnetic force. Eventually, the shifting will stop when enough electrons will shift in a particular direction so that the electric force, in the opposite direction, balances out the magnetic force (<math>qvB=qE</math>). Consequently, in the steady state, <math>E=vB</math> and there is no net force on the bar, so the bar does not require any additional force to keep it moving at a constant velocity, assuming it moves in a frictionless environment.
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In the steady state, a nonzero E-field exists inside the metal bar; however, if the bar is not connected in a circuit, there is no current. This is because the electric force is balanced with the magnetic force, resulting in zero net force on the mobile electrons. The potential difference across the metal bar is then the product of the electric field and the length of the bar.

===Driving Current===
[[File:Circuitmotionalemf.png|thumb|left|alt=Picture demonstrating a possible circuit setup involving a moving conducting bar|A circuit formed with a moving conducting bar on conducting, frictionless rails connected by a resistor]]

[[File:electromagnetforce.png|thumb|alt=sssa|The x and y components of the magnetic force on a mobile electron in the bar]]

If the metal bar is used to form a circuit, where the bar is slid along on two frictionless metal rails that are also connected, then the bar's polarization in the bar mimics a battery and can drive a current.
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When the bar, in the described configuration, has a force applied to it and initially begins to move from rest, the entire lattice of positive ions is pulled in the direction of the bar's velocity. Because the net electric force on mobile electrons in a metal is basically zero, the mobile electrons are left behind for a brief moment, causing a near-instantaneous horizontal polarization in the bar, but are then pulled along with the positive ions due to the electric interaction between the charges.
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Now that the bar, moving within the magnetic field, has a velocity, the direction of the magnetic force can be determined through the right-hand rule. There will be an upward magnetic force on the electrons which will cause the electrons to have a small upward velocity. This is in addition to the horizontal velocity caused by the moving bar. The velocity vector of the electrons would be <Vbar,V,0>. Magnetic force on the electrons is <-e*V*B,-e*Vbar*B, 0>. B is the magnetic field being applied. It is clear that there is a magnetic force on the electrons that is opposite to the horizontal force of the moving bar. This means that the horizontal net force for the bar is Fnet=F-N*V*B*e with N being the number of mobile electrons. If the bar moves at higher speeds, the vertical magnetic force on the electrons becomes greater along with the vertical velocity of the electrons. This in turn, increases the force N*V*B*e to the left of the bar. If the bar continues to accelerate, the horizontal net force will get smaller and smaller until it reaches zero at which point the bar will move at a constant velocity with one force pulling it in one direction and the magnetic force pulling it in the opposite direction. A current now runs through the bar. The mobile electrons in the bar move toward the negatively charged end, rather than the positively charged end, because the continuous depletion of charge means that the electric field is always slightly less than what is needed to balance out the opposing magnetic field. Therefore, the electrons move to the negative end to maintain charge separation and the electric field.
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The potential difference across the bar is still the product of the electric field and the length of the bar. If the bar does have some resistance, then it is treated like a battery with internal resistance. In the circuit, the potential difference round-trip is zero.

===Ideal Conditions===
Motional emf is difficult to observe with light bulbs and batteries because it is relatively small. In order to obtain a sizeable emf, a larger magnetic field (B) needs to be applied over larger regions (L) all the while the bar is moving at greater velocities (V) through the region. If dealing with a wire, adding multiple turns will also help. Even then if you have a large emf, it only lasts for a little amount of time. Also detectors like lightbulbs and compasses are not very sensitive, so in order to actually detect motional emf more sensitive equipment is needed.

==A Mathematical Model==
[[File:RHRmotionalemf.JPG|thumb|alt=Picture showing a right-hand rule applicable to motional emf|A right-hand rule determining the directions of magnetic force, conventional current, and magnetic field]]

The potential difference across the metal bar is <math>\Delta V=emf=EL=v_{bar}BL</math>. If the bar has resistance, then <math>\Delta V=emf-r_{int}I</math>.
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When the bar in the circuit, in steady state, moves a distance <math>\Delta x</math> over a time <math>\Delta t</math>, the work done is <math>F \Delta x</math> and the power supplied is <math>{F \Delta x}/{\Delta t}</math>, or <math>Fv_{bar}</math>. If the rails in the circuit are connected by a resistor, then the power dissipated in the resistor is <math>ILBv_{bar}=I*emf={emf/R}*emf={emf^{2}}/R=RI^{2}</math>.
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Motional emf can be calculated in terms of magnetic flux: <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>.

===Equations to remember===
'''1.''' <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>.
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'''2.''' During steady state, the electric force balances with the magnetic force(<math>qvB=qE</math>), so <math>E=vB</math>.
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'''3.'''We know <math>\Delta V=emf</math>, <math>\Delta V=EL</math>, and <math>E=vB</math>, so <math>\Delta V=v_{bar}BL</math>.
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'''4.''' Power <math>P=I\Delta V</math> so <math>P=I(IR)=I^{2}R</math>.
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==A Computational Model==
A simulation of motional emf can be found [http://www.phy.hk/wiki/englishhtm/Induction.htm here].

==Examples==

===Simple===
[[File:Motionalemfsimple.png|center|alt=Diagram for simple example]]

''Taken from the'' Matter & Interactions ''textbook, variation of P60''.

A neutral iron bar is dragged to the left at speed v through a region with a magnetic field B that points out of the page, as shown above. '''(a)''' In which direction is the magnetic force? '''(b)''' Which diagram (1–5) best shows the state of the bar?
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'''(a)''' The magnetic force points ''upwards''. Use the right hand rule.
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'''(b)''' ''Diagram 3'' shows the correct polarization based on the direction of the magnetic force.
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===Middling===
[[File:21.X.091-moving-bar01.gif|center|alt=Diagram for middle example]]
''Taken from the ''Matter & Interactions ''textbook, variation of P63.''

A neutral metal rod of length '''0.21 m''' slides horizontally at a constant speed of '''7 m/s''' on frictionless insulating rails through a region of uniform magnetic field of magnitude '''0.18 tesla''', directed '''out of the page''' as shown in the diagram. '''(a)''' Is the top of the moving rod positive or negative? '''(b)''' After the initial transient, what is the magnitude of the net force on a mobile electron inside the rod? '''(c)''' What is the magnitude of the electric force on a mobile electron inside the rod? '''(d)''' What is the magnitude of the magnetic force on a mobile electron inside the rod? '''(e)''' What is the magnitude of the potential difference across the rod? '''(f)''' In what direction must you exert a force to keep the rod moving at constant speed?
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'''(a)''' The top of the moving rod is ''negative''. Use the right-hand rule.
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'''(b)''' <math>\left | \overrightarrow{F}_{net} \right |=0 \: N</math> because after the initial transient the system is in steady state and the electric force balances out the magnetic force.
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'''(c)''' First we recognize that the system is in steady state, so <math>E=vB=7 \times 0.18=1.26</math>. Then <math>\left | \overrightarrow{F}_{E} \right |=\left | qE \right |=\left | -1.6 \times 10^{-19}(1.26) \right |=2.016 \times 10^{-19} \: N</math>.
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'''(d)''' In steady state, <math>\left | \overrightarrow{F}_{E} \right |=\left | \overrightarrow{F}_{B} \right |</math>. So, <math>\left | \overrightarrow{F}_{B} \right |=2.016 \times 10^{-19} \: N</math>.
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'''(e)''' <math>\Delta V=EL=v_{bar}BL=7 \times 0.18 \times 0.21=0.2646 \: V</math>.
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'''(f)''' ''No force is needed''. The system is in steady state, so no additional force needs to be applied to keep the rod at constant speed.
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===Difficult===
[[File:Middlemotionalemf.png|center|alt=Picture for difficult example]]

''Taken from the'' Matter & Interactions ''textbook, P64''.
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A metal bar of mass ''M'' and length ''L'' slides with negligible friction but with good electrical contact down between two vertical metal posts, as shown above. The bar falls at a constant speed ''v''. The falling bar and the vertical metal posts have negligible electrical resistance, but the bottom rod is a resistor with resistance ''R''. Throughout the entire region there is a uniform magnetic field of magnitude ''B'' coming straight out of the page. '''(a)''' Calculate the amount of current ''I'' running through the resistor. '''(b)''' What is the direction of the conventional current ''I''? '''(c)''' Calculate the constant speed ''v'' of the falling bar. '''(d)''' What is the direction of the electric force acting on a positive mobile charge? '''(e)''' What is the direction of the magnetic force acting on the bar?
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'''(a)''' The bar is already falling at a constant velocity, so <math>\left | \overrightarrow {F}_{net} \right |=0</math> and <math>E=vB</math>. We know that <math>I={emf}/R</math> and <math>{emf}=EL=vBL</math> so it follows that <math>I={vBL}/R</math>.
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'''(b)''' The magnetic force <math>qvB</math> causes positive charges to move left and negative charges to move right. Conventional current flows from the positively charged end of the bar to the negative end, so it runs ''counterclockwise''.
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'''(c)''' The conventional current ''I'' running through the bar causes a magnetic force upwards of magnitude <math>ILB\sin 90^{\circ}=({vBL}/R)LB={vL^{2}B^{2}}/R</math>. This magnitude must balance out the gravitational force <math>Mg</math> because the magnetic and gravitational forces must cancel for <math>\left | \overrightarrow {F}_{net} \right |=0</math> and the bar to have a constant velocity ''v''. So, setting the forces equal to each other, <math>v={MgR}/{L^{2}B^{2}}</math>.
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Note: We can also find the speed of the falling bar with <math>\left | emf \right |=\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |</math>. Using the loop rule, <math>0=IR - {emf}</math>. So <math>\left | \frac{\mathrm{d} \Phi_ {mag}}{\mathrm{d} t} \right |=IR</math>. We can transform that to <math>IR=\frac{\mathrm{d} (BA)}{\mathrm{d} t}</math>. Since B is constant, <math>IR=B(\frac{\mathrm{d} (A)}{\mathrm{d} t})</math>. We can transform that to <math>IR=B(\frac{\mathrm{d} (Lh)}{\mathrm{d} t})</math>, where h is the distance of the bar above the bottom rod. Since L is constant, <math>IR=BL(\frac{\mathrm{d} (h)}{\mathrm{d} t})</math>. <math>\frac{\mathrm{d} (h)}{\mathrm{d} t})</math> is the velocity of the bar, so <math>IR=BLv</math>. Solving for v, <math>v={MgR}/{L^{2}B^{2}}</math>.
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'''(d)''' The electric force acting on a positive mobile charge points towards the ''right'' in the bar. Using the right hand rule, with B pointing out of the page, v pointing downwards, we know that the magnetic force points to the right. This means that the bar is polarized with positive charges on the left side of the bar and negative charges on the right side. The electric force points from positive charges to negative charges, so the electric force points towards the right.
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'''(e)''' The magnetic force acting on the bar points ''upwards''. We know that current flows from left to right in the bar because positive charges are polarized on the left side of the bar and negative charges are on the right side of the bar. Using the right hand rule, with B pointing out of the page and current pointing to the right, we find that the magnetic force points upwards.

==Connectedness==
The concept of power generated from a mechanical input is how electric generators function. The mechanical energy is commonly supplied by falling water at dams or expanding steam in turbines.
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Motional emf is also applied in the DC motors of airplanes, when mechanical energy is transformed into electrical energy. Additionally, motional emf is applied in breaking systems. When an object moves through a magnetic field, it resists the change(movement) by converting the mechanical energy into electrical energy. So if an object moves with a sufficient amount of force to move, through a magnetic field, it can convert enough mechanical energy to stop a system. This concept is applied in, for example, roller coasters.

==History==
[[File:Faraday-Millikan-Gale-1913.jpg|thumb|alt=Picture of Michael Faraday|Michael Faraday]] The phenomenon of electromagnetic induction, the emf produced from the interactions between a magnetic field and an electric circuit, was discovered independently in 1831 by Michael Faraday and 1832 by Joseph Henry (Faraday published his results first). Many of Faraday's ideas were rejected by the scientists of the day because they had no mathematical basis, but James Clerk Maxwell used these ideas to formulate his electromagnetic theory and Maxwell's equations.
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At the time, it was not well understood how Maxwell's equations related to moving charged objects. J.J. Thomson was the first to attempt to derive, from Maxwell's equations, an equation describing effects on moving charged particles, and Oliver Heaviside built upon this work and fixed errors in Thomson's derivation. In 1892, Hendrik Lorentz finally derived the correct equation called the Lorentz force. The magnetic force component of the Lorentz force describes motional emf, as the magnetic force pushing electrons in the moving charged object induces the emf.

== See also ==
*[[Motional Emf using Faraday's Law]]: An expansion of this concept
*[[Lorentz Force]]: Combining electric and magnetic forces
*[[Generator]]: Real-world application
*[[Right-Hand Rule]]: How it works and other RHRs
===Further reading===
*''Matter & Interactions, Vol 2''
*''The Feynman Lectures on Physics, Vol 2'', Ch 16

===External links===
*[http://web.mit.edu/8.02t/www/materials/StudyGuide/guide10.pdf Guide from MIT]

==References==

*''Matter & Interactions Vol 2''
*MIT OpenCourseWare
*[http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf A Brief History of The Development of Classical Electrodynamics]

[[Category:Fields]]