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Edited by Chris Mickas Fall 2016
'''Paulo Lacombe Spring 2018'''{{Electrodynamics}}
Yiqiao Wu Spring 2017
[[File:Lorentz force-experiment PNr°0128.jpg|thumb|189x189px|Apparatus showing Lorentz force in action on a moving particle]]
In order to find the force acting on a charged particle inside a '''magnetic''' and/or '''electric''' field we need to use the Lorentz force rule. When adding a charged particle to a field the field must exert a force on the particle based on the magnitude and direction of the field and the particle.


'''Edited by Adam Schatz Fall 2017'''
Because the Lorentz force involves both an electric and magnetic field it is considered part of electromagnetism. Lorentz force also completes the net forces acting on a particle together with other forces such as gravitational force.


'''Edited by Connor Meeds Fall 2017'''
== Mathematical Model ==
[[File:Lorentz front.gif|thumb|171x171px|A moving particle in a magnetic field viewed from the side]]


=== '''Complete Formula''' ===


[[File:Headerlorentz.png|400px|thumb|right|Lorentz force diagram]]
:<math>\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})</math>
==The Main Idea==
In the earlier stages of Physics II, students are mostly concerned with electrostatics - the forces generated by and acting upon charges at rest. However, what happens when charges are in motion? This subject introduces another force within the study of the '''Lorentz Force'''.The Lorentz Force is a name for the entire electromagnetic force exerted on a charged particle ''q'' moving with velocity ''v'' through an electric field ''E'' and magnetic field ''B''. In most cases studied throughout college physics courses, the Lorentz Force on a particle is merely contributed by electric and magnetic forces, where other forces acting on the particle are considered negligible. Let's take a look at a simple situation that illustrates this subject in action.


Taking two neutral currents in parallel, we notice something weird happen depending on the direction of the currents in relation to each other:
[[File:Lorentz force electric field.gif|thumb|172x172px|A stationary particle inside an electric field]]
'''F''' is the force acting on the particle


[[File:meeds2.png]]
'''q''' is the charge of the particle


Referencing the diagram above, we find that the parallel currents that run in the same direction undergo an attractive force. Alternatively, parallel currents that run in opposite directions undergo a repulsive force. Furthermore, experimental analysis prove that this resulting force is proportional to the currents (tripling the current in ''one'' of the wires triples the force, while tripling both of the currents produces a force 6x the original). With this in mind, it is clear that the force is proportional to the velocity of a moving charge and points in a direction perpendicular to the moving charge's velocity. This indicates that some kind of magnetic field ''B'' arises from moving charges, and this is a main concept of the Lorentz Force.
'''E''' is the electric field


===A Mathematical Model===
'''v''' is the velocity


The electromagnetic force ''F'' on a charged particle, the Lorentz force (named after the Dutch physicist [http://www.physicsbook.gatech.edu/Hendrik_Lorentz Hendrik A. Lorentz]) is given by:
'''B''' is the magnetic field


<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>
x is the cross product. In this case the result is the vector cross product between the particle velocity and the magnetic field.


where '''<math>q\vec{E}</math>''' is the electric force contributed by an external electric field and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' is the magnetic force contributed by an external magnetic field. As many applications involve vectors, it is valuable to recognize the resulting directions of '''<math>q\vec{E}</math>''' and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' in relation to the particle's charge and velocity within the environment of applied electric/magnetic fields. The resulting electric force vector will always be towards or opposite to the applied electric field, depending on the sign of the charge. For example, an electron undergoing an electric field in the +x direction will receive an electric force in the -x direction. The magnetic force on the particle, however, has a direction perpendicular to both the velocity ''v'' of the particle and the magnetic field ''B'', and has a value proportional to ''q'' and to the magnitude of the vector cross product ''' <math>q\vec{v} ⨯ \vec{B}</math>'''. More specifically, the magnitude of the magnetic force equals ''qvBsinθ'' where ''θ'' is  the angle between ''v'' and ''B''.
==== Electric Force ====
<math>\mathbf{F} = q * \mathbf{E}</math>


====Motion of a Charged Particle in a Uniform Magnetic Field====
==== Magnetic Force ====
<math>\mathbf{F} = q(\mathbf{v} \times \mathbf{B})</math>


A noteworthy result of the Lorentz force is the motion of a charged particle within a uniform magnetic field as the angle between ''v'' and ''B'' varies. If a scenario presents a charged particle with a velocity vector ''v'' perpendicular to the applied magnetic field ''B'' (i.e. θ = 90°), the particle will follow a circular trajectory. The radius of this trajectory can easily be calculated:
=== '''Analysis''' ===
[[File:Right Hand Rule vBF2.PNG|thumb|139x139px|Right Hand Rule]]
Based on the equation we can conclude that the '''electric force''' on the particle is in the same direction as the electric field as long as the particle is positive, the force is in the opposite direction if the particle is negative. The '''magnetic force''' will be perpendicular to the velocity and magnetic field, in other words the normal to the magnetic force and the velocity. This force generally causes the particle to move in a spiral as long as the magnetic force is not parallel to the velocity. In that case the force would be equal to zero.


<math> \vec{F}_{Mag} = \vec{F}_{Centripetal}</math>
In order to determine the direction of the magnetic force in a physical manner we can use the Right Hand Rule. By placing your four main fingers in the direction of the magnetic field and your thumb in the direction of the velocity, the direction of the force is pointed out of your palm as shown in the diagram.


<math> qvB = mv^2/R </math>
== Example ==


<math> R = mv/qB </math>
=== Problem 1 ===
Find the magnitude and direction of the magnetic force on the antiproton.
[[File:Example lorentz force problem.jpg|thumb|Problem 1]]


Suppose that for the following example, a charge is shot into a region filled with a uniform magnetic field coming out of the page:
<math>\vec{v}_p=v<0,-1,0></math>
<math>\hat{r}={\vec{r} \over |r|}={<-d,-d,0> \over \sqrt{(-d)^2+(-d)^2}}={1 \over \sqrt{2}}<-1,-1,0></math>


[[File:Meeds1.png]]
<math>\vec{B_p}={\mu_0 \over 4\pi}*{q\vec{v_p}\times\widehat{r} \over r^2}={\mu_0 \over 4\pi}*{e<0,-v,0>\times{1\over\sqrt{2}}<-1,-1,0> \over 2d^2}</math>


At every instant, the external magnetic field ''B'' points out of the page and thus invokes a magnetic force on the particle perpendicular to the particle's velocity - the force needed to create circular motion. The radius ''R'' can be calculated from the equation above. In addition, it's worth noting that the particle's charge will determine where the particle veers. If the particle were positively charged, the magnetic force would cause the particle to veer downward (due to right hand rule), and vice-versa if negatively charged.
<math>\vec{B_p}={\mu_0ev\over 8\pi\sqrt{2}d^2}*<0,0,-1></math>


In the case that θ is less than 90°, the particle's trajectory will orbit a helix path with a central axis parallel to the field lines.
<math>\vec{F_B}={\mu_0ev\over 8\pi\sqrt{2}d^2}*<1,0,0>\times<0,0,-1>={\mu_0ev\over 8\pi\sqrt{2}d^2}*<0,1,0></math>


If θ is zero, that is, the magnetic field is in the same direction as the particle's velocity, the particle will experience no magnetic force and continue to move normally along the field lines.
== History ==
[[File:Fermilab.jpg|thumb|229x229px|The Fermi National Accelerator Laboratory is a national particle accelerator located in Illinois]]
The Lorentz Force was first derived by Hendrik Lorentz in 1895. It separated the particle from the fields around it putting it into a matter category. This allowed Lorentz to characterize charged particles with both electric and magnetic forces. Even though the Lorentz Force law is an important building block to electromagnetism it cannot account for all electromagnetic phenomenon. For example, it only considers the field acting on a charged particle, ignoring the field created by the particle itself. Therefore, if we were to analyze the forces on a system with both a magnetic and electric field as well as more than one charged particle we would need more equations to calculate the behavior of the charged particles.


===A Computational Model===
One common use of the Lorentz Force law is in particle accelerators. As the name suggest a particle accelerator is an apparatus used to speed up a particle in order to study the behavior of particles in different scenarios. The most famous particle accelerator is the LHC, or large hadron collider in Europe. The LHC uses electromagnetic forces to speed up particles and collide them analyzing the radiation emitted from each collision. Using electromagnetic forces in a loop means that particle accelerators can speed up particles to incredible speeds nearing the speed of light.
 
 
[https://trinket.io/embed/glowscript/43e56e9c64 Here is a visualization] on VPython of a negatively charged particle moving through a constant electric and magnetic field:
 
[[File:Lorentzdiagram.png]]
 
Initially, a negatively charged particle is traveling with initial velocity in the -z direction. There is a constant electric field ''E'' in the -x direction and a constant magnetic field ''B'' in the +y direction. The magnetic force on the negatively charged particle is equal to <math> q\vec{v} ⨯ \vec{B}</math>, or the charge of the particle times the cross product of the particle’s velocity and the magnetic field it travels through. The electric force on the particle is equal to <math>  q\vec{E} </math>, or the charge of the particle times the electric field that the particle travels through. Since the magnetic force on the particle is related to the particle’s velocity ''v'', the magnetic force changes as the the particle’s velocity changes. Conversely, the electric force on the particle is constant. Since the Magnetic force is variable, the Lorentz Force on the particle, or the net force due to magnetic and electric forces on the particle  (<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>) is also variable, and the particle's velocity changes.
 
==Example Problems==
 
===Simple===
 
The electric force on a certain particle is <100,-600,300> N and the magnetic force is <-600,400,0> N. Find the Lorentz force.
 
'''Solution: Lorentz force = <-500,-200,300> N'''
[[File:Soln2.PNG]]
 
===Intermediate===
 
The magnetic force on a proton is 100 N at an angle 30 degrees down from the +x axis. The electric force on the proton is 100 N at an angle 30 degrees up from the +z axis. What is the magnitude of the Lorentz Force on the proton?
 
'''Solution: Lorentz force = 122.5 N'''
 
[[File:Soln1.PNG]]
 
===Difficult===
An electron is traveling with a constant velocity of <0.75c, 0, 0>. You measure the magnetic field to be <0.4, 0.3, 0.5>T everywhere. What is the electric field?
 
'''Solution: E = <0, 1.13e8, -6.75e7> N/C '''
 
Since the electron is traveling at constant velocity, the net force must be zero. Thus, the magnetic field must equal the electric field, or <math> q\vec{v} ⨯ \vec{B}= q\vec{E} </math>. The charge on both sides cancels out to give <math> \vec{v} ⨯ \vec{B}= \vec{E} </math>. Calculating
<math> q\vec{v} ⨯ \vec{B}</math>  =  <0, -1.13e8, 6.75e7>, so the electric field must point in the opposite direction.
 
==Connectedness==
 
 
 
1. Speakers use the Lorentz force of an electromagnet to move a cone that creates sound waves in the air. When current flows through the wires in the electromagnetic in different quantities, the speakers move in unique ways to produce the different sounds that we recognize. Amplifiers for electric guitars and basses work in the same way.
 
2. In today's evolving world, one area of particular interest is sustainable and renewable energy. Wind turbines and hydropower plants work by harnessing the kinetic energy of wind or water and using it to induce an electrical current. The turbines rotate and move a permanent magnet that induces a current in an electromagnet placed inside of the magnet, which is shaped like a hollow cylinder. The induced current is then carried via wires to external sources to provide energy.
 
3. Several industries manufacture products that induce current using the Lorentz Force. For example, electric guitars and basses work by magnetizing the strings and relying on the Lorentz force to create a current in pickups that is then transmitted to an amplifier. Pickups are small electromagnet coils surrounding a magnet that are placed beneath the strings. The strings become magnetized because of the magnet inside the pickup. When they are played and vibrate, they induce current in the electromagnet. The Lorentz force causes the strings to exert forces that move mobile charges and induce the current. The current is then increased through a potentiometer and sent to an amplifier through a cable. In addition, charged particle accelerators like cyclotrons make use of the circular orbit particles experience when ''v'' and ''B'' are perpendicular to each other. For each revolution, a carefully timed electric field offers additional kinetic energy to cause the particles to move in increasingly-larger orbits until a desired energy level is met. These particles are known to be extracted and used in a number of ways, from basic studies of the properties of matter to the medical treatment of cancer.
 
==History==
[[File:hlorentz.jpg|200px|thumb|right|Hendrik Lorentz]]
 
The Lorentz Force is named after Hendrik Lorentz, who derived the formula in the late 19th century following a previous derivation by [[Oliver Heaviside]] in 1889. However, scientists had tried to find formulas for one electromagnetic force for over a hundred years before.Some scientists such as  [[Henry Cavendish]] argued that the magnetic poles of an object could create an electric force on a particle that obeys an inverse-square law. However, the experimental proof was not enough to definitively publish. In 1784,  [[Charles de Coulomb]], using a torsion balance, was able to definitively show through experiment that this was true. After [[Hans Christian Ørsted]] discovered that a magnetic needle is acted on by a voltaic current, [[Andre Marie Ampere]] derived a new formula for the angular dependence of the force between two current elements. However, the force was still given in terms of the properties of the objects involved and the distances between, not in terms of electric and magnetic fields or forces.
 
[[Michael Faraday]] introduced modern ideas of magnetic and electric fields, including their interactions and relations with each other, later to be given full mathematical description by [[William Thomson (Lord Kelvin)]] and [[James Maxwell]]. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, however, it was not initially evident how his equations related to the forces on moving charged objects. [[J.J. Thomson]] was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as <math>\vec{F} = q\vec{E} + \frac{q}{2}\vec{v} ⨯ \vec{B}</math>. Finally, Heaviside and later Lorentz were able to combine the information into the currently accepted Lorentz Force equation.


== See also ==
== See also ==
The [[Hall Effect]] is a special case in which the magnetic and electric forces on a particle or object cancel out, meaning that there is zero net force. Solving these problems involves setting the two forces equal to each other and using given information to find values for <math>\vec{B}</math>, <math>\vec{v}</math>, or <math>\vec{E}</math>.
The Lorentz Force law was used to derive the Laplace force which considers all moving charges in a wire to show the magnetic force acting on the wire.
 
[https://www.youtube.com/watch?v=8QWB8IfNoIs This video] demonstrates a few everyday applications and examples of the Lorentz Force.
 
===Further reading===
 
[[Hall Effect]]
*http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html
*http://www.ittc.ku.edu/~jstiles/220/handouts/section%203_6%20The%20Lorentz%20Force%20Law%20package.pdf
 
===External links===
 
*http://jnaudin.free.fr/lifters/lorentz/
*https://nationalmaglab.org/education/magnet-academy/watch-play/interactive/lorentz-force
*http://web.mit.edu/sahughes/www/8.022/lec10.pdf
 
==References==
 
*Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2.: volume 2.
*Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X.
*Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X.
*Hughs, Scott. “Magnetic Force; Magnetic Fields; Ampere's Law.” MIT.edu. Magnetic Force; Magnetic Fields; Ampere's Law, 29 Nov. 2017, Boston, MIT, Magnetic Force; Magnetic Fields; Ampere's Law.
 
[[Category:Which Category did you place this in?]]

Revision as of 23:00, 18 April 2018

Paulo Lacombe Spring 2018Template:Electrodynamics

Apparatus showing Lorentz force in action on a moving particle

In order to find the force acting on a charged particle inside a magnetic and/or electric field we need to use the Lorentz force rule. When adding a charged particle to a field the field must exert a force on the particle based on the magnitude and direction of the field and the particle.

Because the Lorentz force involves both an electric and magnetic field it is considered part of electromagnetism. Lorentz force also completes the net forces acting on a particle together with other forces such as gravitational force.

Mathematical Model

A moving particle in a magnetic field viewed from the side

Complete Formula

[math]\displaystyle{ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) }[/math]
A stationary particle inside an electric field

F is the force acting on the particle

q is the charge of the particle

E is the electric field

v is the velocity

B is the magnetic field

x is the cross product. In this case the result is the vector cross product between the particle velocity and the magnetic field.

Electric Force

[math]\displaystyle{ \mathbf{F} = q * \mathbf{E} }[/math]

Magnetic Force

[math]\displaystyle{ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) }[/math]

Analysis

Right Hand Rule

Based on the equation we can conclude that the electric force on the particle is in the same direction as the electric field as long as the particle is positive, the force is in the opposite direction if the particle is negative. The magnetic force will be perpendicular to the velocity and magnetic field, in other words the normal to the magnetic force and the velocity. This force generally causes the particle to move in a spiral as long as the magnetic force is not parallel to the velocity. In that case the force would be equal to zero.

In order to determine the direction of the magnetic force in a physical manner we can use the Right Hand Rule. By placing your four main fingers in the direction of the magnetic field and your thumb in the direction of the velocity, the direction of the force is pointed out of your palm as shown in the diagram.

Example

Problem 1

Find the magnitude and direction of the magnetic force on the antiproton.

Problem 1

[math]\displaystyle{ \vec{v}_p=v\lt 0,-1,0\gt }[/math] [math]\displaystyle{ \hat{r}={\vec{r} \over |r|}={\lt -d,-d,0\gt \over \sqrt{(-d)^2+(-d)^2}}={1 \over \sqrt{2}}\lt -1,-1,0\gt }[/math]

[math]\displaystyle{ \vec{B_p}={\mu_0 \over 4\pi}*{q\vec{v_p}\times\widehat{r} \over r^2}={\mu_0 \over 4\pi}*{e\lt 0,-v,0\gt \times{1\over\sqrt{2}}\lt -1,-1,0\gt \over 2d^2} }[/math]

[math]\displaystyle{ \vec{B_p}={\mu_0ev\over 8\pi\sqrt{2}d^2}*\lt 0,0,-1\gt }[/math]

[math]\displaystyle{ \vec{F_B}={\mu_0ev\over 8\pi\sqrt{2}d^2}*\lt 1,0,0\gt \times\lt 0,0,-1\gt ={\mu_0ev\over 8\pi\sqrt{2}d^2}*\lt 0,1,0\gt }[/math]

History

The Fermi National Accelerator Laboratory is a national particle accelerator located in Illinois

The Lorentz Force was first derived by Hendrik Lorentz in 1895. It separated the particle from the fields around it putting it into a matter category. This allowed Lorentz to characterize charged particles with both electric and magnetic forces. Even though the Lorentz Force law is an important building block to electromagnetism it cannot account for all electromagnetic phenomenon. For example, it only considers the field acting on a charged particle, ignoring the field created by the particle itself. Therefore, if we were to analyze the forces on a system with both a magnetic and electric field as well as more than one charged particle we would need more equations to calculate the behavior of the charged particles.

One common use of the Lorentz Force law is in particle accelerators. As the name suggest a particle accelerator is an apparatus used to speed up a particle in order to study the behavior of particles in different scenarios. The most famous particle accelerator is the LHC, or large hadron collider in Europe. The LHC uses electromagnetic forces to speed up particles and collide them analyzing the radiation emitted from each collision. Using electromagnetic forces in a loop means that particle accelerators can speed up particles to incredible speeds nearing the speed of light.

See also

The Lorentz Force law was used to derive the Laplace force which considers all moving charges in a wire to show the magnetic force acting on the wire.

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