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Biot-Savart Law

2020-10-13T17:32:33Z

Eclayton30:

CLAIMED BY AVINASH SIVAKUMAR SPRING 2019
[[File:Knight2_33.f.14.png|400px|thumb|right|A Visual of the Magnetic Field of a Current-Carrying Wire. [http://www4.uwsp.edu/physastr/kmenning/Phys250/Lect19.html]]]The Biot-Savart (pronounced bee-yo sahv-ar) Law quantitatively describes the magnetic field produced by a moving point charge. This law can be viewed as the magnetic counterpart of Coulomb's Law, which quantitatively describes the electric field produced by a point charge. [http://dev.physicslab.org/document.aspx?doctype=3&filename=magnetism_biotsavartlaw.xml] The law is named after Jean-Baptiste Biot and Felix Savart, the two scientists who discovered the law in 1820.

==The Main Idea==

A moving point charge produces a magnetic field in addition to its intrinsic electric field. As opposed to an electric field, which eminates outward in all directions, the magnetic field of a point charge will curl around the point charge.

===A Mathematical Model===

The single moving point charge form of the Biot-Savart Law is <math>\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v \times \hat r}{r^2} </math>. The permeability of free space, or the ease of producing a magnetic field in a vacuum, is a constant <math> 4\pi \times 10^{-7} \frac{T * m^2}{C * \frac{m}{s}} </math>, denoted <math> \mu_0 </math>. The above equation describes the magnetic field vector <math> \vec B </math> at a point <math> \hat r </math> away from the moving point charge with velocity <math> \vec v </math> and charge <math> q </math>. '''Remember''' that <math> \vec r = \vec r_{obs} - \vec r_{source} </math>. '''Note:''' The charge <math> q </math> retains its sign in the Biot-Savart Law. For an electron, <math> q </math> in the Biot-Savart Law would be equal to <math> -1.6 \times 10^{-19} C </math>.

The short wire form of the Biot-Savart Law is <math>\vec B=\frac{\mu_0}{4\pi}\frac{I \Delta\vec l \times \hat r}{r^2} </math>. This equation describes the magnetic field vector <math> \vec B </math> at a point <math> \hat r </math> away from the short wire with length <math> \Delta\vec l </math> and current <math> I </math>.

'''Direction of the Magnetic Field Vector:'''
The direction of the magnetic field vector is different from the direction of the electric field vector. The direction of the magnetic field vector is given by the cross product of two distinct vectors: namely either <math> \vec v </math> or <math> \Delta\vec l </math> and <math> \hat r </math>. The cross product of two vectors is perpendicular to the two original vectors.

Example: Find the direction of the magnetic field of a point charge moving in the <math> +x </math> direction at a point in the <math> -y </math> direction.

Direction of <math> \vec B = \hat v \times \hat r = <1, 0, 0> \times <0, -1, 0> = \begin{bmatrix}\hat x & \hat y & \hat z \\ 1 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix} = <0, 0, -1> </math>, or in the <math> -z </math> direction

You can also calculate the cross product, using formula <math> \vec A\times\vec B = < A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y-A_yB_z> </math>

'''Magnitude of Magnetic Field Vector:'''
The magnitude of the magnetic field vector for a single moving point charge is given by <math> B = \frac{\mu_0}{4\pi} \frac{q v \sin\theta}{r^2} </math>, where <math> q v \sin\theta </math> is the polar form of the cross product and <math> \theta </math> is the angle between the two vectors <math> \vec v </math> and <math> \hat r </math>.

The magnitude of the magnetic field vector for a short wire is given by <math> B = \frac{\mu_0}{4\pi} \frac{I \Delta l \sin\theta}{r^2} </math>, where <math> I \Delta l \sin\theta </math> is the polar form of the cross product and <math> \theta </math> is the angle between the two vectors <math> \Delta\vec l </math> and <math> \hat r </math>.

===A Computational Model===
Check out this link: https://trinket.io/glowscript/93785debae for a look at a moving point charge acted on by a magnetic force. This magnetic force acts on the point charge since the point charge is ''moving'' with a certain velocity. Magnetic force goes hand-in-hand with magnetic field. Just as a point charge can only create a magnetic field if the point charge is moving, a point charge can only feel the force from a magnetic field if the point charge has a velocity. This computational model shows how a moving point charge, which creates its own magnetic field, is acted on by an external magnetic field.

For a computational model of a wire carrying current, see the wiki page on the Biot-Savart Law for Currents [http://www.physicsbook.gatech.edu/Biot-Savart_Law_for_Currents].

===Using the Right-Hand Rule for Biot-Savart Law===

Put your index finger in the direction of the velocity of the particle, and your middle finger out towards the r hat. Make sure these two are 90 degrees away from each other. Now, stick your thumb up. If the particle you are following is positive, the cross product of the velocity of the particle and the r hat is in the direction of your thumb. If the particle is negative, the cross product of the particle and the r hat is in the opposite direction of your thumb.

[[File:20190414 183245.jpg|center|300px]]

==Examples==

'''See below for solutions to these examples.'''

===Example 1===
An electron located at the origin is moving at <math> 2 \times 10^8 \frac{m}{s}</math> in the <math> +x </math> direction. What is the magnetic field at <math> <200, -300, 0> m </math> due to the moving electron?

===Example 2===
A proton is located on the <math> +x </math> axis and is moving with a velocity <math> \vec v </math> in the <math> -y </math> direction. What is the magnitude of the magnetic field due to the moving proton at <math> <0, 0, z> </math> if the angle <math> \theta </math> between <math> \vec v </math> of the proton and <math> \hat r </math> equals <math> 34° </math>?

===Example 3===
An electron located at the origin is moving with velocity <math> \vec v_{electron} </math> in the <math> +z </math> direction. A proton is located directly in front of the electron and is moving with a velocity <math> \vec v_{proton} </math> in the <math> -y </math> direction. What is the magnetic field at the location of the proton due to the moving electron?

===Example 4===
A thin plastic disk with radius <math> 500 cm </math> and negative charge <math> -6 \times 10^{-7} C </math> centered at the origin located in the <math> y-z </math> plane is rotating with a period of <math> 7 s </math> clockwise when viewed from the negative x axis. Determine the magnetic field at <math> \vec d = <-50000, 0, 0> cm. </math>

===Answers===

'''Example 1:'''

[[File:ex1.JPG|center|300px|Example 1]]

First, calculate <math> \vec r </math>, which is <math> \vec r = <200, -300, 0>-<0,0,0>=<200, -300, 0> m </math>. Using <math> \vec r </math>, calculate <math> \hat r </math>:
<math> \hat r = \frac{ <200, -300, 0> }{\sqrt {200^2 + -300^2}}= <0.554, -0.832, 0> </math>

Next, calculate <math> q \vec v </math>. Remember that an electron carries a negative charge: <math> -1.6 \times 10^{-19}* <2 \times 10^8, 0, 0> = <-3.2 \times 10^{-11}, 0, 0> </math>.
Next, compute the cross product, <math> q\vec v\times\hat r </math>. The cross product is given as <math> \vec A\times\vec B = < A_xB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y-A_yB_z> </math> or by computing the determinant of the two vectors in a 3 x 3 matrix. The cross product in this case is <math> <0, 0, 2.65 \times 10^{-11}> </math>.

Finally, divide <math> q\vec v\times\hat r </math> by the magnitude of <math> \vec r </math> squared: <math> \frac{ <0, 0, 2.65 \times 10^{-11}> }{360.55} = <0, 0, 7.34 \times 10^{-14}> </math>. Multiply by the constant <math> 1 \times 10^{-7} </math> given in the formula, and the final answer is <math> \vec B = <0, 0, 7.34 \times 10^{-21}> T </math>

'''Example 2:'''

[[File:ex2.JPG|center|300px|Example 2]]

First, as in example one, calculate <math> \vec r </math>, and then the magnitude of <math>\vec r </math>. <math> \vec r = <0, 0, z> - < x, 0, 0> = <-x, 0, z> </math>. The magnitude of <math>\vec r </math> is then <math> \sqrt{x^2 + z^2} </math>.
We can then multiply the magnitudes of <math> q,\vec v, </math> and <math> sin </math> (<math>\theta </math>) to get the final answer:

<math> B = \frac{\mu_0}{4\pi}\frac{qvsin(34°)}{(\sqrt{x^2 + z^2})^2} </math>

'''Example 3:'''

[[File:ex3.JPG|center|300px|Example 3]]

As seen in the diagram, <math> \hat r </math> from the proton to the electron is parallel to the electron's velocity. Therefore, <math> \vec v_{electron}\times\hat r </math> is zero because the two vectors are parallel (<math> \theta=0 </math>°).

'''Example 4:'''

[[File:Ex4.jpg|center|300px|Example 4]]

To find the overall magnetic field vector at <math> \vec d </math>, the disk must be broken down into many infinitesimally small rings.

Step 1: Find <math> dQ </math>, the amount of charge for an infinitesimal ring

<math> \frac{dQ}{2\pi r dr} = \frac{Q}{\pi R^2} </math>

<math> dQ = \frac{-6 \times 10^{-7}}{5^2\pi} 2\pi r dr = \frac{-12 \times 10^{-7}rdr}{25} </math>

Step 2: Find <math> dI </math>, the current in this infinitesimal ring

<math> dI = \frac{dQ}{T} = \frac {\frac{-12 \times 10^{-7}rdr}{25}}{7} = \frac{-12 \times 10^{-7}rdr}{175} </math>

Step 3: Find <math> d\vec B </math>, the magnetic field vector for the infinitesimal ring

<math> d\vec B = \frac {\mu_0}{4\pi} \frac {2 dI \pi R^2}{x^3} \hat x </math>

<math> d\vec B (<-500, 0, 0>) = \frac {\mu_0}{4\pi} \frac {2 \frac{-12 \times 10^{-7}rdr}{175} \pi 5^2}{-500^3} \hat x = \frac{24 \times 10^{-14}rdr\pi}{7*500^3} \hat x </math>

Step 4: Find <math> \vec B </math>, the magnetic field vector over the entire disk

<math> \vec B = \int\limits_0^R\ d\vec B </math>

<math> \vec B(<-500, 0, 0>) = \int\limits_0^5\ \frac{24 \times 10^{-14}rdr\pi}{7*500^3} \hat x = \frac {25}{2} \frac{24 \times 10^{-14} \pi}{7*500^3} T </math>

==Applications==
===Magnetic Response===
The Biot-Savart law has applications in nuclear magnetic resonance (NMR) spectroscopy, used to measure the chemical signals given off by compounds. The law can be used to calculate the magnetic responses at the atomic or molecular level, provided that the current density can be obtained mathematically. For more about NMR spectroscopy, see the wiki page [https://en.wikipedia.org/wiki/Nuclear_magnetic_resonance] [[File:Vortex.jpg|200px|thumb|right|In this example,[http://www.science20.com/news_articles/tendex_and_vortex_lines_new_way_visualize_warped_space_and_time-78171], the yellow arms represent whirring space from a black hole and the red lines represent vortex lines. Astrophysicists may use the Biot-Savart Law to calculate the velocity.]]
===Aerodynamics===
In aerodynamics, the Biot-Savart law may be used to calculate the velocity induced by vortex lines, which are lines that are everywhere tangent to the vorticity vector. A vorticity vector is a pseudovector field that describes the tendency of something to rotate; in other words, the vorticity vector is the curl of the velocity field of a fluid.

===Medical Technology===
Aside from applications in aerospace engineering and chemistry, the Biot-Savart law also plays an important role in magnetic resonance imaging (MRI). MRIs are a crucial piece of technology in the medical field; they are used for diagnosing cancer, detecting stress fractures, analyzing musculoskeletal injuries, and much more. An MRI machine is essentially one large magnetic field, and the hydrogen proton's axes in the human body all line up when a person is placed in an MRI. The magnetic field in an MRI is created by electric coils with moving electrons, which carry a current through the coils. This current is produced by exposing moving electrons to the electric field in the coils. if the moving electrons did not create magnetic fields, then no current would run through the coils, and MRIs would not exist.

==History==

Felix Savart was born on June 30, 1791 in Mezieres, France to a family with a strong association with military engineering schools. While completing his formal education in 1808 at the university in Metz, Savart decided to pursue medicine and become a physician. After serving a short stint in Napoleon's army in the the first engineering battalion, he resumed his medical training and graduated from Strasbourg in 1816. During his medical studies, Savart became interested in first century Roman writer Aulus Cornelius Celsus and his famous medical book ''De medicinia''. Savart began working on a translation and set up a medical practice in Metz in 1817, but he gradually became more interested in physics rather than patients, particularly in sound and acoustics. He began building violins as a way to explore the form of the instrument through mathematical principles.

In 1819, Savart officially closed the doors of his medical practice and went to Paris to find a publisher for the translation of ''De medicina''. While there, he attended a lecture on acoustics by Jean-Baptiste Biot at the Faculty of Sciences. The two met there and began collaboration, when in 1820, Hans Christian Oersted published his findings regarding a compass needle's behavior when placed near a current-carrying wire: the needle pointed at right angles to the wire. Biot and Savart began looking more closely into the field produced by a current-carrying wire, and by using the oscillation of a magnetic dipole to determine the strength of the field close to a current-carrying wire, they discovered what is now called the Biot-Savart Law [[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Savart.html]].

== See also ==
Right hand rule [[http://www.physicsbook.gatech.edu/Right-Hand_Rule]]
Biot-Savart Law for Currents [[http://www.physicsbook.gatech.edu/Biot-Savart_Law_for_Currents]]
Jean-Baptiste Biot [[http://www.physicsbook.gatech.edu/Jean-Baptiste_Biot]].

===External links===

"Teach Engineering" on Biot-Savart [https://www.teachengineering.org/view_lesson.php?url=collection/van_/lessons/van_mri_lesson_5/van_mri_lesson_5.xml]
Derivation and Examples [http://study.com/academy/lesson/the-biot-savart-law-derivation-examples.html]
Biot-Savart and Ampere's Law on MIT OpenCourse [http://ocw.mit.edu/high-school/physics/exam-prep/magnetic-fields/biot-savarts-law-amperes-law/]

==References==

Matter & Interactions vol. II

[[Category:Which Category did you place this in?]]

Biot-Savart Law for Currents

2020-10-07T19:11:21Z

Eclayton30: /* Current Loop Integration Example */

Vaishnavi Ramanan - Fall 2018

==Biot-Savart Law==

The Biot-Savart Law can be used for more than just single moving charges; a notable application of this law is its ability to calculate the magnetic field for an extremely large number of charges - an example of thousands of charges moving together is within a current carrying wire (current being the amount of charges moving over a specific amount of time).

When using Biot-Savart Law to find the magnetic field of a short wire, we can extend this concept to a variety of different shapes - long current carrying wires, current carrying loops, etc.

*One main point to note is that the application of the Biot-Savart law is specifically for steady state current (current that is not changing over time ----- <math> dI/dt = 0 </math> )!

====A Mathematical Model====
First We start off with the original version of the Biot-Savart Law.
<math>\vec B=\frac{\mu_0}{4 \pi } \frac{q\vec v\times\hat r}{r^2}</math>, where <math> \frac{\mu_0}{4 \pi } = 1 \times 10^{-7}\frac{Tm^2}{Cm/s}, </math>

Because we are dealing with a portion of wire <math>\mathrm{d}\boldsymbol{\ell}</math> long with an Area A containing n moving particles with charge q, we find that the total number of moving charges is equal to |q|(nAv) which is also equal to I, the current in the wire.

<math>B = \frac{\mu_0I}{4\pi}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>

Because the shape of the current carrying wire can vary from a straight wire to a loop, we must integrate over the region of the wire.

<math>\Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>

The key point is that there are <math> nA\Delta l </math> electrons in a short length of wire, each moving with average speed <math>\vec v </math>, so that the sum of all the <math> q\vec v </math> contributions is <math> nA \Delta l|q|\vec v = I\Delta l. </math>

When applying the Biot-Savart Law to a Long Straight Wire, we follow a set of steps,

Step 1: Cut Up the Distribution into Pieces (<math> \Delta l </math>) and Draw <math> \Delta B </math>.

Step 2: Write an Expression for the Magnetic Field Due to One Piece.

Step 3: Add Up the Contributions of All the Pieces (each <math> \Delta l </math> adds together to make the entirety of the length).

Step 4: Check the Result.

Where the Magnetic Field of a Straight Wire is shown by,

<math> B = \frac{\mu_0}{4\pi}\frac{LI}{r(r^2 + (L/2)^2)^{1/2}} </math> for length <math> L </math>, conventional current <math> I </math>, a perpendicular distance <math> r </math> from the center of the wire, or,
<math> B = \frac{\mu_0}{4\pi}\frac{2I}{r} </math> if <math> L\>>\>> r. </math>

==Right Hand Rule== [[File:IMG_2467.jpg|200px|thumb|alt text]]

When using the Biot-Savart Law for Currents, it is crucial to understand the direction of the magnetic field created by a current.

In order to get the direction of the Magnetic Field, as shown in the image on the right, we:

1. Point our thumb in the direction of the current, and

2. automatically curl our fingers around in one direction.

This curling direction is the direction of the magnetic field itself.

==Long Wire Integration Example==
[[File:IMG_2469.jpg|200px|thumb|alt text]]
====For a long wire of length L positioned along the x axis with current flowing in the positive x direction (as shown by image on right)====
First, we start off with our adjusted Biot-Savart Formula for a slice of wire.

<math>\Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>

Second, we must find <math>r</math>, the vector pointing from the source to the observation location. In this case, we will choose an observation location y above the rod.

<math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0> </math>. which has a magnitude of <math>\sqrt(x^2+y^2)</math>

We see that <math>\hat r = \frac{r}{|r|} </math> .

<math>\hat r = \frac{<-x,y,0>}{\sqrt(x^2+y^2))}</math>

We then have to express <math> \Delta \boldsymbol{\ell}</math> in terms of our variable of integration, x.
<math> \Delta \boldsymbol{\ell}</math> = <math> \Delta x<1,0,0,></math>

Our new equation after substituting our new variables is
<math> \Delta B = \frac{\mu_0I\Delta x<1,0,0,>}{4\pi(x^2+y^2)} \times \frac{<-x,y,0>}{\sqrt(x^2+y^2))} </math>

Finding the cross product of the above vectors gives us a product in the +z direction.
<math> \Delta B = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})} </math>

We are finally ready to integrate. Because we are integrating the entire rod our limits are

<math> \int\limits_{-L/2}^{L/2}\ </math> <math> \int\limits_{-L/2}^{L/2}\ = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})} </math>

We find that our final answer is <math> B= \frac{\mu_0}{4\pi}\frac{LI}{y\sqrt(y^2+(L/2)^2)}\hat z </math>

==Current Loop Integration Example==
'''Saved for Elizabeth Clayton 10/7/2020'''

==A Computational Model==
The following link shows the magnetic field produced by small segments of wire in a loop individually.
For a long straight wire, we see that there is a circular magnetic field surrounding the wire with current. The following link does a stepwise visual of the contributions of each part of the wire at an observation location a distance r from the wire.

http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-Bwire-with-r

We see that along the axis of the wire, each contribution not on the axis is negated due to symmetry and the resulting magnetic field is all along the wire.

http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-B-loop-with-r-dB.

== See also ==
[Right Hand Rule[http://www.physicsbook.gatech.edu/Right-Hand_Rule]]

[Direction of magnetic fields[http://www.physicsbook.gatech.edu/Direction_of_Magnetic_Field]]
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

==External Links==
[[Category:Which Category did you place this in?]]