Biot-Savart Law - Revision history

Jluce8 at 14:44, 8 April 2025

2025-04-08T14:44:31Z

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			CLAIMED BY Jonathan Luce Spring 2025

	CLAIMED BY Grant Reidy Fall 2024		CLAIMED BY Grant Reidy Fall 2024

Greidy3: Added vPython Link

2024-11-22T22:47:12Z

Added vPython Link

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	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]].		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]].

			== VPython Demo==

			[Interactive VPython Biot-Savart Demo](https://trinket.io/glowscript/877d2b5b77)

	== See also ==		== See also ==

Greidy3: Made clerical edits and added clarifications to wording on certain items on the page.

2024-11-22T22:42:28Z

Made clerical edits and added clarifications to wording on certain items on the page.

← Older revision		Revision as of 18:42, 22 November 2024
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			CLAIMED BY Grant Reidy Fall 2024

	CLAIMED BY Alexander Arredondo Fall 2023		CLAIMED BY Alexander Arredondo Fall 2023

	CLAIMED BY Olivia Wang Fall 2023		CLAIMED BY Olivia Wang Fall 2023
	[[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.		[[File:Knight2_33.f.14.png\|400px\|thumb\|right\|Visualization of the magnetic fields generated around 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 serves as the magnetic analog to Coulomb's Law, which describes electrical forces between static charges. [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==		==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 moving point charge produces a magnetic field in addition to its intrinsic electric field. Unlike the electric field that radiates outward symmetrically from a point charge, the magnetic field forms circular loops around the moving charge.

	The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb's law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing. The relationship between the magnetic field ~~contribution and its source current element is called the Biot-Savart law~~.		The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb's law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing. The Biot-Savart Law mathematically expresses the relationship between a current element and the resulting magnetic field it produces.

	The direction of the magnetic field contribution follows the right hand rule illustrated for a straight wire. This direction arises from the vector product nature of the dependence upon electric current.		The direction of the magnetic field contribution follows the right hand rule illustrated for a straight wire. This direction arises from the vector product nature of the dependence upon electric current.
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	===A Mathematical Model===		===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 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> helps locate the observation point relative to the source of the magnetic field. '''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>.		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>.
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	'''Magnitude of Magnetic Field Vector:'''		'''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 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>. Remember that this formula specifies spherical coordinates, where <math> \theta </math> is the angle between the velocity vector and the radial vector <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>.		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>.
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	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>		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>

			The determinant method effectively calculates the three-dimensional cross-product, which ensures the magnetic field vector is perpendicular to both original vectors.

	'''Example 2:'''		'''Example 2:'''

Aarredondo6 at 18:47, 28 November 2023

2023-11-28T18:47:09Z

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	CLAIMED BY ~~AVINASH SIVAKUMAR SPRING 2019~~		CLAIMED BY Alexander Arredondo Fall 2023

	CLAIMED BY Olivia Wang Fall 2023		CLAIMED BY Olivia Wang Fall 2023

Ywang4155: /* Solving Biot-Savart Problem strategy */

2023-11-27T19:51:04Z

Solving Biot-Savart Problem strategy

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	===Solving Biot-Savart Problem strategy===		===Solving Biot-Savart Problem strategy===
	To solve Biot-Savart law problem, the following steps are helpful:		To solve Biot-Savart law problem, the following steps are helpful:

	1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing 𝐁⃗ and 𝐝𝐥⃗ ,Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law.		1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing 𝐁⃗ and 𝐝𝐥⃗ ,Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law.

Ywang4155 at 19:50, 27 November 2023

2023-11-27T19:50:38Z

← Older revision		Revision as of 15:50, 27 November 2023
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	To solve Biot-Savart law problem, the following steps are helpful:		To solve Biot-Savart law problem, the following steps are helpful:
	1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing 𝐁⃗ and 𝐝𝐥⃗ ,Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law.		1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing 𝐁⃗ and 𝐝𝐥⃗ ,Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law.

	2. Draw the current element length 𝑑𝐥⃗ and the unit vector 𝐫ˆ, noting that 𝑑𝐥⃗ points in the direction of the current and 𝐫ˆ points from the current element toward the point where the field is desired.		2. Draw the current element length 𝑑𝐥⃗ and the unit vector 𝐫ˆ, noting that 𝑑𝐥⃗ points in the direction of the current and 𝐫ˆ points from the current element toward the point where the field is desired.

	3. Calculate the cross product 𝑑𝐥⃗ ×𝐫ˆ. The resultant vector gives the direction of the magnetic field according to the Biot-Savart law.		3. Calculate the cross product 𝑑𝐥⃗ ×𝐫ˆ. The resultant vector gives the direction of the magnetic field according to the Biot-Savart law.

	4. Use Equation and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration.		4. Use Equation and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration.

	5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part.		5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part.

Ywang4155 at 19:50, 27 November 2023

2023-11-27T19:50:05Z

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

			===Solving Biot-Savart Problem strategy===
			To solve Biot-Savart law problem, the following steps are helpful:
			1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing 𝐁⃗ and 𝐝𝐥⃗ ,Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law.
			2. Draw the current element length 𝑑𝐥⃗ and the unit vector 𝐫ˆ, noting that 𝑑𝐥⃗ points in the direction of the current and 𝐫ˆ points from the current element toward the point where the field is desired.
			3. Calculate the cross product 𝑑𝐥⃗ ×𝐫ˆ. The resultant vector gives the direction of the magnetic field according to the Biot-Savart law.
			4. Use Equation and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration.
			5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part.

	==Examples==		==Examples==

Ywang4155 at 18:47, 27 November 2023

2023-11-27T18:47:07Z

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	===Sagged Conductors for Three Phase typical tower===		===Sagged Conductors for Three Phase typical tower===
	Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower		Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower
	https://onlinelibrary.wiley.com/doi/10.1002/cae.20365		https://onlinelibrary.wiley.com/doi/10.1002/cae.20365 (2016)
	The Biot-Savart law is a fundamental principle in electromagnetism that describes the magnetic field produced by a steady current. It is commonly used to calculate the magnetic field around a current-carrying conductor.		The Biot-Savart law is a fundamental principle in electromagnetism that describes the magnetic field produced by a steady current. It is commonly used to calculate the magnetic field around a current-carrying conductor.
	When applying the Biot-Savart law to sagged conductors in a three-phase transmission line on a typical tower, several factors need to be considered. Sagged conductors are often used in overhead transmission lines to accommodate thermal expansion and contraction as well as mechanical loads caused by wind.		When applying the Biot-Savart law to sagged conductors in a three-phase transmission line on a typical tower, several factors need to be considered. Sagged conductors are often used in overhead transmission lines to accommodate thermal expansion and contraction as well as mechanical loads caused by wind.
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	Matter & Interactions vol. II		Matter & Interactions vol. II

			Khawaja, Arsalan & Huang, Qi. (2016). Characteristic estimation of high voltage transmission line conductors with simultaneous magnetic field and current measurements. 1-6. 10.1109/I2MTC.2016.7520595.

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

Ywang4155 at 18:45, 27 November 2023

2023-11-27T18:45:49Z

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	==Applications==		==Applications==
			===Sagged Conductors for Three Phase typical tower===
	Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower		Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower
	https://onlinelibrary.wiley.com/doi/10.1002/cae.20365		https://onlinelibrary.wiley.com/doi/10.1002/cae.20365
			The Biot-Savart law is a fundamental principle in electromagnetism that describes the magnetic field produced by a steady current. It is commonly used to calculate the magnetic field around a current-carrying conductor.
			When applying the Biot-Savart law to sagged conductors in a three-phase transmission line on a typical tower, several factors need to be considered. Sagged conductors are often used in overhead transmission lines to accommodate thermal expansion and contraction as well as mechanical loads caused by wind.
	===Magnetic Response===		===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.]]		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.]]

Ywang4155 at 18:40, 27 November 2023

2023-11-27T18:40:34Z

← Older revision		Revision as of 14:40, 27 November 2023
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	==Applications==		==Applications==
	Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower		Application on Biot-Savart law for Sagged Conductors for Three Phase typical tower
	~~[[File~~:~~Biot-Savart_Law_Application~~.~~png\|center\|150px\|application on Biot-Savart]]~~		https://onlinelibrary.wiley.com/doi/10.1002/cae.20365
	===Magnetic Response===		===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.]]		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.]]