Magnetic Field of a Loop - Revision history

Joogps at 02:47, 29 April 2026

2026-04-29T02:47:45Z

← Older revision		Revision as of 22:47, 28 April 2026
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	For more on how inductors use these principles, see [[Inductors]].		For more on how inductors use these principles, see [[Inductors]].

	~~https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop#Electromagnetic_Induction~~		<br clear="all">

	==Connectedness==		==Connectedness==

Joogps at 02:47, 29 April 2026

2026-04-29T02:47:33Z

← Older revision		Revision as of 22:47, 28 April 2026
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	For more on how inductors use these principles, see [[Inductors]].		For more on how inductors use these principles, see [[Inductors]].

			https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop#Electromagnetic_Induction

	==Connectedness==		==Connectedness==

	Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.		Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

Joogps: /* Electromagnetic Induction */

2026-04-29T02:46:47Z

Electromagnetic Induction

← Older revision		Revision as of 22:46, 28 April 2026
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	==Electromagnetic Induction==		==Electromagnetic Induction==



	[[File:Induction_experiment.png\|thumb\|300px\|Faraday's electromagnetic induction experiment. Source: Wikimedia Commons, public domain.]]		[[File:Induction_experiment.png\|thumb\|300px\|Faraday's electromagnetic induction experiment. Source: Wikimedia Commons, public domain.]]
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	The negative sign reflects '''Lenz's Law''': the induced current flows in a direction that opposes the change in flux that caused it. For example, if the magnetic flux through a loop is increasing, the induced current will flow in the direction that creates a magnetic field opposing that increase — effectively trying to keep the flux constant. This is consistent with the right hand rule discussed above. If you know the direction of the induced magnetic field (opposing the change), curling the fingers of your right hand in the direction of the induced current will confirm its flow direction around the loop.		The negative sign reflects '''Lenz's Law''': the induced current flows in a direction that opposes the change in flux that caused it. For example, if the magnetic flux through a loop is increasing, the induced current will flow in the direction that creates a magnetic field opposing that increase — effectively trying to keep the flux constant. This is consistent with the right hand rule discussed above. If you know the direction of the induced magnetic field (opposing the change), curling the fingers of your right hand in the direction of the induced current will confirm its flow direction around the loop.

			[[File:Itaipu-Wasserkraftwerk_Generator.JPG\|left\|thumb\|300px\|One of the 700 MW generators inside the Itaipu hydroelectric dam. Source: Wikimedia Commons, CC BY-SA 3.0.]]

	This principle is exactly how electric generators work. In a simple generator, a loop of wire (or a coil with <math>N</math> turns) is rotated inside a uniform external magnetic field. As the loop spins, the angle <math>\theta</math> between the field and the loop's normal vector changes continuously, causing the flux <math>\Phi_B = B A \cos\theta</math> to vary sinusoidally with time. By Faraday's Law, this produces an alternating emf. For a coil of <math>N</math> turns rotating at angular velocity <math>\omega</math>, the induced emf is:		This principle is exactly how electric generators work. In a simple generator, a loop of wire (or a coil with <math>N</math> turns) is rotated inside a uniform external magnetic field. As the loop spins, the angle <math>\theta</math> between the field and the loop's normal vector changes continuously, causing the flux <math>\Phi_B = B A \cos\theta</math> to vary sinusoidally with time. By Faraday's Law, this produces an alternating emf. For a coil of <math>N</math> turns rotating at angular velocity <math>\omega</math>, the induced emf is:

	::<math>\mathcal{E} = N B A \omega \sin(\omega t)</math>		::<math>\mathcal{E} = N B A \omega \sin(\omega t)</math>


	~~[[File:Itaipu-Wasserkraftwerk_Generator.JPG\|left\|thumb\|300px\|One of the 700 MW generators inside the Itaipu hydroelectric dam. Source: Wikimedia Commons, CC BY-SA 3.0.]]~~

	This is the origin of alternating current (AC) electricity. The same loop geometry analyzed in the earlier sections of this page directly determines how much voltage a generator produces. Increasing the number of loops <math>N</math>, the area of each loop, the strength of the external magnetic field, or the rotation speed all increase the output. Power plants, whether they burn fuel, use nuclear reactions, or harness wind and water, all ultimately spin a coil inside a magnetic field to generate electricity using this principle.		This is the origin of alternating current (AC) electricity. The same loop geometry analyzed in the earlier sections of this page directly determines how much voltage a generator produces. Increasing the number of loops <math>N</math>, the area of each loop, the strength of the external magnetic field, or the rotation speed all increase the output. Power plants, whether they burn fuel, use nuclear reactions, or harness wind and water, all ultimately spin a coil inside a magnetic field to generate electricity using this principle.

	One of the most striking real-world examples of electromagnetic induction at scale is the Itaipu Dam, a binational hydroelectric plant on the Paraná River between Brazil and Paraguay. The dam's 20 generating units each produce 700 MW by channeling river water through massive turbines that spin generator coils inside strong magnetic fields, directly applying the mathematical relationship described above. With a total installed capacity of 14 GW, Itaipu is one of the largest hydroelectric plants in the world and demonstrates how the physics of a single current-carrying loop scales up to power entire countries.		One of the most striking real-world examples of electromagnetic induction at scale is the Itaipu Dam, a binational hydroelectric plant on the Paraná River between Brazil and Paraguay. The dam's 20 generating units each produce 700 MW by channeling river water through massive turbines that spin generator coils inside strong magnetic fields, directly applying the mathematical relationship described above. With a total installed capacity of 14 GW, Itaipu is one of the largest hydroelectric plants in the world and demonstrates how the physics of a single current-carrying loop scales up to power entire countries.

			For more on how inductors use these principles, see [[Inductors]].

	==Connectedness==		==Connectedness==

Joogps at 02:45, 29 April 2026

2026-04-29T02:45:06Z

← Older revision		Revision as of 22:45, 28 April 2026
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	*[[Biot-Savart Law for Currents]]<br>		*[[Biot-Savart Law for Currents]]<br>
	*[[Current]]<br>		*[[Current]]<br>
			*[[Inductors]]<br>

	==References==		==References==

Joogps: /* Electromagnetic Induction */

2026-04-29T02:42:33Z

Electromagnetic Induction

← Older revision		Revision as of 22:42, 28 April 2026
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	==Electromagnetic Induction==		==Electromagnetic Induction==

			[[File:Induction_experiment.png\|thumb\|300px\|Faraday's electromagnetic induction experiment. Source: Wikimedia Commons, public domain.]]

	The previous sections describe how a current flowing through a loop produces a magnetic field. The reverse process is equally important: a changing magnetic field through a loop can produce a current. This phenomenon, known as electromagnetic induction, was discovered by Michael Faraday in 1831 and is the basis for nearly all modern electricity generation.		The previous sections describe how a current flowing through a loop produces a magnetic field. The reverse process is equally important: a changing magnetic field through a loop can produce a current. This phenomenon, known as electromagnetic induction, was discovered by Michael Faraday in 1831 and is the basis for nearly all modern electricity generation.

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	::<math>\mathcal{E} = N B A \omega \sin(\omega t)</math>		::<math>\mathcal{E} = N B A \omega \sin(\omega t)</math>


			[[File:Itaipu-Wasserkraftwerk_Generator.JPG\|left\|thumb\|300px\|One of the 700 MW generators inside the Itaipu hydroelectric dam. Source: Wikimedia Commons, CC BY-SA 3.0.]]

	This is the origin of alternating current (AC) electricity. The same loop geometry analyzed in the earlier sections of this page directly determines how much voltage a generator produces. Increasing the number of loops <math>N</math>, the area of each loop, the strength of the external magnetic field, or the rotation speed all increase the output. Power plants, whether they burn fuel, use nuclear reactions, or harness wind and water, all ultimately spin a coil inside a magnetic field to generate electricity using this principle.		This is the origin of alternating current (AC) electricity. The same loop geometry analyzed in the earlier sections of this page directly determines how much voltage a generator produces. Increasing the number of loops <math>N</math>, the area of each loop, the strength of the external magnetic field, or the rotation speed all increase the output. Power plants, whether they burn fuel, use nuclear reactions, or harness wind and water, all ultimately spin a coil inside a magnetic field to generate electricity using this principle.

			One of the most striking real-world examples of electromagnetic induction at scale is the Itaipu Dam, a binational hydroelectric plant on the Paraná River between Brazil and Paraguay. The dam's 20 generating units each produce 700 MW by channeling river water through massive turbines that spin generator coils inside strong magnetic fields, directly applying the mathematical relationship described above. With a total installed capacity of 14 GW, Itaipu is one of the largest hydroelectric plants in the world and demonstrates how the physics of a single current-carrying loop scales up to power entire countries.

	==Connectedness==		==Connectedness==

Joogps at 02:32, 29 April 2026

2026-04-29T02:32:37Z

@@ Line 1: / Line 1: @@
-== Last Edited By Ava Williamson ==
+== Last Edited By João Pozzobon (Spring 2026) ==
 ==Main Idea==
 [[File: Mag Field in Loop.PNG|thumb|250px|Image depicting the magnetic field coming from a loop with a current.]]
@@ Line 121: / Line 121: @@
 ::::::<math>r = \sqrt{\frac{wh}{\pi}}</math>
+==Electromagnetic Induction==
 ==Connectedness==

Jkinoshita3 at 05:31, 24 November 2024

2024-11-24T05:31:36Z

← Older revision		Revision as of 01:31, 24 November 2024
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	~~== Claimed Joshua Kinoshita Fall 2024 ==~~
	== Last Edited By Ava Williamson ==		== Last Edited By Ava Williamson ==
	==Main Idea==		==Main Idea==

Jkinoshita3 at 05:31, 24 November 2024

2024-11-24T05:31:07Z

← Older revision		Revision as of 01:31, 24 November 2024
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			== Claimed Joshua Kinoshita Fall 2024 ==
	== Last Edited By Ava Williamson ==		== Last Edited By Ava Williamson ==
	==Main Idea==		==Main Idea==
	~~=Claimed By Joshua Kinoshita Fall 2024=~~
	[[File: Mag Field in Loop.PNG\|thumb\|250px\|Image depicting the magnetic field coming from a loop with a current.]]		[[File: Mag Field in Loop.PNG\|thumb\|250px\|Image depicting the magnetic field coming from a loop with a current.]]

Jkinoshita3 at 05:26, 24 November 2024

2024-11-24T05:26:44Z

← Older revision		Revision as of 01:26, 24 November 2024
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	== Last Edited By Ava Williamson ==		== Last Edited By Ava Williamson ==
	==Main Idea==		==Main Idea==
			=Claimed By Joshua Kinoshita Fall 2024=
	[[File: Mag Field in Loop.PNG\|thumb\|250px\|Image depicting the magnetic field coming from a loop with a current.]]		[[File: Mag Field in Loop.PNG\|thumb\|250px\|Image depicting the magnetic field coming from a loop with a current.]]

Awilliamson40 at 23:40, 15 November 2020

2020-11-15T23:40:44Z

← Older revision		Revision as of 19:40, 15 November 2020
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	===Computational Model===		===Computational Model===

	[Click to Run the Computational Model: https://www.glowscript.org/#/user/awilliamson40/folder/Public/program/Bfieldring]		[Click to Run the Computational Model of the Magnetic Field produced by a Loop on Axis: https://www.glowscript.org/#/user/awilliamson40/folder/Public/program/Bfieldring]

	==Examples==		==Examples==
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	::::First, It is important to recognize that the general equation		::::First, It is important to recognize that the general equation
	::::::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3}</math>		::::::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3}</math>
	::::is the one that will be used in this problem because the distance <math>d</math> is very large compared to the dimensions of the loops and it is perpendicular to the plane of the loops.		::::is the one that will be used in this problem because the distance <math>d</math> is very large compared to the dimensions of the loops, and it is perpendicular to the plane of the loops.
	::::		::::
	::::Next, Notice that the equation above can be rearranged as		::::Next, Notice that the equation above can be rearranged as
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	::::		::::
	::::The equation for the magnitude of the magnetic field for Loop 1 can be written as		::::The equation for the magnitude of the magnetic field for Loop 1 can be written as
	::::::<math>B_1 = \frac{\mu_0}{4 \pi} \frac{2 ~~\pi~~ I ~~r^2~~}{d^3}</math>		::::::<math>B_1 = \frac{\mu_0}{4 \pi} \frac{2 I}{d^3} {( \pi r^2 )}</math>
	::::where <math>r</math> is the unknown constant.		::::where <math>r</math> is the unknown constant.
	::::The equation for the magnitude of the magnetic field for Loop 2 can be written as		::::The equation for the magnitude of the magnetic field for Loop 2 can be written as
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	:::: where <math>r</math> is the unknown constant.		:::: where <math>r</math> is the unknown constant.
	::::		::::
	::::The problem states that the magnitudes of the magnetic fields are equal for a distance <math>d</math> above the center of each loop. This means we can set <math>B_1 = B_2</math> and solve for r.		::::The problem states that the magnitudes of the magnetic fields are equal for a distance <math>d</math> above the center of each loop. This means we can set <math>B_1 = B_2</math> and solve for <math>r</math>.
	::::::<math>B_1 = B_2</math>		::::::<math>B_1 = B_2</math>
	::::::<math>\frac{\mu_0}{4 \pi} \frac{2 ~~\pi~~ I ~~r^2~~}{d^3} = \frac{\mu_0}{4 \pi} \frac{2 I}{d^3} {(w h)}</math>		::::::<math>\frac{\mu_0}{4 \pi} \frac{2 I}{d^3} {( \pi r^2 )} = \frac{\mu_0}{4 \pi} \frac{2 I}{d^3} {(w h)}</math>
	::::Canceling out variables, we discover		::::Canceling out variables, we discover
	::::::<math>r^2 = wh</math>		::::::<math>r^2 = \frac{wh}{\pi}</math>
	::::Leading to the final answer		::::Leading to the final answer
	::::::<math>r = \sqrt{wh}</math>		::::::<math>r = \sqrt{\frac{wh}{\pi}}</math>


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	==History==		==History==
	~~::'''Insert History here'''~~		Hans Christian Ørsted discovered that electric currents produce a magnetic field while experimenting with currents changing the magnetic fields of nearby magnets. André-Marie Ampère found that objects with electrical currents attract each other similar to magnets, so he then hypothesized that magnetic fields are due to loops of electric current. Ampère then found that the magnetic dipole moment is essentially equal to the conventional current times the cross sectional area of a loop. The magnetic moment for a magnet was found to relate to the current in a loop in the following manner <math>\mu(magnet) = IA</math> where <math>A</math> is the cross-sectional area of the magnet.

	==See also==		==See also==
	===Further Reading===		===Further Reading===
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	*Matter and Interactions Vol. II		*Matter and Interactions Vol. II
	*HyperPhysics, Magnetic Field of a Loop		*HyperPhysics, Magnetic Field of a Loop
			*Wikipedia, Magnetic moment [https://en.wikipedia.org/wiki/Magnetic_moment#Amperian_loop_model]
			*Glowscript code and "Difficult Problem" created by Ava Williamson

	[[Category:Fields]]		[[Category:Fields]]