Path Independence of Electric Potential - Revision history

Ishan Khambete at 03:28, 4 December 2024

2024-12-04T03:28:29Z

← Older revision		Revision as of 23:28, 3 December 2024
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	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

	[[File:PathI.jpg\|~~400px\|thumb\|Figure 1: Path Independence~~]]		[[File:PathI.jpg\|200px]]


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	===2nd Computational Model===		===2nd Computational Model===

	<iframe src="https://trinket.io/embed/~~python~~/~~abc123~~" width="~~600~~" height="~~400~~"></iframe>		<iframe src="https://trinket.io/embed/glowscript/632d6d4225eb?outputOnly=true&start=result" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

			==Examples==

			===Simple Example===

			[[File:Example_Problem.jpg\|200px]]
			[[File:Solution_Part_1.jpg\|200px]]
			[[File:Solution_Part_2.jpg\|200px]]


	~~==Examples==~~
	===Example 1===		===Example 1===
	This a basic example of using path independence to simplify the calculation of potential difference.		This a basic example of using path independence to simplify the calculation of potential difference.
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	<math> \Delta V = -\frac{qs}{4\pi\epsilon_0d^2} </math></div>		<math> \Delta V = -\frac{qs}{4\pi\epsilon_0d^2} </math></div>




	==Relation to Multivariable Calculus==		==Relation to Multivariable Calculus==

Ishan Khambete at 01:42, 4 December 2024

2024-12-04T01:42:57Z

← Older revision		Revision as of 21:42, 3 December 2024
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	which breaks up the potential difference into the potential difference across the x component, then the y component, then the z component. If you can break it into 3 different paths this way, then why not any other way?		which breaks up the potential difference into the potential difference across the x component, then the y component, then the z component. If you can break it into 3 different paths this way, then why not any other way?

	==Path Independence==		==Why is Path Independence Useful==
	Now that we know some logical reasoning behind why potential has the quality of path independence, what is it?		Now that we know some logical reasoning behind why potential has the quality of path independence, what is it?

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	Moreover, path independence allows us to do some tricks when calculating potential difference. Calculating potential difference for a path through constant electric field is much simpler than calculating potential difference for a path through changing electric field - the latter requires you to integrate an electric field expression, which often times can get messy. By taking advantage of path independence, it's often possible to draw a path from one point to another such that there's constant electric field, and in cases where you do have a changing field, drawing an effective path is often a critical part of getting the solution.		Moreover, path independence allows us to do some tricks when calculating potential difference. Calculating potential difference for a path through constant electric field is much simpler than calculating potential difference for a path through changing electric field - the latter requires you to integrate an electric field expression, which often times can get messy. By taking advantage of path independence, it's often possible to draw a path from one point to another such that there's constant electric field, and in cases where you do have a changing field, drawing an effective path is often a critical part of getting the solution.

			==Models==

	===Mathematical Model===		===Mathematical Model===
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	The above is a visual and computational representation of the idea of path independence. Two different particles, one blue and one red, will begin in the bottom corner and take different paths to reach the top corner. The blue particle will take a direct straight-line path while the red particle will take a more indirect path. After each step forward (done by clicking on the scene), the change in potential from the last step and total change in potential will be printed for each particle. Note that at the end of the program, when both particles have reached the top corner, their total changes will be the same! Point A is marked by a green sphere, Point B is marked by a red sphere, moving negative particle 1 is marked by a blue sphere and trail, while moving negative particle 2 is marked by a white sphere and trail. Remember to scroll down in the display window to make sure you've seen the most recent printed value!		The above is a visual and computational representation of the idea of path independence. Two different particles, one blue and one red, will begin in the bottom corner and take different paths to reach the top corner. The blue particle will take a direct straight-line path while the red particle will take a more indirect path. After each step forward (done by clicking on the scene), the change in potential from the last step and total change in potential will be printed for each particle. Note that at the end of the program, when both particles have reached the top corner, their total changes will be the same! Point A is marked by a green sphere, Point B is marked by a red sphere, moving negative particle 1 is marked by a blue sphere and trail, while moving negative particle 2 is marked by a white sphere and trail. Remember to scroll down in the display window to make sure you've seen the most recent printed value!

			===2nd Computational Model===

			<iframe src="https://trinket.io/embed/python/abc123" width="600" height="400"></iframe>


	==Examples==		==Examples==

Ishan Khambete at 00:58, 4 December 2024

2024-12-04T00:58:35Z

← Older revision		Revision as of 20:58, 3 December 2024
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	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

	[[File:~~Img 8031~~.~~JPG~~\|400px\|thumb\|Figure 1: Path Independence]]		[[File:PathI.jpg\|400px\|thumb\|Figure 1: Path Independence]]

Ishan Khambete at 00:52, 4 December 2024

2024-12-04T00:52:16Z

← Older revision		Revision as of 20:52, 3 December 2024
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	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

	[[File:Img ~~8041~~.JPG\|400px\|thumb\|Figure 1: Path Independence]]		[[File:Img 8031.JPG\|400px\|thumb\|Figure 1: Path Independence]]

Ishan Khambete at 00:51, 4 December 2024

2024-12-04T00:51:41Z

@@ Line 1: / Line 1: @@
-''Edited by Josh Whitley Spring 2016, Emilie Pourchet Fall 2016, William Nute Fall 2017''
+'''CLAIMED BY ISHAN KHAMBETE FALL 2024'''
 See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.
 ==Why is Potential Path Independent?==

Ishan Khambete at 23:34, 3 December 2024

2024-12-03T23:34:36Z

← Older revision		Revision as of 19:34, 3 December 2024
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	'''CLAIMED BY WILLIAM NUTE FALL 2017'''		'''CLAIMED BY WILLIAM NUTE FALL 2017'''
			'''CLAIMED BY ISHAN KHAMBETE FALL 2024'''

	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

Keerthi Padmanabhan at 20:14, 16 April 2023

2023-04-16T20:14:05Z

← Older revision		Revision as of 16:14, 16 April 2023
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	''Edited by Josh Whitley Spring 2016, Emilie Pourchet Fall 2016, William Nute Fall 2017~~, Keerthi Padmanabhan Spring 2023~~''		''Edited by Josh Whitley Spring 2016, Emilie Pourchet Fall 2016, William Nute Fall 2017''

	'''CLAIMED BY WILLIAM NUTE FALL 2017'''		'''CLAIMED BY WILLIAM NUTE FALL 2017'''
	~~Claimed by Keerthi Padmanabhan, Spring 2023~~

	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

	~~[[File:PathIndependenceDiagram.jpeg]]~~

Keerthi Padmanabhan at 20:08, 16 April 2023

2023-04-16T20:08:41Z

← Older revision		Revision as of 16:08, 16 April 2023
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	See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.		See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

	[[File:~~40274~~.~~jpg\|frameless\|upright=3\|center~~]]		[[File:PathIndependenceDiagram.jpeg]]

Keerthi Padmanabhan at 20:04, 16 April 2023

2023-04-16T20:04:57Z

← Older revision		Revision as of 16:04, 16 April 2023
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	''Edited by Josh Whitley Spring 2016, Emilie Pourchet Fall 2016, William Nute Fall 2017''		''Edited by Josh Whitley Spring 2016, Emilie Pourchet Fall 2016, William Nute Fall 2017, Keerthi Padmanabhan Spring 2023''

	'''CLAIMED BY WILLIAM NUTE FALL 2017'''		'''CLAIMED BY WILLIAM NUTE FALL 2017'''
			Claimed by Keerthi Padmanabhan, Spring 2023

			See the picture drawn below. It is a clear example of path independence. If someone were to travel from point A to C to D to B, it would essentially be the same as traveling straight from A to B when we think about potential.

			[[File:40274.jpg\|frameless\|upright=3\|center]]


	Potential difference is a scalar quantity; this means it has no direction and only a magnitude. Moreover, potential difference is also a "path independent" quantity. This means that given two different observation locations, the potential difference between those two locations will always be the same no matter what path is taken between them (similar to work, only here we use electric fields rather than forces). The path independent nature of potential helps to simplify electric potential calculations and enables us to solve problems more easily by allowing us to define paths to simplify calculations.		Potential difference is a scalar quantity; this means it has no direction and only a magnitude. Moreover, potential difference is also a "path independent" quantity. This means that given two different observation locations, the potential difference between those two locations will always be the same no matter what path is taken between them (similar to work, only here we use electric fields rather than forces). The path independent nature of potential helps to simplify electric potential calculations and enables us to solve problems more easily by allowing us to define paths to simplify calculations.

Wnute3 at 01:51, 10 November 2017

2017-11-10T01:51:57Z

← Older revision		Revision as of 21:51, 9 November 2017
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	Potential difference is a scalar quantity; this means it has no direction and only a magnitude. Moreover, potential difference is also a "path independent" quantity. This means that given two different observation locations, the potential difference between those two locations will always be the same no matter what path is taken between them (similar to work, only here we use electric fields rather than forces). The path independent nature of potential helps to simplify electric potential calculations and enables us to solve problems more easily by allowing us to define paths to simplify calculations.		Potential difference is a scalar quantity; this means it has no direction and only a magnitude. Moreover, potential difference is also a "path independent" quantity. This means that given two different observation locations, the potential difference between those two locations will always be the same no matter what path is taken between them (similar to work, only here we use electric fields rather than forces). The path independent nature of potential helps to simplify electric potential calculations and enables us to solve problems more easily by allowing us to define paths to simplify calculations.


	==Why is Potential Path Independent?==		==Why is Potential Path Independent?==
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	===Mathematical Model===		===Mathematical Model===

	In more formal notation, you could see something like this:		In more formal notation, you could see something like this:

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	===Example 2===		===Example 2===

	[[File:Path-indep3.png\|frameless\|upright=3\|center]]		[[File:Path-indep3.png\|frameless\|upright=3\|center]]

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	Volta found that, in the presence of significant electrical potential between two metals, electrical charge can flow through a metal wire (and through frog legs, salinated brine, etc). The analogy of the time was that current flowed through wire similarly to water in a pipe. Because of this discovery, Volta lives on through the concept of voltage and the associated unit of measurement - the volt.		Volta found that, in the presence of significant electrical potential between two metals, electrical charge can flow through a metal wire (and through frog legs, salinated brine, etc). The analogy of the time was that current flowed through wire similarly to water in a pipe. Because of this discovery, Volta lives on through the concept of voltage and the associated unit of measurement - the volt.



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