Gauss's Law - Revision history

Ielsissi3: I added a summary table. I think this section can be of great help to a student, especially before a test, when they need a quick recap/reminder.

2025-04-13T16:59:56Z

I added a summary table. I think this section can be of great help to a student, especially before a test, when they need a quick recap/reminder.

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-'''Claimed by James Swope (Fall 2020), Edited by Nabiha Ahsan (Fall 2021), Edited by Tyler Morgan(Fall 2023), Edited by Anshu Dendukuri(Spring 2024)'''
+'''Ismail Elsissi (ielsissi3) Spring 2025'''
 <!--I apologize for my apparent lack of contribution. I had hoped to create a great deal of new content (specifically a computational model and a few hard examples), however, I ended up running out of time. I was also unable to finish tidying up/rewriting the article. Please don't judge me too critically.
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 Gauss's Law is a very powerful law that spans a diverse array of fields, with applications in physics, mathematics, chemistry, and engineering, among others. Along with [[James Maxwell]]'s other three equations, Gauss's Law forms the foundation of classical electrodynamics.
 ==The Main Idea==

Yparmar6 at 21:42, 23 November 2024

2024-11-23T21:42:02Z

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	Gauss is also known for his advancements in the astronomy field. While the dwarf planet Ceres was discovered by Giuseppe Piazza (Italian astronomer), Gauss was able to predict the position for its rediscovery. Due to the methodology behind this rediscovery, Gauss's Method, there was also significant advancements in the theory of the motion of planetoids, which introduced the Gaussian gravitation constant.		Gauss is also known for his advancements in the astronomy field. While the dwarf planet Ceres was discovered by Giuseppe Piazza (Italian astronomer), Gauss was able to predict the position for its rediscovery. Due to the methodology behind this rediscovery, Gauss's Method, there was also significant advancements in the theory of the motion of planetoids, which introduced the Gaussian gravitation constant.

			The foundation for Gauss's law was laid by Isaac Newton, who formulated the inverse-square law of gravitation in 1687. This idea of forces diminishing with the square of distance inspired analogous formulations for electrostatics. In 1785, Charles-Augustin de Coulomb experimentally confirmed that electric forces between charges follow an inverse-square law. The concept of a field—an invisible influence extending through space—began to take shape in the late 18th and early 19th centuries.
			Mathematicians and physicists, including Joseph Priestley and others, laid groundwork for understanding how charges influence surrounding regions. Gauss formulated the law in his personal notes in 1835. These notes contained his ideas on how electric fields could be mathematically related to charges through surface integrals. His reluctance to publish might have been due to the lack of experimental confirmation at the time or his tendency to prioritize mathematical rigor over immediate dissemination. The law became widely known as part of James Clerk Maxwell’s unification of electromagnetic theory in the 1860s. Maxwell included Gauss’s law as one of his four foundational equations in Maxwell’s Equations. Although Maxwell credited earlier physicists, including Gauss, Faraday, and Ampère, the incorporation of Gauss’s law into a cohesive electromagnetic theory marked its formal introduction into mainstream physics. By the late 19th century, Gauss’s law was being taught in academic settings as a fundamental part of electrostatics and electromagnetism. It gained recognition in its integral and differential forms, largely through Maxwell’s equations and the broader development of vector calculus.

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

Yparmar6 at 21:36, 23 November 2024

2024-11-23T21:36:49Z

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	[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg\|\|Portrait of Gauss]]		[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg\|\|Portrait of Gauss]]

	Gauss's Law was first 'discovered' in 1773, by the great mathematician Joseph-Louis Lagrange. It was further developed by Carl Friedrich Gauss in 1813.		Gauss's Law was first 'discovered' in 1773, by the great mathematician Joseph-Louis Lagrange. It was further developed by Carl Friedrich Gauss in 1813. The law was inspired by Coulomb's law.



			Gauss was a prolific German mathematician and physicist. He made significant contributions to a variety of fields in mathematics and physics. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians".

			Gauss is also known for his advancements in the astronomy field. While the dwarf planet Ceres was discovered by Giuseppe Piazza (Italian astronomer), Gauss was able to predict the position for its rediscovery. Due to the methodology behind this rediscovery, Gauss's Method, there was also significant advancements in the theory of the motion of planetoids, which introduced the Gaussian gravitation constant.


	Gauss was a prolific German mathematician and physicist. He made significant contributions to a variety of fields in mathematics and physics. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians".

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

Adendukuri7 at 23:54, 14 April 2024

2024-04-14T23:54:20Z

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	'''Claimed by James Swope (Fall 2020), Edited by Nabiha Ahsan (Fall 2021), Edited by Tyler Morgan(Fall 2023)'''		'''Claimed by James Swope (Fall 2020), Edited by Nabiha Ahsan (Fall 2021), Edited by Tyler Morgan(Fall 2023), Edited by Anshu Dendukuri(Spring 2024)'''
	<!--I apologize for my apparent lack of contribution. I had hoped to create a great deal of new content (specifically a computational model and a few hard examples), however, I ended up running out of time. I was also unable to finish tidying up/rewriting the article. Please don't judge me too critically.		<!--I apologize for my apparent lack of contribution. I had hoped to create a great deal of new content (specifically a computational model and a few hard examples), however, I ended up running out of time. I was also unable to finish tidying up/rewriting the article. Please don't judge me too critically.

Tmorgan65: /* Easy Example */

2023-11-27T02:35:22Z

Easy Example

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	==Examples==		==Examples==

	===Easy Example===		===Easy Examples===
			====Easy Example 1====

	Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).		Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).
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	Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.		Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

	5. Consider a point charge <math> Q = +2 \, \text{μC} </math> enclosed by a spherical Gaussian surface with a radius of <math> r = 10 \, \text{cm} </math>. Calculate the electric flux through the surface.		====Easy Example 2====
			Consider a point charge <math> Q = +2 \, \text{μC} </math> enclosed by a spherical Gaussian surface with a radius of <math> r = 10 \, \text{cm} </math>. Calculate the electric flux through the surface.

	Solution:		Solution:
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	<math> \Phi_{el} = \frac{2 \times 10^{-6} \, \text{C}}{8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2} </math>		<math> \Phi_{el} = \frac{2 \times 10^{-6} \, \text{C}}{8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2} </math>


	===Intermediate Examples===		===Intermediate Examples===

Tmorgan65: /* Hard Example 2: */

2023-11-27T02:33:56Z

Hard Example 2:

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	To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.		To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.

	====~~Hard~~ Example 2:====		====Example 2====

	A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where <math>\lambda_0</math> and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.		A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where <math>\lambda_0</math> and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.

Tmorgan65: /* Example 1: */

2023-11-27T02:33:36Z

Example 1:

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	===Hard Examples===		===Hard Examples===

	====Example 1:====		====Example 1====

	Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.		Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.

Tmorgan65: /* Example 1: */

2023-11-27T02:33:20Z

Example 1:

← Older revision		Revision as of 22:33, 26 November 2023
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	===Hard Examples===		===Hard Examples===

	====~~Hard~~ Example 1:====		====Example 1:====

	Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.		Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.

Tmorgan65: /* Solution */

2023-11-27T02:32:51Z

Solution

← Older revision		Revision as of 22:32, 26 November 2023
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	Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.		Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.

	=====Solution=====		=====Solution:=====
	To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.		To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.

Tmorgan65: Created new easy/hard problems

2023-11-27T02:02:42Z

Created new easy/hard problems

← Older revision		Revision as of 22:02, 26 November 2023
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	====Hard Example 2:====		====Hard Example 2:====

	A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where \lambda_0 and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.		A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where <math>\lambda_0</math> and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.

	=====Solution:=====		=====Solution:=====