Charged Ring - Revision history

Gorj: /* Simplified Abstract Model */

2024-04-17T03:07:11Z

Simplified Abstract Model

← Older revision		Revision as of 23:07, 16 April 2024
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	This electric field model is dependent upon three variables: Q (electric charge), R (radius of the ring), and z (distance away from the center of the ring). This model is correct, i.e. it is completely accurate in all situations in the perfect world. We can create a more simplified model based on this model, given certain assumptions. This section will specifically explore how we can simplify this model when we assume that the distance from the ring is significantly larger than the radius of the ring (z >> R).		This electric field model is dependent upon three variables: Q (electric charge), R (radius of the ring), and z (distance away from the center of the ring). This model is correct, i.e. it is completely accurate in all situations in the perfect world. We can create a more simplified model based on this model, given certain assumptions. This section will specifically explore how we can simplify this model when we assume that the distance from the ring is significantly larger than the radius of the ring (z >> R).

	We can analyze this situation intuitively with an analogy. Imagine you are holding a ring directly in front of you, a couple of inches from your face. You would clearly see that the object is, in fact, ring-shaped. No surprise here. Now imagine that the ring is a mile away from your face. In this case, you may not even see the ring since it is so far away. And if you did, the shape would seem more like a point ~~in space, rather than identifying that it is~~ ring-~~shaped~~. This analogy is identical to our assumption of our simplified model; the ring's radius is so much significantly smaller than the distance. From the observer's point of view, it seems more like a point instead of a ring. This is ~~the~~ fundamental ~~point of~~ our simplified model. We will treat the ring as a point charge, an object with no radius.		We can analyze this situation intuitively with an analogy. Imagine you are holding a ring directly in front of you, a couple of inches from your face. You would clearly see that the object is, in fact, ring-shaped. No surprise here. Now imagine that the ring is a mile away from your face. In this case, you may not even see the ring since it is so far away. And if you did, the shape would seem more like a point. You would not be able to identify the ring-shape property. This analogy is identical to our assumption of our simplified model; the ring's radius is so much significantly smaller than the distance. From the observer's point of view, it seems more like a point instead of a ring. This is fundamental to our simplified model. We will treat the ring as a point charge, an object with no radius.

	We can analyze this situation logically and mathematically by exploring the extreme case: when the distance from the ring approaches infinity. Since the radius will remain finite, the scale between the distance and the radius also approaches infinity. As a result, the relative effect the radius will have on the model is essentially zero. Thus, we can completely discard radius from our simplified model. As a result, we will only have two variables in our simplified model instead of three: Q (electric charge) and z (distance from ring). We can derive this simplified model mathematically using limits.		We can analyze this situation logically and mathematically by exploring the extreme case: when the distance from the ring approaches infinity. Since the radius will remain finite, the scale between the distance and the radius also approaches infinity. As a result, the relative effect the radius will have on the model is essentially zero. Thus, we can completely discard the radius from our simplified model. As a result, we will only have two variables in our simplified model instead of three: Q (electric charge) and z (distance from the ring). We can derive this simplified model mathematically using limits.

	:<math>\vec{E}=\lim_{R\to0}\left(\frac{1}{4\pi\epsilon_{0}}\frac{Qz}{(z^{2}+R^{2})^{3/2}}\hat{z}\right)		:<math>\vec{E}=\lim_{R\to0}\left(\frac{1}{4\pi\epsilon_{0}}\frac{Qz}{(z^{2}+R^{2})^{3/2}}\hat{z}\right)

Gorj at 03:04, 17 April 2024

2024-04-17T03:04:07Z

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 ==The Main Idea==
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 [https://trinket.io/glowscript/707d492e19 This simulation] shows the result of the computation for a ring composed of 2000 electrons. This is why the vector is pointing into the ring rather than out of the ring, which would happen for a ring composed of positively charged points.
 ==Examples==

Sydney: /* Easy */

2020-05-05T20:48:20Z

Easy

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	==Examples==		==Examples==

	===~~Easy~~===		===Simple===

	1.) Find the function <math>E(x)</math> that represents the magnitude of the electric field along the center axis due to a uniformly charged ring of radius 0.04 m with total charge 8 C.		1.) Find the function <math>E(x)</math> that represents the magnitude of the electric field along the center axis due to a uniformly charged ring of radius 0.04 m with total charge 8 C.

Sydney: /* See also */

2020-05-05T20:47:53Z

Sydney: /* See also */

2020-05-05T20:47:15Z

Sydney: /* External Links */

2020-05-05T20:45:32Z

External Links

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	All images produced by the author		All images produced by the author

	==External Links==		===External Links===

	https://www.youtube.com/watch?v=80mM3kSTZcE		https://www.youtube.com/watch?v=80mM3kSTZcE

Sydney: /* References */

2020-05-05T20:44:47Z

References

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	[[Charged Capacitor]]		[[Charged Capacitor]]

	==References==		===References===

	"Electric Field on the Axis of a Ring of Charge". University of Delaware Physics Library. Adapted from Stephen Kevan's lecture on Electric Fields and Charge Distribution. April 8, 1996. http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html		"Electric Field on the Axis of a Ring of Charge". University of Delaware Physics Library. Adapted from Stephen Kevan's lecture on Electric Fields and Charge Distribution. April 8, 1996. http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html

	Chabay, R., & Sherwood, B. (2015). Matter and Interactions (4th ed., Vol. 2, pp. 597-599). Wiley.		Chabay, R., & Sherwood, B. (2015). Matter and Interactions (4th ed., Vol. 2, pp. 597-599). Wiley.


	All images produced by the author		All images produced by the author

Sydney: /* History */

2020-05-05T20:43:06Z

History

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	</div></div>		</div></div>

	==~~History~~==		==Connectedness==

	~~Added~~ by AYESHA AHUJA ~~SPRING~~ 2020		Edited by AYESHA AHUJA Spring 2020

	The ~~first known time when charged rings were utilized~~ to ~~describe electromagnetism was during Faraday’s discovery~~ of ~~electrical induction. To conduct his experiments~~, ~~Faraday used~~ a ring ~~made~~ of ~~iron that was attached~~ to a ~~battery. When this makeshift circuit was turned on it manipulated~~ a ~~compass needle by deflecting it. He showcased an induced current in~~ the ~~ring~~.		The idea of charge density is somewhat analogous to the idea of the mass density, which is useful in a variety of contexts, including the computation of the center of mass, the first and second moments of mass, which are useful in statics and rigid body dynamics. Instead of computing the uniform distribution of the mass of this ring, we are computing the uniform distribution of charge. This concept is also useful in visualizing what happens within a wire in a steady state circuit. The wire can be viewed to be a continuous length of rings of charge, which act as a channel through which electrons are transported, and that the electric field of these rings of charge pushes the electrons within the wire.

	~~===Connectedness===~~		It is also important in the context of Maxwell's equations, specifically in Gauss's Law and in the Maxwell-Faraday equation, which are concerned with electron flux and induced fields, respectively.

	~~Edited by AYESHA AHUJA Spring 2020~~		==History==

	The idea of charge density is somewhat analogous to the idea of the mass density, which is useful in a variety of contexts, including the computation of the center of mass, the first and second moments of mass, which are useful in statics and rigid body dynamics. Instead of computing the uniform distribution of the mass of this ring, we are computing the uniform distribution of charge. This concept is also useful in visualizing what happens within a wire in a steady state circuit. The wire can be viewed to be a continuous length of rings of charge, which act as a channel through which electrons are transported, and that the electric field of these rings of charge pushes the electrons within the wire.		Added by AYESHA AHUJA SPRING 2020

	~~It is also important in the context~~ of ~~Maxwell's equations~~, ~~specifically in Gauss's Law and~~ in the ~~Maxwell-Faraday equation, which are concerned with electron flux and induced fields, respectively~~.		The first known time when charged rings were utilized to describe electromagnetism was during Faraday’s discovery of electrical induction. To conduct his experiments, Faraday used a ring made of iron that was attached to a battery. When this makeshift circuit was turned on it manipulated a compass needle by deflecting it. He showcased an induced current in the ring.

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

Sydney: /* Connectedness */

2020-05-05T20:42:23Z

Connectedness

← Older revision		Revision as of 16:42, 5 May 2020
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	The first known time when charged rings were utilized to describe electromagnetism was during Faraday’s discovery of electrical induction. To conduct his experiments, Faraday used a ring made of iron that was attached to a battery. When this makeshift circuit was turned on it manipulated a compass needle by deflecting it. He showcased an induced current in the ring.		The first known time when charged rings were utilized to describe electromagnetism was during Faraday’s discovery of electrical induction. To conduct his experiments, Faraday used a ring made of iron that was attached to a battery. When this makeshift circuit was turned on it manipulated a compass needle by deflecting it. He showcased an induced current in the ring.

	==Connectedness==		===Connectedness===

	Edited by AYESHA AHUJA Spring 2020		Edited by AYESHA AHUJA Spring 2020

Sydney: /* Examples */

2020-05-05T20:41:30Z

Examples

← Older revision		Revision as of 16:41, 5 May 2020
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	1.) Find the function <math>E(x)</math> that represents the magnitude of the electric field along the center axis due to a uniformly charged ring of radius 0.04 m with total charge 8 C.		1.) Find the function <math>E(x)</math> that represents the magnitude of the electric field along the center axis due to a uniformly charged ring of radius 0.04 m with total charge 8 C.

	~~'''~~Solution~~'''~~		<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
			<div style="font-weight:bold;line-height:1.6;">Solution</div>
			<div class="mw-collapsible-content">

	We can use the derived formula for the electric field due to a charged ring. Only the horizontal components of the electric field will remain as the rest will cancel out.		We can use the derived formula for the electric field due to a charged ring. Only the horizontal components of the electric field will remain as the rest will cancel out.

	<math>E(x) = \frac{1}{4\pi\epsilon_{0}}\frac{(8 C)*x}{((0.04 m)^2 + x^2)^\frac{3}{2}}</math>		<math>E(x) = \frac{1}{4\pi\epsilon_{0}}\frac{(8 C)*x}{((0.04 m)^2 + x^2)^\frac{3}{2}}</math>

			</div></div>

	===Middling===		===Middling===
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	d. 100 cm		d. 100 cm

	''Solution''		<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
			<div style="font-weight:bold;line-height:1.6;">Solution</div>
			<div class="mw-collapsible-content">

	Use the derived formula, and convert the given distances into meters and the charge into Coulombs. Plug these into <math>x</math>		Use the derived formula, and convert the given distances into meters and the charge into Coulombs. Plug these into <math>x</math>
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	d. 8.06e5 N/C		d. 8.06e5 N/C

			</div></div>

	===Difficult===		===Difficult===
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	Find the electric field at a distance <math>z</math> along the center axis away from a uniformly charged semicircular ring of radius R and total charge Q.		Find the electric field at a distance <math>z</math> along the center axis away from a uniformly charged semicircular ring of radius R and total charge Q.

	''Solution''		<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
			<div style="font-weight:bold;line-height:1.6;">Solution</div>
			<div class="mw-collapsible-content">

	We can tweak the same integral for a uniformly charged ring by changing the limits of integration.		We can tweak the same integral for a uniformly charged ring by changing the limits of integration.
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	<math>\vec{E} = \frac{1}{4\pi\epsilon_{0}}\frac{Q}{\pi(R^{2}+z^{2})^{3/2}}(0\hat{x}+2R\hat{y}+{\pi}z\hat{z})</math>		<math>\vec{E} = \frac{1}{4\pi\epsilon_{0}}\frac{Q}{\pi(R^{2}+z^{2})^{3/2}}(0\hat{x}+2R\hat{y}+{\pi}z\hat{z})</math>

			</div></div>

	==History==		==History==

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

	===Further ~~Reading~~===		===Further reading===

	*[[Charged Rod]]		*[[Charged Rod]]
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	*[[Charged Capacitor]]		*[[Charged Capacitor]]

	===External ~~Links~~===		===External links===

	*https://www.youtube.com/watch?v=80mM3kSTZcE		*https://www.youtube.com/watch?v=80mM3kSTZcE

← Older revision		Revision as of 16:47, 5 May 2020
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	== See also ==		== See also ==

	~~[[Charged Rod]]~~		===Further Reading===

	[[Charged ~~Disk~~]]		*[[Charged Rod]]

	[[Charged ~~Spherical Shell~~]]		*[[Charged Disk]]

	[[Charged ~~Capacitor~~]]		*[[Charged Spherical Shell]]

	~~===References===~~		*[[Charged Capacitor]]

	"Electric Field on the Axis of a Ring of Charge". University of Delaware Physics Library. Adapted from Stephen Kevan's lecture on Electric Fields and Charge Distribution. April 8, 1996. http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html		===External Links===

	~~Chabay, R~~.~~, & Sherwood, B. (2015). Matter and Interactions (4th ed., Vol. 2, pp. 597-599). Wiley~~.		*https://www.youtube.com/watch?v=80mM3kSTZcE

	~~All images produced by the author~~		*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

	~~===External Links===~~		*http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html

	~~https://www.youtube.com/watch?v~~=~~80mM3kSTZcE~~		==References==

	http://~~hyperphysics~~.~~phy-astr~~.~~gsu~~.edu/~~hbase~~/~~electric~~/~~elelin~~.html		*"Electric Field on the Axis of a Ring of Charge". University of Delaware Physics Library. Adapted from Stephen Kevan's lecture on Electric Fields and Charge Distribution. April 8, 1996. http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html

	~~http://www~~.~~physics~~.~~udel~~.~~edu/~watson/phys208/exercises/kevan/efield1~~.~~html~~		*Chabay, R., & Sherwood, B. (2015). Matter and Interactions (4th ed., Vol. 2, pp. 597-599). Wiley.

			*All images produced by the author

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