Potential Energy for a Magnetic Dipole - Revision history

Thenderlong3 at 23:01, 3 December 2015

2015-12-03T23:01:47Z

← Older revision		Revision as of 19:01, 3 December 2015
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	<p>[http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment Magnetic Dipole Moment]</p>		<p>[http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment Magnetic Dipole Moment]</p>
	<p>[http://www.physicsbook.gatech.edu/Magnetic_Field Magnetic Field]</p>		<p>[http://www.physicsbook.gatech.edu/Magnetic_Field Magnetic Field]</p>
			<p>[http://www.physicsbook.gatech.edu/Work Work]</p>

	== References ==		== References ==
	Chabay, Ruth, and Bruce Sherwood. "Magnetic Force." Matter & Interactions. Fourth ed. Vol. 2. Hoboken: John Wiler & Sons, 2015. 829-833. Print		Chabay, Ruth, and Bruce Sherwood. "Magnetic Force." Matter & Interactions. Fourth ed. Vol. 2. Hoboken: John Wiler & Sons, 2015. 829-833. Print

Thenderlong3: /* References */

2015-12-03T22:46:13Z

References

← Older revision		Revision as of 18:46, 3 December 2015
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	== References ==		== References ==
			Chabay, Ruth, and Bruce Sherwood. "Magnetic Force." Matter & Interactions. Fourth ed. Vol. 2. Hoboken: John Wiler & Sons, 2015. 829-833. Print


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

Thenderlong3 at 21:53, 3 December 2015

2015-12-03T21:53:28Z

← Older revision		Revision as of 17:53, 3 December 2015
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	When finding the potential energy keep in mind that there is a negative sign in front of the dot product of dipole moment and magnetic field. That being said System 4 would have the largest potential energy because the dot product of the two vectors would yield a negative value in the y-component. This system would be very unstable and if we were to move the dipole moment ever so slightly, it would immediately turn to point the same exact way as the magnetic field. System 1 & 2 would have the same potential energy of 0. When finding the dot product of System 3, we would get a positive value, but with the negative sign in the front of the formula, we have a negative potential energy. So system 3 shows the most stable and desired state. So the ranking of the potential energies would be 4 > 1 = 2 > 3.		When finding the potential energy keep in mind that there is a negative sign in front of the dot product of dipole moment and magnetic field. That being said System 4 would have the largest potential energy because the dot product of the two vectors would yield a negative value in the y-component. This system would be very unstable and if we were to move the dipole moment ever so slightly, it would immediately turn to point the same exact way as the magnetic field. System 1 & 2 would have the same potential energy of 0. When finding the dot product of System 3, we would get a positive value, but with the negative sign in the front of the formula, we have a negative potential energy. So system 3 shows the most stable and desired state. So the ranking of the potential energies would be 4 > 1 = 2 > 3.

	~~==Connectedness==~~
	~~#How is this topic connected to something that you are interested in?~~
	~~#How is it connected to your major?~~
	~~#Is there an interesting industrial application?~~

	~~==History==~~

	~~Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.~~

	== See also ==		== See also ==
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	<p>[http://www.physicsbook.gatech.edu/Magnetic_Field Magnetic Field]</p>		<p>[http://www.physicsbook.gatech.edu/Magnetic_Field Magnetic Field]</p>

			== References ==

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

	~~Books, Articles or other print media on this topic~~

	~~This section contains the the references you used while writing this page~~

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

Thenderlong3: /* Examples */

2015-12-03T20:38:23Z

Examples

← Older revision		Revision as of 16:38, 3 December 2015
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	As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.		As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.

			----

	Now let's look at the dipole moments and the magnetic field below. How would we go about ranking the potential energy from of each system from greatest to smallest.		Now let's look at the dipole moments and the magnetic field below. How would we go about ranking the potential energy from of each system from greatest to smallest.

Thenderlong3: /* Examples */

2015-12-03T20:34:57Z

Examples

← Older revision		Revision as of 16:34, 3 December 2015
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	==Examples==		==Examples==

	~~Be sure to show all steps in your solution and include diagrams whenever possible~~		Let's now envision two magnets interacting with each other.

	~~===Simple===~~		Knowing what we know now about the potential energy of a magnetic dipole, we can see why magnets move each other. Systems tend to favor states of low potential energy so when a magnet is inched closer to another magnet, their dipoles will align so that the potential energy takes on a negative value (state with the lowest potential energy). This action is illustrated below.
	~~===Middling===~~
	=~~==Difficult===~~		[[File:physics_potential_energy4.png]]

			As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.

			Now let's look at the dipole moments and the magnetic field below. How would we go about ranking the potential energy from of each system from greatest to smallest.

			[[File:physics_potential_energy5.png]]

			When finding the potential energy keep in mind that there is a negative sign in front of the dot product of dipole moment and magnetic field. That being said System 4 would have the largest potential energy because the dot product of the two vectors would yield a negative value in the y-component. This system would be very unstable and if we were to move the dipole moment ever so slightly, it would immediately turn to point the same exact way as the magnetic field. System 1 & 2 would have the same potential energy of 0. When finding the dot product of System 3, we would get a positive value, but with the negative sign in the front of the formula, we have a negative potential energy. So system 3 shows the most stable and desired state. So the ranking of the potential energies would be 4 > 1 = 2 > 3.

	==Connectedness==		==Connectedness==

Thenderlong3: /* A Mathematical Model */

2015-12-03T20:22:39Z

A Mathematical Model

← Older revision		Revision as of 16:22, 3 December 2015
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	Potential energy for a dipole can also be written as the dot product <math>U_m = -\overrightarrow{μ} \bullet \overrightarrow{B}\ </math>.		Potential energy for a dipole can also be written as the dot product <math>U_m = -\overrightarrow{μ} \bullet \overrightarrow{B}\ </math>.

	~~Let's now envision two magnets interacting with each other.~~

	Knowing what we know now about the potential energy of a magnetic dipole, we can see why magnets move each other. Systems tend to favor states of low potential energy so when a magnet is inched closer to another magnet, their dipoles will align so that the potential energy takes on a negative value (state with the lowest potential energy). This action is illustrated below.

	~~[[File:physics_potential_energy4.png]]~~

	~~As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.~~

	==Examples==		==Examples==

Thenderlong3: /* A Mathematical Model */

2015-12-03T20:13:29Z

A Mathematical Model

← Older revision		Revision as of 16:13, 3 December 2015
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	<p><math>Work = ΔU_m = \int\limits_{θ_i}^{θ_f}\ 2IwBsinθ(\frac{h}{2}dθ) = IwhB \int\limits_{θ_i}^{θ_f}\ sinθ dθ </math>		<p><math>Work = ΔU_m = \int\limits_{θ_i}^{θ_f}\ 2IwBsinθ(\frac{h}{2}dθ) = IwhB \int\limits_{θ_i}^{θ_f}\ sinθ dθ </math>
	</p>		</p>
	Where <math>IwBsinθ</math> is the perpendicular force exerted ~~by us~~ on only one side of the circuit. We are pushing on both sides so we multiply that by 2. <math>(\frac{h}{2}dθ)</math> is the distance that the force is exerted.		Where <math>IwBsinθ</math> is the perpendicular force exerted on only one side of the circuit. We are pushing on both sides so we multiply that by 2. <math>(\frac{h}{2}dθ)</math> is the distance that the force is exerted.
	<p><math>ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i]</math>		<p><math>ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i]</math>
	</p>		</p>

Thenderlong3 at 20:12, 3 December 2015

2015-12-03T20:12:36Z

← Older revision		Revision as of 16:12, 3 December 2015
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	As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.		As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.

	~~===A Computational Model===~~

	~~How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]~~

	==Examples==		==Examples==
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	== See also ==		== See also ==

	~~Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into~~ that ~~context?~~		Here are some other resources that may help.
			<p>[http://www.physicsbook.gatech.edu/Potential_Energy Potential Energy]</p>
			<p>[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html Magnetic Forces]</p>
			<p>[http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment Magnetic Dipole Moment]</p>
			<p>[http://www.physicsbook.gatech.edu/Magnetic_Field Magnetic Field]</p>


	===Further reading===		===Further reading===

	Books, Articles or other print media on this topic		Books, Articles or other print media on this topic

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

	~~Internet resources on this topic~~

	~~==References==~~

	This section contains the the references you used while writing this page		This section contains the the references you used while writing this page

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

Thenderlong3: /* A Mathematical Model */

2015-12-03T19:32:39Z

A Mathematical Model

← Older revision		Revision as of 15:32, 3 December 2015
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	[[File:physics_potential_energy4.png]]		[[File:physics_potential_energy4.png]]

			As the magnet on the left is held and inched closer to the magnet on the right, the magnet on the right will turn so that the magnetic dipoles align with each other.

	===A Computational Model===		===A Computational Model===

Thenderlong3: /* A Mathematical Model */

2015-12-03T19:26:51Z

A Mathematical Model

← Older revision		Revision as of 15:26, 3 December 2015
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	<p><math>Work = ΔU_m = \int\limits_{θ_i}^{θ_f}\ 2IwBsinθ(\frac{h}{2}dθ) = IwhB \int\limits_{θ_i}^{θ_f}\ sinθ dθ </math>		<p><math>Work = ΔU_m = \int\limits_{θ_i}^{θ_f}\ 2IwBsinθ(\frac{h}{2}dθ) = IwhB \int\limits_{θ_i}^{θ_f}\ sinθ dθ </math>
	</p>		</p>
			Where <math>IwBsinθ</math> is the perpendicular force exerted by us on only one side of the circuit. We are pushing on both sides so we multiply that by 2. <math>(\frac{h}{2}dθ)</math> is the distance that the force is exerted.
	<p><math>ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i]</math>		<p><math>ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i]</math>
	</p>		</p>
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	</math>		</math>
	</p>		</p>
			so the potential energy for a dipole is
			<math>
			U_m = -μBcosθ
			</math>.

			Potential energy for a dipole can also be written as the dot product <math>U_m = -\overrightarrow{μ} \bullet \overrightarrow{B}\ </math>.

			Let's now envision two magnets interacting with each other.

			Knowing what we know now about the potential energy of a magnetic dipole, we can see why magnets move each other. Systems tend to favor states of low potential energy so when a magnet is inched closer to another magnet, their dipoles will align so that the potential energy takes on a negative value (state with the lowest potential energy). This action is illustrated below.

	What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.		[[File:physics_potential_energy4.png]]

	===A Computational Model===		===A Computational Model===