Potential Energy for a Magnetic Dipole: Difference between revisions
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As we push the circuit to rotate a angle of dθ around it's horizontal axis, the total distance that the force is exerted on one side of the circuit is equal to (h/2)*dθ. Keep in mind though that we are exerted force on both sides of the circuit. In the diagram this is indicated as Force by Us. We will have to determine the total amount of work done by using an integral because the force exerted depends on the value of θ and θ increases as we push the circuit more and more out of alignment. The work done by pushing the circuit out of alignment changes the magnetic potential energy of the system. | As we push the circuit to rotate a angle of dθ around it's horizontal axis, the total distance that the force is exerted on one side of the circuit is equal to (h/2)*dθ. Keep in mind though that we are exerted force on both sides of the circuit. In the diagram this is indicated as Force by Us. We will have to determine the total amount of work done by using an integral because the force exerted depends on the value of θ and θ increases as we push the circuit more and more out of alignment. The work done by pushing the circuit out of alignment changes the magnetic potential energy of the system. | ||
<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><math> ΔU_m = IwhB[-cosθ] | <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><math>ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i]</math> | |||
</p> | |||
<p> | |||
<math> | |||
ΔU_m = Δ(-μBcosθ) since μ = IA = Iwh | |||
</math> | |||
</p> | |||
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. | 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. | ||
Revision as of 14:46, 3 December 2015
Page Claimed by Thomas Henderlong
The Main Idea
If a magnetic dipole is allowed to pivot freely without any outside forces, then it will match it's alignment with the applied magnetic field. The system favors a state of lower potential so the aligned magnetic dipole is associated with a lower potential energy in the applied magnetic field. Think of how a magnets interact with each other when inched closer to each other. They'll align in a way so that the north and south ends point in the same direction or have the north end of one magnet touching the south end of the other magnet (state of lowest potential).
A Mathematical Model
By calculating how much work it takes to move a magnetic dipole out of alignment, we can determine the increased potential energy for that dipole.
Let's take a look at a rectangular circuit on a horizontal axle in a magnetic field pictured below.
Now let's look at this system from the side.
We can see that the magnetic force acting on the circuit pushes the circuit out horizontally. The magnetic force is trying to make the circuit's magnetic dipole moment, μ, become parallel with the magnetic field. We can measure the amount of work it takes to move the circuit from an initial angle of θi to a final angle of θf.
As we push the circuit to rotate a angle of dθ around it's horizontal axis, the total distance that the force is exerted on one side of the circuit is equal to (h/2)*dθ. Keep in mind though that we are exerted force on both sides of the circuit. In the diagram this is indicated as Force by Us. We will have to determine the total amount of work done by using an integral because the force exerted depends on the value of θ and θ increases as we push the circuit more and more out of alignment. The work done by pushing the circuit out of alignment changes the magnetic potential energy of the system.
[math]\displaystyle{ Work = ΔU_m = \int\limits_{θ_i}^{θ_f}\ 2IwBsinθ(\frac{h}{2}dθ) = IwhB \int\limits_{θ_i}^{θ_f}\ sinθ dθ }[/math]
[math]\displaystyle{ ΔU_m = IwhB[-cosθ]_{θ_i}^{θ_f} = -IwhB[cosθ_f - cosθ_i] }[/math]
[math]\displaystyle{ ΔU_m = Δ(-μBcosθ) since μ = IA = Iwh }[/math]
What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\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.
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
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