I took electromagnetism a while ago, but now that I took Lagrangian and Hamiltonian mechanics, this question came up to me when I imagined an electric dipole in the presence of a uniform electric field. Do I have to consider the potential energy generated by the potential of the dipole? Also, due to the external electric field, there will be a torque and thus, a kinetic energy from the rotation of the dipole. Honestly I don't know if my assumptions are right.
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4$\begingroup$ Before getting to dipoles: can you write down the Lagrangian for a single point charge in an external electric field and derive the equations of motion? $\endgroup$– AndrewCommented Mar 19 at 4:06
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$\begingroup$ Does this help? physics.stackexchange.com/questions/801427/… . In your case the vector and scalar potentials will be due to something other than another dipole, but otherwise, the expressions should be the same $\endgroup$– CryoCommented Mar 19 at 20:46
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1 Answer
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Consider a free to move stationary dipole consisting of two charges, q and -q (of masses m1 and m2 respectively), and the distance between them being d. Let the dipole moment of the dipole be p.
Consider a uniform (not necessarily constant) electric field E such that initially, the angle between E and p is θ0. (See the figure). Ignore gravity (or assume the rotation to be on a horizontal plane).
After some time t=t, let the angle between p and E be θ.
Note that, there is no net force on the dipole, so the body will move about its center of mass.
The Electric potential energy at that moment(in that position), by definition, is
U(θ)=−p.E
The Kinetic energy, owing to the rotational motion of the dipole about its C.O.M, is as shown below.
(I am new here and I am not able to type this out, therefore the image, sorry for the inconvenience)
With the Electric Potential energy and Rotational Kinetic energy, you can proceed to formulate the Lagrangian.
(If you want to know more about the forces, torques, and the derivation of the potential energy formula, you may want to modify the question accordingly, and I shall edit the my answer and complete it)
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4$\begingroup$ Hello! Regarding typing out equations, for future questions/answers, you can find a basic MathJax/LaTeX tutorial at MathJax basic tutorial and quick reference. $\endgroup$– jng224Commented Mar 19 at 9:30
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$\begingroup$ Thanks a lot @jng224. Most appreciated. I have my JEE Advance Examination in a month so I am actually busy preparing. After that, I will surely modify my answer and type out the equations. $\endgroup$ Commented Mar 19 at 15:23