A dipole subject to a homogenous electric field.A lot of molecules can be seen in a first approximation as two charges with a fixed separation. In this problem you will learn what happens if such a molecule is exposed to a constant electric field, especially, what kind of interesting movements one can observe. You will also be able to apply some of your knowledge of mechanics.

A processing molecule in an external electric field.

Problem Statement

We want to assume that both charges are equal (no net charge). Furthermore they shall be separated by a fixed distance \(R\) and have two not necessarily equal masses \(m_1\) and \(m_2\). Now there is a constant electrical field which may be parallel to the \(z\)-axis.

What happens to the molecule in the field? Describe its movement qualitatively!

You may want to proceed as follows:

  1. find the potential energy of the molecule in the electric field,
  2. determine conserved quantities using the Lagrange function and
  3. discuss the movement in terms of this conserved quantity.


Background: Symmetries and Conserved Quantities

The Lagrange function \(L\left(q_k,\dot{q}_k,t\right)\) is given in terms of generalized coordinates \(q_k\) and velocities \(\dot{q}_k\). Conserved quantities are related to symmetries of the Lagrange function. If for some \(l\)\[ \begin{eqnarray}\frac{\partial L\left(q_{k},\dot{q}_{k},t\right)}{\partial q_{l}}&=&0 \ , \mathrm{then}\quad p_l \equiv  \frac{\partial L\left(q_{k},\dot{q}_{k},t\right)}{\partial\dot{q}_{l}}\end{eqnarray}\]is a conserved quantity. Do you remember why, regarding the Euler-Lagrange equation?

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