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## Question: Dipole Moment from constant electric field

In [1] it is stated that the force per unit angle on a dipole system as illustrated in Fig.1 is

where . The text was also referring to torques, and it wasn’t clear to me if the result was the torque or the force. Derive the result to resolve any doubt (in retrospect dimensional analysis would also have worked).

## Answer

Let’s put the electric field in the direction (), so that the potential acting on charge is given implicitly by

or

Our positions, and velocities are

Our kinetic energy is

For our potential energies we require the component of the position vectors, which are

Our potentials are

Our system Lagrangian, after dropping the constant reference potential that doesn’t effect the dynamics is

For this problem we had two equal masses and equal magnitude charges and

Putting these together, with , we have the result stated in the text

# References

[1] E.A. Jackson. *Equilibrium statistical mechanics*. Dover Pubns, 2000.