My test paper asked me which way a dipole should be orientated in an electric field to minimize its potential energy. My answer was that the dipole should lie parallel to the electric field with the positive end of the dipole pointing in the direction of the electric field because that way, the positive end would be closer to the negative "attracting" of the field, and similarly the negative end of the dipole would be closer to the positive side of the field (i.e. the side from which the electric field is coming). Is this the right way of thinking about it?
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2 Answers
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That's a pretty good way to look at it. To be more mathematically explicit, notice that the energy of an electric dipole (see here) with dipole moment $\mathbf p$ in an electric field $\mathbf E$ is $$ U = -\mathbf p\cdot\mathbf E = -|\mathbf p||\mathbf E|\cos\theta $$ This expression is minimized when $\cos\theta = 1$, which is when the angle between the dipole moment vector (which for a physical dipole points from the negative to the positive charge) and the field is $0$. This is precisely the condition for the dipole to be aligned with the electric field as you described.
If the dipole is not aligned with the field, then it will experience a torque that tends to align it with the field. You can see why this happens in the physical dipole; the positive charge feels a force in the direction of the field, while the negative charge feels a force in the direction opposite the field, and these both tend to rotate the dipole to align in with the field.
answered Feb 6, 2013 at 6:41
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$\begingroup$ But even if it is anti-parallel aligned with the electric field the torque is zero. So why is it more stable when it is parallel? $\endgroup$ Commented Jan 10, 2020 at 13:54
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1$\begingroup$ @adosar When the dipole is anti-parallel, the potential energy as a function of $\theta$ attains a local maximum instead of a local minimum. You can verify this by plotting $-\cos\theta$ versus $\theta$. This means that although the dipole will be in static equilibrium when exactly anti-parallel, that equilibrium will be unstable; any perturbation, no matter how small, will cause a torque tending to align it with the field. $\endgroup$ Commented Jan 10, 2020 at 20:50
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$\begingroup$ Thanks for the answer. I got it now! $\endgroup$ Commented Jan 12, 2020 at 16:29
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$V=-PE \cos \theta$. From this expression, we can deduce that the potential energy is dependent on the angle of orientation, which varies between $-pE$ when the dipole is aligned with the field and $pE$ when the dipole is aligned against the field. Because the energy is least when the field and the dipole are parallel to each other, it follows that this is the most favored orientation. i.e. the dipole tends to align itself with the field.
