Physics Asked on September 2, 2021
I am looking at the following derivation of the potential energy of a dipole in a uniform electric field, paraphrased from phys.libretexts.org:
Consider an electric dipole $p$ placed in a uniform electric field
$E$. There is a torque on the dipole of magnitude $p E sinθ$ . In
order to increase $θ$ by $δθ$ you would have to do work amounting to
$p E sin{θ}, δθ$. The amount of work you would have to do to
increase the angle between $p$ and $E$ from $0$ to $θ$ would be the
integral of this from $0$ to $θ$, which is $pE ,(1−cosθ)$. This is
the potential energy of the dipole, provided one takes the potential
energy to be zero when $p$ and $E$ are parallel. In many
applications, writers find it convenient to take the potential energy
(P.E.) to be zero when $p$ and $E$ perpendicular. In that case, the
potential energy is$$U= −p E ,cosθ=−textbf{p}⋅textbf{E}.$$ This is negative when $θ$
is acute and positive when $θ$ is obtuse. You should verify that the
product of $p$ and $E$ does have the dimensions of energy.
Now in this derivation, I do not understand from where the $theta$ should be measured, because $theta$ here is actually the angle between $p$ and $E$, so when the dipole move angles $dtheta$, angle between them should be obtuse and not acute, and so the derivation is done when $theta$ is greater than $frac{pi}{2}.$ But it says the exact opposite. Could someone tell me what is happening?
The angle $theta$ is angle between $textbf{p}$ and $textbf{E}$. I think what you are asking is, "What orientation of the dipole should the initial angle, say $theta_0$, correspond to?". If you integrate the differential of the work done in moving the dipole from an initial angle $theta_0$ to a final angle $theta$ w.r.t. $textbf{E}$, you get the expression for the potential energy stored in the dipole on the right-hand-side. On the left-hand-side, you get the difference in energy, so $$U - U_0= -p E, (cos{theta} - cos{theta_0}).$$ If you compare the terms in the expression above you will see that we can take $U_0 equiv -pE cos{theta_0}.$ This is nothing but the $zero$ of the potential energy, which we can take to be anything (as long as we remain consistent). Typically, we take it to be 0 since it is the simplest choice, so this corresponds to assigning $theta_0 = frac{pi}{2}.$
It is customary to measure angles positive in the counter-clockwise sense, so if you take $theta_0 = frac{pi}{2}$ to be the zero of the potential energy, then this means the energy of the dipole is positive only when $theta > frac{pi}{2}$.
On the other hand, you could define $U_0 equiv -p,E$, which would mean $theta_0 = 0$ (i.e. the dipole is aligned with the field). In this case, the energy you measure will be positive for $theta >0$.
Both conventions are equally valid, and it depends on your preference which one you choose, but first one is more popular.
Answered by Yejus on September 2, 2021
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