What is potential energy of an electric dipole without any external field?
If its 0 then why? Wouldn't the charges get attracted thus do work so they must have some potential energy.
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If the dipole is formed from two point charges $q$ and $-q$ a distance $d$ apart it has dipole moment $p=qd$. The potential energy of this system of charges is $$ U=-\frac{q^2}{4\pi\epsilon_0 d} $$ Unfortunately this potential $U$ cannot be written in terms of $p$ only. I can try to elimiate $q$, giving $$ U=-\frac{p^2}{4\pi\epsilon_0 d^3} $$ But I can't eliminate $d$ from this equation. Therefore, the electrostatic potential energy of a dipole is not uniquely determined by the dipole moment $p$. The smaller the dipole physically is, the more negative potential it has.
A similar problem occurs when we try to ask what the potential energy of a point charge in free space is. We can consider the point charge to be a sphere of charge of radius $R$, then its potential energy is
$$
\frac{3q^2}{20\pi\epsilon_0 R}
$$
Just like with the dipole, the point charge's potential energy is not uniquely determined by its charge $q$. If you ask me "how much energy did it take to assemble this very small sphere of charge $q$, I will say "I don't know unless you tell me how big it is." Same answer goes for a dipole.
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$\begingroup$ So a dipole existing without any external electric field influence do have a potential energy. I got confused before because i searched on net and found its equal to 0. $\endgroup$ Commented Feb 12 at 8:27
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1$\begingroup$ @AmitRai I think you need a more sophisticated understanding of what potential is before we can move forward. (1) the potential energy of a system can always be redefined by shifting it by a constant. If you're considering a dipole that is in some way "fixed" - i.e. it cannot change its dipole moment - and you only want to model the motion (rotation and translation) of that dipole, it may be convenient to define the potential of that dipole in free space as zero. My answer calculates the energy it would take to construct a dipole from two point charges. $\endgroup$– AXensenCommented Feb 12 at 18:59
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1$\begingroup$ @AmitRai So I am not saying that my dipole is fixed - I can freely change its dipole moment, and I want to know how much energy it takes to do so. I'm trying to address your question "wouldn't the charges get attracted and thus do work"... yes, this idea you had relates to the energy it would take to assemble a dipole, but it doesn't change how the dipole will move in an electric field if you assume the distance between the charges can never change. So this concept doesn't affect the potential describing the motion of the dipole in an external electric field. $\endgroup$– AXensenCommented Feb 12 at 19:01