# Doubt in understanding the potential energy of dipole in external electric field?

When a dipole is placed in external electric field it experiences a torque $$\vec \tau = \vec p \ \times \vec E$$ whose magnitude is $$||\vec \tau|| = ||\vec p|| \cdot ||\vec E|| \cdot sin\theta$$ On calculating the potential energy: $$W=-\Delta U$$ $$\therefore W = -\int pEsin\theta\cdot d\theta$$ Which gives: $$W = pEcos\theta$$ But its wrong and is given as $$-pEcos\theta$$ in many text books. Please Explain.

• The value of a potential energy depends on where you choose the reference position. Notice that the expression given by the book is zero when the angle is 90 degrees. It increases for larger angles and decreases for smaller (going negative). You need to specify limits on your integral. – R.W. Bird Oct 17 '19 at 18:17

Let me start from the beginning

Wnet = $$\Delta$$K.E

Wconservative+Wnon-conser.+Wexternal = $$\Delta$$K.E

Wnon-conser. = Zero [No non-conservative force is acting ]

$$\Delta$$K.E = zero [Assumption here is that the dipole is rotated very slowly therefore change amounts to zero]

Therefore the equation reduces to

-Wconservative = Wexternal

-d(Wconservative) = $$\Gamma$$*d$$\theta$$

-$$\int$$d(Wconservative) = $$\int$$ $$\Gamma$$*d$$\theta$$

-Wconservative = $$\int$$ p*E sin$$\theta$$ d$$\theta$$

-Wconservative = p*E $$\int$$ sin$$\theta$$ d$$\theta$$

-Wconservative = p*E(-cos$$\theta$$)

$$\Delta$$U = -p*E cos$$\theta$$ [ Work done by conservative force is negative of change in potential energy of system.]

It would make sense to integrate having definite limits then comparing the equations on both sides to understand what the potential of the dipole at and angle would be, I haven't done that in hopes that you can carry it from here(that is if you want it as such). Either way the result would be the same.