Understanding the Potential Energy of a Dipole in an Electric Field

[DavideGenoa]

Hi, friends! I read that the torque exerced by a uniform electric field $\mathbf{E}$ on a dipole with moment $\mathbf{p}$ is $\boldsymbol{\tau}=\mathbf{p}\times\mathbf{E}$. Then the book, Gettys' Physics 2, explain that the work made to rotate the dipole around a fixed point is

$\int_{\theta_1}^{\theta_2}\tau d\theta=\int_{\theta_1}^{\theta_2}pE\sin\theta d\theta=-pE\cos\theta_2-(-pE\cos\theta_1)=\Delta U$
​

and therefore potential energy can be defined as $U=-\mathbf{p}\cdot\mathbf{E}$.

Well, there is a major obstacle to my comprehension of that: I think that $\int_{\theta_1}^{\theta_2}\tau d\theta=\int_{\theta_1}^{\theta_2}pE\sin\theta d\theta$ (where I think $\theta$ to be the oriented angle from $\mathbf{p}$ to $\mathbf{E}$) is the work done by the electric fied while the dipole rotates from an angle $\theta_1$ with the direction of $\mathbf{E}$ to an angle $\theta_2$ with $\mathbf{E}$, but I know the definition of the variation of potential energy $\Delta U$ as the opposite of the work done by a conservative force field: $\Delta U=-W_{\text{conservative}}$.

Or isn't $\int_{\theta_1}^{\theta_2}\tau d\theta$ the work done by the electric field? I would think that it is, because the work done by a force $\mathbf{F}$ to move along the circumference $\gamma$ parametrized by $\mathbf{r}:[0,2\pi]\to\mathbb{R}^3$ is $\int_{\theta_1}^{\theta_2} \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(\theta)d\theta$. Here $\mathbf{F}\cdot \mathbf{r}'=F_t R$ where $F_t$ is the tangential component of the force (positive if and only if counterclockwisely oriented) and $R$ the distance of the moved object from the centre of the circle and, if I correctly understand, $RF_t$ precisely is the torque $\tau_z$ with respect to the centre of rotation of a rotating body, therefore the work done by the forces acting on the rotating body are $W_{\text{tot}}=\int_{\theta_1}^{\theta_2} \sum\tau_z d\theta$. Or am I wrong?

I heatily thank you for any answer!​

 

[DavideGenoa]

I have understood: in the formula for $\Delta U$ the angle $\theta$ is from $\mathbf{E}$ to $\mathbf{p}$, i.e $\tau$ is the opposite of the torque $\tau_z$.