# Understanding the Potential Energy of a Dipole in an Electric Field

• DavideGenoa
In summary, the conversation discusses the torque exerted by a uniform electric field on a dipole and the work required to rotate the dipole around a fixed point. The potential energy can be defined as the negative of the dot product of the dipole moment and the electric field. There is confusion about the definition of work and the angle used in the formula for potential energy.
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!​

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##.

## What is potential energy?

Potential energy is the energy that an object possesses due to its position or configuration. It is stored energy that has the potential to be converted into other forms of energy.

## What is a dipole?

A dipole is a pair of equal and opposite charges separated by a distance. This creates an electric field between them.

## How is potential energy related to a dipole?

The potential energy of a dipole is the amount of energy required to separate the two charges from each other. It is directly proportional to the magnitude of the charges and inversely proportional to the distance between them.

## How is the potential energy of a dipole calculated?

The potential energy of a dipole can be calculated using the formula U = -pE, where U is the potential energy, p is the dipole moment, and E is the electric field strength.

## Can the potential energy of a dipole be changed?

Yes, the potential energy of a dipole can be changed by changing the distance between the charges or by changing the magnitude of the charges. It can also be affected by external electric fields.

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