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Can an electric dipole placed in a non uniform electric field (say as that of an infinite line charge) ever experience a zero net force and simultaneously a zero net torque on it? I know about the following cases:-

  1. zero net force but non-zero torque:- when the dipole is placed parallel to infinite line charge.

  2. non-zero net force but zero torque :- when the dipole is placed perpendicular to the line charge.

  3. non zero net force and non zero net torque :- when the dipole is placed in any other oblique position.

But is the fourth case possible in any other kind of non uniform field?

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Trivially, you could just have the field be uniform in a finite region around the dipole and not uniform elsewhere, so that the electric field as a whole technically isn't uniform, but this might not be the spirit of the question you're asking. Fortunately, you can just as easily construct situations in which:

  • the electric field is non-uniform and smooth, and
  • there is at least one point where an electric dipole will simultaneously experience no torque and no force.

The torque $\vec{\tau}$ on the dipole is given by:

$$\vec{\tau}=\vec{p}\times\vec{E}$$

where $\vec{p}$ is the electric dipole moment vector. Likewise, the force $\vec{F}$ on the dipole is given by:

$$\vec{F}=\vec{p}\cdot\nabla\vec{E}$$

To enforce zero torque, we need only require that $\vec{p}$ and $\vec{E}$ are parallel at the position of the dipole.

For simplicity's sake, let's say that $\vec{E}$ points in the same direction everywhere, and that $\vec{p}$ is parallel to it. Let's call that direction the $\hat{x}$ direction. In other words, let's say that $\vec{E}=E(\vec{r})\hat{x}$ and $\vec{p}=p\hat{x}$. Then we have that

$$\vec{\tau}=0$$

by construction, and

$$\vec{F}=p\frac{\partial E(\vec{r})}{\partial x}$$

by virtue of everything pointing in the same direction. Wherever $\frac{\partial E(\vec{r})}{\partial x}=0$, a dipole will experience both zero force and zero torque.

For example, the field given by $\vec{E}(\vec{r})=kx^3\hat{x}$ has $\frac{\partial E}{\partial x}=3kx^2$, so a dipole will experience no force and no torque when $x=0$.

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  • $\begingroup$ Isn't the expression F= P. dE/dR valid only for a short dipole? $\endgroup$ – Devanshu Pandey Nov 26 '19 at 14:11
  • $\begingroup$ @DevanshuPandey Yes, this answer assumes an ideal electric dipole, as do the underlying expressions of force and torque. $\endgroup$ – probably_someone Nov 26 '19 at 14:17
  • $\begingroup$ So what if the dipole is quite large just like two opposite charges connected to a massless rod? Would the above said case be then possible? $\endgroup$ – Devanshu Pandey Nov 26 '19 at 14:22
  • $\begingroup$ @DevanshuPandey Then any $E(x)$ which is symmetric about $x=0$ will work, if the dipole is placed along the $x$-axis with its center at $x=0$. For example, $E(x)=kx^2$ will work. But at this point, you're no longer using the dipole formulas; rather, you're just considering the force on each individual charge. $\endgroup$ – probably_someone Nov 26 '19 at 14:36

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