# An electric dipole placed in a non-uniform electric field

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?

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 $$x=0$$.

• Isn't the expression F= P. dE/dR valid only for a short dipole? – Devanshu Pandey Nov 26 '19 at 14:11
• @DevanshuPandey Yes, this answer assumes an ideal electric dipole, as do the underlying expressions of force and torque. – probably_someone Nov 26 '19 at 14:17
• 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? – Devanshu Pandey Nov 26 '19 at 14:22
• @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. – probably_someone Nov 26 '19 at 14:36