Electrostatics > The potential of an electric field. Conductors in a constant electric field > Problem 6.3.34

Statement

$6.3.34.$ The hoop uniformly charged with a positive charge rests on four rollers and
can rotate. One section of the hoop passes through a hole made in parallel
differently charged plates. According to the inventor, the area of the hoop
located between the plates will be attracted to the negative plate and repelled
from the positive one. There is no field outside the plates. Thus, the rotation
of the hoop will be maintained even if there is resistance to movement — a
perpetual motion machine is obtained. What is the inventor’s mistake? Prove
that the moment of forces acting on such a hoop in any electrostatic field is
zero.

Solution

The error here is ignoring the conservative nature of the electrostatic field, in addition to the symmetry of the ring, of course.

the incorrect assumption is that:

The electric field is strictly confined between the plates and is zero outside them.
And the net force on the portion of the ring located between the plates produces an unbalanced torque.

but this is a mistake .

Consider a rigid ring with uniform linear charge density $\lambda > 0$, radius a, which can rotate freely about its axis. It is immersed in an arbitrary external electrostatic field
$\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r})$.

We want to show that the resultant torque of the electric forces about the center of the ring,

$\mathbf{M},$ is identically zero.

The electrostatic potential energy of the ring in the external field is:

$U = \int_{\text{ring}} V(\mathbf{r})\, dq = \lambda \oint V(\mathbf{r})\, dl$

where $dq = \lambda dl$ and the integral is taken over the circumference of the ring.

If we rotate the ring by an angle $\theta$ about its axis, the charge distribution in space changes because the ring is a rigid object. The new potential energy will be a function of $\theta: U(\theta)$.
the total torque of the forces on a rigid system that can rotate is:

$M = -\frac{dU}{d\theta}$

(where M is the component of the torque along the rotation axis,).

Now, the charge distribution of the ring is invariant under rotation: when the ring is rotated by an arbitrary angle, the spatial charge distribution is exactly the same as at the beginning (because the charge is uniformly distributed around the ring).
Since the external field $\mathbf{E}(\mathbf{r}$) is fixed (does not change with time), the potential energy U does not depend on$ \theta:$

$U(\theta) = \text{constant} \quad \Rightarrow \quad \frac{dU}{d\theta} = 0$

Therefore,

${M = 0}$

There is no net torque to accelerate the rotation. Any initial motion would be damped by losses (friction, air resistance) until it stops.

Answer

The invention fails because, in an electrostatic field, a rigid body that is uniformly charged and has rotational symmetry cannot experience a net torque that sustains perpetual motion. The net force on the section between the plates is exactly compensated by forces on the rest of the ring, due to the actual fringing field. The total torque of the electrostatic forces on the ring is zero, which prevents continuous rotation without an external energy input.
so we have
$\boxed{M = 0}$