Правка разделов «Statement», «Solution», «Answer»

Alexphysics правка от 15 июн. 2026 г., 03:29

правка #18789 предыдущая #18788 ← раньше позже →

en/6.3.34.md +30 −18

		@@ -1,47 +1,59 @@
			### Statement

		−	$6.3.34.$ [~~Insert the problem 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 incorrect assumption is that:

		−	1. The electric field is strictly confined between the plates and is zero outside them.
		−	~~2. T~~he net force on the portion of the ring located between the plates produces an unbalanced torque.
		+	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.

		−	What actually happens is that, although the field is concentrated between the plates, the field lines at the edges curve and fields exist outside the interplate region.
		−	These fields then act on the rest of the ring and generate forces that exactly cancel any net torque.
		+	but this is a mistake .

		−	~~Furthermore, since the electrostatic field is conservative, the potential energy of the ring cannot change over a complete rotation, so no net work can be extracted cyclically.~~
		+	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
		+	$\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r})$.

		−	~~Consider a rigid ring with uniform linear charge density \lambda > 0, radius a, which can rotat~~e fre~~ely 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,

		−	~~We want to show that the resultant torque of the electric forces about the center of the ring,~~ \mathbf{M}, is identically zero.
		+	$\mathbf{M},$ 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
		+	$U = \int_{\text{ring}} V(\mathbf{r})\, dq = \lambda \oint V(\mathbf{r})\, dl$

		−	where dq = \lambda dl and the integral is taken over the circumference of the ring.
		+	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)~~. From mechanics, the total torque of the forces on a rigid system that can rotate is:~~
		+	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}
		+	$M = -\frac{dU}{d\theta}$

		−	(where M is the component of the torque along the rotation axis, ~~with the appropriate sign~~).
		+	(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:
		+	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
		+	$U(\theta) = \text{constant} \quad \Rightarrow \quad \frac{dU}{d\theta} = 0$

			Therefore,

		−	{M = 0}
		+	${M = 0}$

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

			#### Answer

		−	~~[Insert a concise answer or boxed result]~~
		+	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
		+	$\boxed{M = 0}$

	@@ -1,47 +1,59 @@
	### Statement		### Statement

	$6.3.34.$ [~~Insert the problem 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		### Solution

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

	1. The electric field is strictly confined between the plates and is zero outside them.		The electric field is strictly confined between the plates and is zero outside them.
	~~2. T~~he net force on the portion of the ring located between the plates produces an unbalanced torque.		And the net force on the portion of the ring located between the plates produces an unbalanced torque.

	What actually happens is that, although the field is concentrated between the plates, the field lines at the edges curve and fields exist outside the interplate region.		but this is a mistake .
	These fields then act on the rest of the ring and generate forces that exactly cancel any net torque.

	~~Furthermore, since the electrostatic field is conservative, the potential energy of the ring cannot change over a complete rotation, so no net work can be extracted cyclically.~~		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
			$\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r})$.

	~~Consider a rigid ring with uniform linear charge density \lambda > 0, radius a, which can rotat~~e fre~~ely 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,

	~~We want to show that the resultant torque of the electric forces about the center of the ring,~~ \mathbf{M}, is identically zero.		$\mathbf{M},$ is identically zero.

	The electrostatic potential energy of the ring in the external field is:		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		$U = \int_{\text{ring}} V(\mathbf{r})\, dq = \lambda \oint V(\mathbf{r})\, dl$

	where dq = \lambda dl and the integral is taken over the circumference of the ring.		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)~~. From mechanics, the total torque of the forces on a rigid system that can rotate is:~~		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}		$M = -\frac{dU}{d\theta}$

	(where M is the component of the torque along the rotation axis, ~~with the appropriate sign~~).		(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).		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:		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		$U(\theta) = \text{constant} \quad \Rightarrow \quad \frac{dU}{d\theta} = 0$

	Therefore,		Therefore,

	{M = 0}		${M = 0}$

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

	#### Answer		#### Answer

	~~[Insert a concise answer or boxed result]~~		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
			$\boxed{M = 0}$