Why does the given conserved quantity mean the motion is on a cone?

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deuteron
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An electrone moves in a magnetic field ##B(\vec r)=g \frac {\vec r}{|\vec r|^3}##. Why does the conservation of the quantity $$\vec J=\vec r \times\vec p +eg\frac {\vec r}{|\vec r|}$$ mean that the motion is on the surface of a cone?
 
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  • #2
Is this homework?
 
  • #3
You multiplying ##\vec{J}## with ##\vec{r}## gives
$$\vec{r} \cdot \vec{J}=e g r.$$
Now use spherical coordinates with ##\vec{J}/J## as the polar axis. Then the equation implies
$$J x_3 =e g r \; \Rightarrow \; \cos \vartheta=\frac{x_3}{r}=\frac{e g}{J}=\text{const},$$
which is the (implicit equation of a cone).

In the spherical coordinates you thus have
$$\vec{r}=\begin{pmatrix} r \sin \vartheta \cos \varphi \\ r \sin \vartheta \sin \varphi \\ e g r/J \end{pmatrix},$$
which describes a cone since ##\vartheta=\text{const}##.
 
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Related to Why does the given conserved quantity mean the motion is on a cone?

Why does the conservation of angular momentum imply motion on a cone?

When angular momentum is conserved, the vector representing angular momentum remains constant in both magnitude and direction. As a result, the motion of the object must occur in such a way that it maintains a constant angle with this fixed vector, leading to a conical trajectory.

How does the conservation of energy relate to motion on a cone?

The conservation of energy ensures that the kinetic and potential energy of the system remains constant. In the context of motion on a cone, this means that the object moves in a way that balances these energies, often resulting in a stable, periodic motion that traces out a cone.

What role does the central force play in conical motion?

A central force, which always points towards a fixed point (such as gravity towards the center of the Earth), ensures that the object is constrained to move in a path where the distance to the center remains constant. This constraint, combined with the conservation of angular momentum, results in motion on a cone.

Can you explain the geometry of conical motion in simple terms?

Imagine spinning a string with a weight at the end. The weight traces out a circular path, but if you lift or lower your hand, the path of the weight becomes a cone. Similarly, in many physical systems, the combination of forces and conserved quantities leads to a conical path.

Are there real-world examples of conical motion due to conserved quantities?

Yes, one of the most common examples is the precession of a spinning top. Due to the conservation of angular momentum, the axis of the spinning top traces out a cone. Another example is the motion of planets and satellites under gravitational forces, where their orbits can be conical sections.

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