Question

Question #532da

Answer 1

This is the mathematical expression of the fact that Force is the gradient of Potential Energy.

Explanation:

A potential energy can be defined for any purely conservative force. Potential energy is a scalar and Force is a vector.

The force vector `\vec{F_{}}` is related to this scalar potential energy `U` as `\vec{F_{}} = -\nabla U`, where `\nabla U` is the gradient of this potential energy.

When a magnet of dipole moment ( `\vec{\mu_{}}`) is placed in an external magnetic field (`\vec{B_{}}`), it has a potential energy of : `U=-\vec{\mu_{}}.\vec{B_{}}`.

In the Stern-Gerlach Experiment the external magnetic field is directed parallel to `z` axis and is inhomogeneous. i.e its value changes with `z`.
`\vec{B_{}}=B(z)\hat{z}` `\qquad` `\vec{F_{}}=-\nabla U \approx \mu_z\frac{\partialB_z}{\partial z}`

The approximately symbol is there because it is not practical to get an inhomogeneous magnetix field with purely `z` component alone. In a practical setup the `x` and `y` components are non-zero but negligible.

When silver atoms carrying a magnetic moment enters this inhomogeneous magnetic field, it will experience a force depending on the relative strengths and directions of its magnetic moment vector (`\vec{\mu_{}}`) and the applied inhomogeneous magnetic field `\vec{B_{}}`.