# Question #532da

Feb 9, 2016

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 $\setminus \vec{{F}_{}}$ is related to this scalar potential energy $U$ as $\setminus \vec{{F}_{}} = - \setminus \nabla U$, where $\setminus \nabla U$ is the gradient of this potential energy.

When a magnet of dipole moment ( $\setminus \vec{\setminus {\mu}_{}}$) is placed in an external magnetic field ($\setminus \vec{{B}_{}}$), it has a potential energy of : $U = - \setminus \vec{\setminus {\mu}_{}} . \setminus \vec{{B}_{}}$.

In the Stern-Gerlach Experiment the external magnetic field is directed parallel to $z$ axis and is inhomogeneous. i.e its value changes with $z$.
$\setminus \vec{{B}_{}} = B \left(z\right) \setminus \hat{z}$$\setminus q \quad$ $\setminus \vec{{F}_{}} = - \setminus \nabla U \setminus \approx \setminus {\mu}_{z} \setminus \frac{\setminus \partial {B}_{z}}{\setminus \partial z}$

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 ($\setminus \vec{\setminus {\mu}_{}}$) and the applied inhomogeneous magnetic field $\setminus \vec{{B}_{}}$.