Question #532da

1 Answer
Feb 9, 2016

Answer:

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_{}}#.