Force On A Current Carrying Conductor
3 min read
Force on a Current Carrying Conductor: A conductor has free electrons which can move in the presence of a field. Since a magnetic field exerts a force on a charge moving with a velocity
, it also exerts a force on a conductor carrying a current.
Consider a conducting wire carrying a current . The current density at any point in the wire is given by'
![\begin{displaymath}\vec J = ne\vec v\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image004_0006.gif)
where
is the number density of electrons having a charge
each and
is the average drift velocity at that point.
![$n$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image005_0005.gif)
![$e$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image006_0006.gif)
![$\vec v$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image002_0008.gif)
Consider a section of length
of the wire. If
is the cross sectional area of the section oriented perpendicular to the direction of
, the force on the electrons in this section is
![\begin{displaymath}d\vec F = dq\vec v\times\vec B\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image010_0003.gif)
![$dl$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image007_0004.gif)
![$A$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image008_0004.gif)
![$\vec J$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image009_0004.gif)
![\begin{displaymath}d\vec F = dq\vec v\times\vec B\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image010_0003.gif)
where
is the amount of charge in the section
![$dq$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image011_0002.gif)
![\begin{displaymath}dq = neAdl\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/main3_clip_image012_0003.gif)
![](https://raw.githubusercontent.com/xibsked/menka/master/books/physics-for-engineers-2/64428cdc393c35eef7a5a3712f53769b1.png)
Thus the force on the conductor in this section is
If represents a vector whose magnitude is the length of the segment and whose direction is along the direction of
, we may rewrite the above as
The net force on the conductor is given by summing over all the length elements. If denotes a unit vector in the direction of the current, then