Question

A straight conductor of uniform cross section carries a time varying current, which varies at the rate $dI/dt=I$. If $s$ is the specific charge that is carried by each charge carrier of the conductor and $\ell$ is the length of the conductor, then the total force experienced by all the charge carriers per unit length of the conductor due to their drift velocities only is

• A. $F=\dot{I}s$
• B. $F=\frac{\dot{I}}{2\sqrt{Is}}$
• C. $F=\frac{\dot{I}}{s}$
• D. $F=\frac{2I\dot{I}}{s}$

Answer 1

The correct option is C $F=\frac{\dot{I}}{s}$

Momentum of each charge carrier moving with a drift velocity $v$ is $mv$. Total number of charge carriers in the sample is $N=n\left(A\ell \right)$, where $n$ is the number of charge carriers per unit volume and $A$ is area of cross section of the conductor.
Total momentum is $p=N\left(mv\right)=nA\ell mv$
Further, we have
$v=I/neA$
$p=nA\ell m\left[\frac{I}{neA}\right]=\ell \left[\frac{m}{e}\right]I$
Since $F=\frac{dp}{dt}$
$F=\frac{\ell }{S}\dot{I}or\frac{F}{\ell }=\frac{\dot{I}}{s}[\therefore s=specificcharge=e/m]$