Question

A point charge q moves with velocity v along a straight line. Displacement current density at a point located at a distance r from charge and in the line of motion of the charge will be

• A. $\frac{q{v}^2}{4\pi r^2}$
• B. $\frac{qv}{r^3}$
• C. $\frac{qv}{4\pi r^3}$
• D. $\frac{qv}{2\pi r^3}$

Answer 1

The correct option is D $\frac{qv}{2\pi r^3}$

$i_d={\epsilon }_0\frac{d}{dt}EA$
$i_{density}=\frac{i_d}{A}={\epsilon }_0\frac{d}{dt}E$ ,where $E=\frac{kq}{r^2}$
$i_{density}={\epsilon }_0\frac{d}{dt}\frac{kq}{r^2}={\epsilon }_{0}kq\frac{d}{dt}{r}^{-2}=\frac{-2{\epsilon }_{0}kq}{r^3}v$
By placing $k=\frac{1}{4\pi {\epsilon }_0}$,
$i_{density}=\frac{qv}{2\pi r^3}$