Question

Two long concentric cylindrical conductors of radii $a$ and $b\left(b<a\right)$ are maintained at a potential difference $V$ and carry equal and opposite currents $I$ . An electron with a particular velocity $u$ parallel to the axis will travel undeviated in the evacuated region between the conductors. Then $u=$

• A. $\frac{4\pi \mathrm{V}}{{\mu }_{0}I\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
• B. $\frac{2\pi V}{{\mu }_{0}Iln\frac{a}{b}}$
• C. $\frac{2\pi \mathrm{V}}{{\mu }_{0}I\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
• D. $\frac{2\pi \mathrm{V}}{{\mu }_{0}I\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$

Answer 1

Answer: (B)

$E=\frac{\lambda }{2\pi {ϵ}_{0}r}$
p.d. $V=\frac{\lambda }{2\pi {\epsilon }_0}\ln \left(\frac{\mathrm{a}}{\mathrm{b}}\right)$
$\mathrm{E}=\frac{\mathrm{V}}{r\ln \left(\frac{\mathrm{a}}{\mathrm{b}}\right)}$
$B$ varies with $r$
$B=\frac{{\mu }_{0}I}{2\pi r}$
$B$ and $E$ are perpendicular to each other


$\therefore quB=qE$
(If particle travel undeviated)
$u=\frac{E}{B}=\frac{2\pi V}{{\mu }_{0}I\ln \left(\frac{a}{b}\right)}$