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. $\frac{2 \pi V}{\mu _{0} I ln \frac{a}{b}}$
• 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)

$E=\frac{\lambda }{2 \pi \epsilon _{0} r}$
p.d. $\quad V=\frac{\lambda}{2 \pi \varepsilon_0} \ln \left(\frac{\mathrm{a}}{\mathrm{b}}\right)$
$\mathrm{E}=\frac{\mathrm{V}}{r \ln \left(\frac{\mathrm{a}}{\mathrm{b}}\right)}$
$B$ varies with $r$
$B=\frac{\mu _{0} I}{2 \pi r}$
$B$ and $E$ are perpendicular to each other


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