Question

Two long coaxial cylindrical metal tubes (inner radius $a$ , outer radius $b$ ) stand vertically in a tank of dielectric oil (of mass density $\rho$ , dielectric constant $K$ ). The inner one is maintained at potential $V$ and the outer one is grounded. To what equilibrium height ( $h$ ) does the oil rise in the space between the tubes? [Assume this height ( $h$ ) as an equilibrium height]

• A. $\frac{{\epsilon }_{0}2V^2(K-1)}{g\rho \left(b^2-a^2\right)\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
• B. $\frac{{\epsilon }_0{V}^2(K-1)}{\rho \left(b^2-a^2\right)g\ln \left(\frac{b}{a}\right)}$
• C. $\frac{4{\epsilon }_0{V}^2(K-1)}{g\rho \left(b^2-a^2\right)\ln \left(\frac{b}{a}\right)}$
• D. $\frac{6{\epsilon }_0{\mathrm{V}}^2(\mathrm{K}-1)}{\rho \left({\mathrm{b}}^2-{\mathrm{a}}^2\right)\mathrm{g}\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$

Answer 1

Answer: (B)

$Gravitationalforce=mg$
$=\rho \pi \left(b^2-a^2\right)gh$
Net upward force $F=\frac{dU}{dh}$
$=\frac{1}{2}{V}^2\frac{dC}{dh}$
where $h$ is the height of liquid.
Now we calculate $C$ as a function of $h$




$C_{eq}=C_{Air}+C_K$
$=\frac{2\pi {\epsilon }_0(L-h)}{\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}+\frac{2\pi {\epsilon }_0\mathrm{Kh}}{\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
$C=\frac{2\pi {\epsilon }_0\{L+(K-1)h\}}{\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
$\mathrm{F}=\frac{1}{2}{\mathrm{V}}^2\times \frac{\mathrm{dC}}{\mathrm{dh}}=\frac{1}{2}{\mathrm{V}}^2\times \frac{2\pi {\epsilon }_0(\mathrm{K}-1)}{\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$
$\therefore \mathrm{F}=\mathrm{mg}$
$\frac{\frac{1}{2}{\mathrm{V}}^2\times 2\pi {\epsilon }_0(\mathrm{K}-1)}{\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}=\rho \pi \left({\mathrm{b}}^2-{\mathrm{a}}^2\right)\mathrm{gh}$
$h=\frac{\pi {\epsilon }_0{V}^2(K-1)}{\rho \pi \left(b^2-a^2\right)g\ln \left(\frac{b}{a}\right)}$
$=\frac{{\epsilon }_0{\mathrm{V}}^2(\mathrm{K}-1)}{\rho \left({\mathrm{b}}^2-{\mathrm{a}}^2\right)\mathrm{g}\ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)}$