Question

The capacitor plates are fixed on an inclined plane and connected to a battery of e.m.f. $E$. The capacitor plates have area $A$, length $l$ and the distance between them is $d$. A dielectric slab of mass $m$ and dielectric constant $k$ is inserted into the capacitor and tied to a mass $M$ by a massless string as shown in the figure. Find the value of $M$ for which the slab will stay in equilibrium. There is no friction between slab and plates.

• A. $\frac{m}{2}+\frac{E^2{\epsilon }_{0}A(k-1)}{2lgd}$
• B. $\frac{m}{2}-\frac{E^2{\epsilon }_{0}A(k-1)}{2lgd}$
• C. $\frac{m}{2}+\frac{E^2{\epsilon }_{0}A(k-1)}{2lgd}$
• D. $\frac{m}{2}-\frac{E^2{\epsilon }_{0}A(k-1)}{lgd}$

Answer 1

The correct option is A $\frac{m}{2}+\frac{E^2{\epsilon }_{0}A(k-1)}{2lgd}$

From given, we can draw FBD as
the force applied by the plates of capacitor on dielectric is given by:
$F_c=\frac{E^2(k-1)}{2l}\frac{{\epsilon }_{0}A}{d}$

From balancing condition:

$Mg=\frac{E^2(k-1)}{2l}\frac{{\epsilon }_{0}A}{d}+mg\times \frac{1}{2}$

$M=\frac{E^2(k-1){\epsilon }_{0}A}{2lgd}+\frac{mg}{2g}$

$M=\frac{m}{2}+\frac{E^2(k-1){\epsilon }_{0}A}{2lgd}$