Question

In the figure shown a parallel plate capacitor is connected across a source of emf $\epsilon$. The plates are square shaped with edge $\ell$ and separated by a distance $d$. A dielectric slab of dielectric constant $k$ and thickness $d$ is inserted between the plates with constant speed $v$. A current of equal the resistance of connecting wires, ignore the resistance of connecting wires. A current of equal strength is passed through a coil of resistance $R$ and dipped into water kept in a bucket at $0^{o}C$. What would be the temperature of the water at the end of $5$ minutes if mass of the water is $m$ and specific heat of the water is $c$?

Answer 1

Consider that length $x$ of the dielectric is inside the capacitor. The capacitance of the system is
$C=\frac{{\epsilon }_{0}kx\ell }{d}+\frac{{\epsilon }_0\ell (\ell -x)}{d}=\frac{{\epsilon }_0\ell }{d}\left[\right(k-1)x+\ell ]$
Charge on the capacitor
$q=\frac{{\epsilon }_0\ell }{d}\left[\right(k-1)x+\ell ]\epsilon$
$⟹\frac{dq}{dt}=\frac{{\epsilon }_0\ell }{d}(k-1)\epsilon \frac{dx}{dt}$
$I=\frac{{\epsilon }_0\ell \epsilon v(k-1)}{d}$
$\because$ Heat produced in time $t=I^{2}Rt$
Let temperature of the water increase by $T$
$⟹mcT=I^{2}Rt$
$⟹T=\frac{I^{2}Rt}{mc}$