Question

A fully charged capacitor has a capacitance C. It is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat s and mass m. If the temperature of the block is raised by $\Delta T$ , the potential difference V across the capacitor is:
A. $\dfrac{{ms\Delta T}}{C}$
B. $\sqrt {\dfrac{{ms\Delta T}}{C}}$
C. $\sqrt {\dfrac{{2ms\Delta T}}{C}}$
D. $\dfrac{{ms\Delta T}}{s}$

Answer 1

The capacitor is initially charged and it is discharged. The energy lost during the discharging of the capacitor will be gained to raise the temperature of the block. Since, there is no external energy loss, the energy lost by the capacitor will be equal to the energy required to raise the temperature of the block.

Complete step by step answer:
Let us understand what is happening during the discharge of the capacitor. Initially, the capacitor is fully charged.The energy stored in the capacitor is given as:
$E = \dfrac{1}{2}C{V^2}$
Where $E$ is the energy stored in the capacitor;$C$ is the capacitance of the capacitor and $V$ is the potential difference across the capacitor.
When the capacitor is discharging, energy is released in the process.This energy is responsible to raise the temperature of the block.The energy required to raise the temperature of the block by $\Delta T$ is given as:
$Q = ms\Delta T$
Where $Q$ is the energy required to raise the temperature of the block;$m$ is the mass of the block and $s$ is the specific heat capacity.
Both the energy must be equal, therefore we have
$E = Q$
$\Rightarrow \dfrac{1}{2}C{V^2} = ms\Delta T$
$\therefore V = \sqrt {\dfrac{{2ms\Delta T}}{C}}$
Therefore, the potential difference across the capacitor is $\sqrt {\dfrac{{2ms\Delta T}}{C}}$

Hence,option C is the correct option.

Note: Do remember the formula of the energy stored in the capacitor and the energy required to raise the temperature of a block by $\Delta T$ .Remember that when there is no heat loss, the energy of the system is conserved.The energy stored in a capacitor is nothing but the electric potential energy and is related to the voltage and charge on the capacitor. If the capacitance of a conductor is C, then it is initially uncharged and it acquires a potential difference V when connected to a battery.