Question

A $450\,\mu F$ capacitor is charged to $295\,{\text{V}}$ . Then a wire is connected between the plates. Thermal energy is produced as the capacitor discharges if all of the energy that was stored goes into the heating the wire is approximately $20x\,{\text{J}}$ . What is the value of $x$ ?

Answer 1

First we will find the expression for the energy of a capacitor. After we find the magnitude of energy, we will equate with the one given in the question, to find the value of $x$ accordingly.

Formula used:
Energy of a capacitor is given by:
$U = \dfrac{1}{2}C{V^2}$ …… (1)
Where,
$U$ indicates the energy stored in the capacitor.
$C$ indicates the capacity of the capacitor.
$V$ indicates the potential difference.

Complete step by step answer:
In the given question, we are supplied the following data:
There is a capacitor whose capacity is $450\,\mu F$ .The capacitor is charged to a voltage of $295\,{\text{V}}$ .A wire is connected between the plates which discharges the capacitor and the energy of the plates is converted to heat up the wire.The heat energy found is approximately $20x\,{\text{J}}$ .We are required to find the value of $x$ .

To begin with, we will need to find the energy of the capacitor. For that we also have a formula which was mentioned above. Substituting the required values in the equation (1), we get:
U = \dfrac{1}{2}C{V^2} \\\ \Rightarrow U = \dfrac{1}{2} \times 450 \times {10^{ - 6}} \times {295^2} \\\ \Rightarrow U = 19.6\,{\text{J}} \\\ \Rightarrow U \sim 20\,{\text{J}} \\\
Therefore, the energy of the capacitor is found to be $20\,{\text{J}}$ .
So, now, we equate the following:
$20x\,{\text{J}} = 20\,{\text{J}} \\\ \Rightarrow x = \dfrac{{20}}{{20}} \\\ \therefore x = 1$

Hence, the value of $x$ is $1$.

Note: It is important to remember that since energy can neither be destroyed nor be created, it can only be transformed from one form to the other. The energy between the plates of the capacitor is used to heat up the wire, by transferring all the energy.