Animated Solution for Physics - Electrostatics: The work done in placing a charge of $8\times 10^{-18}\text{ C}$ on a condenser of capacity $100\mu \text{F}$ is

Imagine you are trying to pack a lot of people into a small room. The more people you push in, the harder they push back. Charging a capacitor is exactly like that! As you place more and more charge onto the plates of a capacitor, the existing charge repels the new charge. The work you do against this electrostatic repulsion doesn't just vanish; it gets stored inside the capacitor as electrostatic potential energy.

In this problem, we are asked to find the work done in placing a specific amount of charge on a condenser (which is just an older term for a capacitor).

Let's look at the data we have been given:

1. The charge to be placed on the capacitor is $q=8\times 10^{-18}\text{ C}$. 2. The capacity (or capacitance) of the condenser is $C=100\mu \text{F}$.

Before we jump into any formulas, we must ensure all our units are in the standard SI format. The capacitance is given in microfarads ($\mu \text{F}$), so we need to convert it to farads ($\text{F}$):

The work done $W$ in charging a capacitor is equal to the energy $U$ stored in it. Depending on what variables are known, we can express this energy in three different ways:

Since we are given the charge $q$ and the capacitance $C$, the most direct tool for our job is:

Now, let's carefully substitute our values into the formula. Watch out for the powers of 10, as this is where silly mistakes often happen!

First, let's square the numerator. The square of $8$ is $64$, and when you square an exponent, you multiply the power by $2$. So, $(10^{-18}{)}^2$ becomes $10^{-36}$.

Now, we can separate the numbers and the powers of 10 to make the division easier:

Dividing $64$ by $2$ gives us $32$. For the exponents, when dividing, we subtract the denominator's power from the numerator's power: $-36-(-4)=-32$.

And there we have it! The work done in placing the charge on the condenser is $32\times 10^{-32}\text{ J}$. This perfectly matches option (d).