Animated Solution for Physics - Electrostatics: Two points $P$ and $Q$ are maintained at the potentials of $10\text{ V}$ and $-4\text{ V}$, respectively. The work done in moving 100 electrons from $P$ to $Q$ is

Imagine you are standing in an electric field. You have two distinct points, $P$ and $Q$. The potential at point $P$ is $V_P=10\text{ V}$, and the potential at point $Q$ is $V_Q=-4\text{ V}$.

Our mission is to move a specific amount of charge from point $P$ to point $Q$. The charge we are moving consists of $100$ electrons.

To find the work done by an external agent in moving a charge $Q$ between two points in an electric field, we use the fundamental relationship:

Here, $\Delta V$ is the change in potential, which is always the final potential minus the initial potential. Since we are moving from $P$ to $Q$, the change in potential is:

Now, let's talk about the charge $Q$. We are moving $100$ electrons. Remember, electrons carry a negative charge! The charge of a single electron is $e=-1.6\times 10^{-19}\text{ C}$. Therefore, the total charge $Q$ is:

Let's carefully substitute all our values into the master equation.

First, let's simplify the potential difference:

Now, let's simplify the total charge:

Multiplying these two together:

Notice how the two negative signs cancel each other out, resulting in a positive value.

The positive sign is physically significant. It tells us that the external agent must actively do work to move these electrons. Why? Because electrons naturally want to move towards higher potentials (from $-4\text{ V}$ to $10\text{ V}$). Forcing them to move to a lower potential requires external effort!