Animated Solution for Physics - Electrostatics: Two point charges $+8q$ and $-2q$ are located at $x=0$ and $x=L$, respectively. The location of a point on the x-axis at which the net electric field due to these two point charges is zero, is

Balancing Electric Fields

Imagine a microscopic tug-of-war between two invisible forces. On one side, we have a powerful positive charge, $+8q$, anchored firmly at the origin ($x=0$). On the other side, a weaker negative charge, $-2q$, is stationed at a distance $L$ ($x=L$). Our mission is to find the exact coordinate on the x-axis where these two opposing electric fields perfectly cancel each other out. This magical location is known as the null point.

Before we dive into any equations, we must use our physical intuition to locate the general region of this null point. The net electric field ${\vec{E}}_{\text{net}}$ is the vector sum of the fields produced by both charges. For the net field to be zero, the individual fields must be equal in magnitude but strictly opposite in direction.

Let's break the x-axis into three distinct regions: 1. Between the charges ($0<x<L$): Here, the positive $+8q$ charge pushes the electric field to the right, and the negative $-2q$ charge pulls the electric field to the right. Since both vectors point in the same direction, they will always add up. A null point is impossible here. 2. To the left of the origin ($x<0$): In this region, the fields do point in opposite directions. However, the $+8q$ charge is not only closer but also four times stronger in magnitude than the $-2q$ charge. The field from $+8q$ will always overpower the field from $-2q$. No null point here either. 3. To the right of $x=L$ ($x>L$): Here, the fields oppose each other. The point is closer to the weaker $-2q$ charge, which compensates for its smaller magnitude. The stronger $+8q$ charge is further away. This is the only region where the two fields can perfectly balance!

Now that we know the null point lies at some coordinate $x>L$, we can set up our mathematical condition. We equate the magnitudes of the two electric fields:

Substituting the formula for the electric field of a point charge, we get:

Notice how beautifully this equation simplifies. The constants $\frac{1}{4\pi {ϵ}_0}$ and the charge variable $q$ are present on both sides. We can cancel them out immediately, leaving us with a clean algebraic relationship:

At this stage, it might be tempting to cross-multiply and expand the squared terms, which would lead to a messy quadratic equation. But there is a much more elegant path. Since both sides are perfect squares, we can simply take the square root of both sides:

Now, we cross-multiply to solve for $x$:

Subtracting $x$ from both sides and moving $2L$ over, we arrive at our final destination:

And there we have it! The electric field vanishes exactly at the coordinate $x=2L$. By combining physical intuition with elegant algebra, we bypassed complex quadratics and found the null point with absolute certainty.