Animated Solution for Physics - Electrostatics: Positive and negative point charges of equal magnitude are kept at $(0,0,a/2)$ and $(0,0,-a/2)$, respectively. The work done by the electric field when another positive point charge in moved from $(-a,0,0)$ to $(0,a,0)$ is

The Zero Work Journey: Navigating the Equatorial Plane of a Dipole

Imagine a 3D coordinate system. We place a positive charge $+q$ at $(0,0,a/2)$ and a negative charge $-q$ at $(0,0,-a/2)$. This arrangement of two equal and opposite charges separated by a small distance forms an electric dipole aligned along the $z$-axis.

Now, we are tasked with moving a test charge from an initial point $A$ to a final point $B$. Let's look closely at their coordinates: $A$ is at $(-a,0,0)$ and $B$ is at $(0,a,0)$.

What do points $A$ and $B$ have in common? Their $z$-coordinates are both zero! This means they lie entirely within the $xy$-plane.

For a dipole aligned along the $z$-axis, the $xy$-plane is its equatorial plane. The defining characteristic of the equatorial plane is that every single point on it is equidistant from the positive charge $+q$ and the negative charge $-q$.

Because electric potential is a scalar quantity given by $V=\frac{1}{4\pi {ϵ}_0}\frac{q}{r}$, the positive potential from $+q$ is exactly canceled out by the negative potential from $-q$ at every point on this plane. Therefore, the net electric potential everywhere on the equatorial plane is exactly zero.

Since both our initial point $A$ and final point $B$ lie on this equatorial plane, the potential at both points is zero:

The work done by the electric field in moving a charge $q_{\text{test}}$ is given by the negative change in potential energy:

Substituting our values, we get:

Furthermore, the electrostatic force is a conservative force. This means the work done is completely independent of the path taken between $A$ and $B$. Whether you move the charge in a straight line, a semi-circle, or a complex zigzag, as long as it starts at $A$ and ends at $B$, the work done by the electric field will always be zero.