Animated Solution for Physics - Electrostatics: Figure shows lines of constant potential in a region in which an electric field is present. The values of the potential are written in brackets. Of the points $A$, $B$ and $C$, the magnitude of the electric field is greatest at the point…… .

The journey to mastering electrostatics often involves developing a deep visual intuition for electric fields and potentials. This problem is a beautiful exercise in exactly that—translating a visual map of equipotential lines into a physical understanding of electric field strength.

Imagine you are looking at a topographic map of a mountain. The lines on the map represent regions of constant elevation. If the lines are close together, the slope is steep. If they are far apart, the terrain is relatively flat.

Equipotential lines work in the exact same way, but instead of gravitational potential (height), they represent electric potential (voltage). In our given figure, we see a series of curved lines, each labeled with a specific voltage: $50\text{V}$, $40\text{V}$, $30\text{V}$, $20\text{V}$, and $10\text{V}$.

We are asked to determine where the electric field is the strongest among three specific points: $A$, $B$, and $C$.

To crack this, we need to bridge the gap between potential and electric field. The fundamental relationship is given by the potential gradient:

In a more practical, discrete form, the magnitude of the electric field can be approximated as:

Here, $\Delta V$ is the potential difference between two adjacent equipotential lines, and $\Delta r$ is the perpendicular distance between them.

Notice something crucial about our map: the potential difference $\Delta V$ between any two adjacent lines is exactly $10\text{V}$. It is a constant!

Since $\Delta V$ is constant, the magnitude of the electric field $|E|$ is inversely proportional to the spacing $\Delta r$:

This is our golden rule. A smaller distance $\Delta r$ means a stronger electric field.

Now, let's simply look at the points $A$, $B$, and $C$. At point $A$, the lines are relatively far apart. At point $C$, they are also quite spaced out. But look at point $B$! The equipotential lines are crowded very close together. The distance $\Delta r$ is the absolute smallest at this location.

Because the spacing is minimum at point $B$, the potential gradient is the steepest. Therefore, the magnitude of the electric field is greatest at point $B$.

Bonus Insight: The electric field always points from higher potential to lower potential, and it crosses the equipotential lines at exactly $90^{\circ }$. At point $B$, the electric field vector would point roughly towards the bottom left, slicing perpendicularly through the tightly packed lines.