Animated Solution for Physics - Electrostatics: Three positive charges of equal value $q$ are placed at the vertices of an equilateral triangle. The resulting lines of force should be sketched as in

A Symmetrical Trio

Imagine three identical positive charges, each of magnitude $+q$, sitting perfectly at the corners of an equilateral triangle. This highly symmetric arrangement creates a fascinating and complex electric field in the surrounding space. Our goal is to visualize this invisible force field by sketching its electric lines of force.

To do this correctly, we must rely on the fundamental laws of electrostatics rather than just guessing.

Before we evaluate the given options, let us recall the golden rules that govern electric field lines:

1. Origin and Termination: Electric field lines always originate from positive charges and terminate at negative charges. If there are no negative charges nearby, they must travel outwards to infinity. 2. No Closed Loops: This is a crucial property. Electrostatic field lines never form closed loops. This is a direct consequence of the conservative nature of the electrostatic field. If they formed a closed loop, the work done in moving a test charge along that loop would be non-zero, which violates the principle of energy conservation. 3. No Intersections: Two field lines can never cross each other. If they did, it would imply two different directions of the electric field at a single point, which is physically impossible. 4. Lateral Repulsion: Field lines traveling in the same direction exert a lateral pressure on each other, causing them to spread apart.

Armed with these rules, let us critically examine the provided sketches:

Option (a) shows the field lines forming continuous closed loops between the three charges. As we just established, electrostatic fields are conservative, making closed loops strictly impossible. This option is immediately discarded.

Option (b) depicts the field lines originating from the positive charges and terminating abruptly at the center of the triangle. However, field lines can only terminate on a negative charge or at infinity. Since the center is just empty space, this representation is physically flawed.

Option (d) displays closed concentric circles around the charges. These circular patterns actually represent equipotential surfaces, not electric field lines. Electric field lines must always be perpendicular to these surfaces.

This leaves us with Option (c), which is the perfect representation. Notice how the lines originate from each positive charge and travel outwards towards infinity.

Because like charges repel, their field lines also exert a lateral pressure on each other. This causes the lines in the region between the charges to bend away from the center.

Right at the centroid of the equilateral triangle, the electric field vectors from all three charges are equal in magnitude and separated by $120^{\circ }$. They perfectly cancel each other out, creating a neutral point where the net electric field is exactly zero ($E_{net}=0$). Option (c) beautifully captures this outward flow, the lateral repulsion, and the central void of the neutral point.