Three identical charges \(q\) form an equilateral triangle of side \(a\) with two charges on the \(x\) -axis and one on the positive \(y\) -axis. (a) Find an expression for the electric field at points on the \(y\) -axis above the uppermost charge. (b) Show that your result reduces to the field of a point charge \(3 q\) for \(y \gg a\).

The concept of an electric field at a point is akin to feeling the wind on your face; just as you can feel the wind's direction and strength without seeing it, an electric field represents the influence an electric charge exerts on other charges around it, without them being in contact. At any given point in space, this electric field can be visualized as a vector that has both magnitude and direction. Lone charges create an electric field radiating outward, and the strength of this field decreases with the square of the distance from the charge.
Imagine being near a single charge; you’d feel a ‘force’ symbolizing the electric field's strength that pushes or pulls on other charges within the vicinity. Mathematically, it's described by the equation:

• \(E = k\frac{|q|}{r^2}\)

, where \(E\) is the electric field strength, \(k\) is Coulomb’s constant, \(q\) is the charge creating the field, and \(r\) is the distance from the charge to the point of interest. For the homework question, the electric field at any point on the y-axis is determined uniquely by the geometrical setup and the contributions from each charge involved.

The superposition principle is a critical concept that helps us simplify the complex interactions of multiple charges. It states that the total electric field created by several charges is the vector sum of the fields created by each charge individually.
Think of it like having multiple speakers in a room; each one adds to the overall sound level, and the sounds blend together seamlessly. Similarly, in electrostatics, you 'hear'—or rather, detect—the combined 'volume'—that is, electric field strength—from all charges. To calculate the total electric field at a point, we add up the vectors of the electric fields from each charge. This is precisely what was done in the exercise: after determining the electric field contributions from each charge using the equation for a single charge's field, these vectors are added up. However, because electric fields are vectors, direction matters, and this is why \(E_B\) and \(E_C\) result in cancellation in the x direction, while their y components add up to give the resultant electric field \(E_P\).

The behavior of point charges is elegantly simple yet profoundly fundamental in understanding electric fields. A point charge is an idealized charge with no spatial extent; it’s treated as if all of its charge is concentrated at a single point. This simplification allows for handling electric fields through basic algebra and calculus. In the context of the homework exercise, the electric fields due to each of the point charges contribute to the total electric field at a point above them.
With point charges arrayed in space, like our three charges forming a triangle, their individual fields can be combined using vectors. When evaluating the electric field of multiple point charges, it is essential to note the linearity of the electric field, which means the resulting field is the sum of fields from individual charges, consistent with the superposition principle. Hence, even though we deal with three charges, by adding their electric fields and considering their symmetrical positioning and cancellation of certain components, we distill their collective effect into a singular, comprehensible expression.