Question

Two particles, each carrying positive charge q, are on the y axis, one at y=+a and the other at y=−a. calculate electric field for any point on the x axis

Answer 1

The electric field produced by a single particle of charge q is given by:
$E(r)=k_e \frac{q}{r^2}$
where $k_e$ is the Coulomb's constant, and r is the distance of the point we are measuring the field from the particle.

The problem asks to calculate the electric field for any point on the x-axis. Let's start with the charge located at y=+a. Let's take a generic point on the x-axis: this will have coordinates (x,0). Therefore, the distance of this point from the charge will be 
$r= \sqrt{x^2+a^2}$
And so, if we substitute this into the  formula of the electric field, we have
$E_1 = k_e \frac{q}{x^2+a^2}$

For the second charge, the electric field is the same. In fact, the  second charge is also +q. It is located at y=-a, therefore the distance from the generic point x will be
$r= \sqrt{x^2+(-a)^2} = \sqrt{x^2+a^2}$
And so, the field generated by charge 2 is
$E_2=k_e \frac{q}{x^2+a^2}$

so, the total field is
$E_{tot}(x) = E_1 + E_2=2k_e \frac{q}{x^2+a^2}$