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

The field for any point on the x-axis will determined as

The given parameters;

charge of each particle, = qthe position of one charge = + aposition of the second charge = -a

The electric field between these charge is determined by applying Coulomb's law as shown below;

where;

r is the distance between the charges

The resultant distance at any point in the x-axis is calculated as follows;

The electric field is calculated as follows;

Thus, the field for any point on the x-axis will determined as

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The electric field produced by a single particle of charge q is given by:

where  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 

And so, if we substitute this into the  formula of the electric field, we have


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

And so, the field generated by charge 2 is


so, the total field is