## The Electric Field of two Point Charges

Sometimes it happens that a thing is more than the sum of its parts. What about two charges? Can their respective electric field behave fundamentally different in some way than just a single charge? In this problem you will learn about two main concepts in electromagnetics - the superposition principle and the dipole.

## Problem Statement

Two electric charges, $$q_1=+q$$ and $$q_2=-q$$, are placed on the $$x$$ axis separated by a distance $$d$$. Using Coulomb's law and the superposition principle, what is the magnitude and direction of the electric field on the $$y$$ axis? What happens if both charges are equal? Draw a schematic of the fields for both cases in the $$x,y$$-plane in a field line plot.

## Background: The Superposition Principle

The superposition principle plays a mayor role in (linear) electrodynamics. It allows the calculation of electromagnetic fields with arbitrary charge distributions.
One configuration is of particular interest - two separated point charges of opposite charge. In the limit of vanishing separation, it is called dipole. Its field fundamentally differs from that of just a single charge even though it is just the sum of the charge. The dipole as a concept is extremely important throughout electrodynamics. It is applied for example explaining the emission of electromagnetic radiation or as a model for molecules, see The Precessing Dipole Molecule. This problem will guide us in this direction.