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.

Schematic of the electric field of two opposite point charges with a certain separationProblem 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.

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