How do we know the point of 0 electric field is on the axis?

[laser]

Homework Statement

E=0

Relevant Equations

Electric field is a vector

In many homework problems I've encountered, they all seem to assume the electric field = 0 point is along the axis of the two charges. Intuitively it kind of makes sense, but I'm looking for a solid justification for it. In other words, why can't it be off the axis of the two charges? When setting up the problem, everyone seems to just assume it is on the axis.

Couldn't find any information about this in the book/online. Thanks!

 

[Gordianus]

Draw the electric field at points outside the líne that joins q1 and q2

 

[kuruman]

The only way to add two vectors to get zero is if they have equal magnitudes and are antiparallel.

 

[Orodruin]

… and the only places where they are antiparallel is where the two charges and the point of zero field all lie on the same line.