when do I use absolute value brackets in the electric field equation E=k(q)/r^2 ?

Question

when do I use the absolute value brackets on the point charge?

 

E = k (q)/r^2 

 

vs

 

E = k |q|/r^2

 

 ?

Answer 1

If you only want the magnitude of the electric field at a distance r from a point source charge q, use the absolute value.  If you want the vector E, then you need to account for the sign of q in a coordinate system.  In this case, the coordinate system of choice has its origin at the position of the source point q.  Suppose you want E at a field point r given by 

 

r = re_r

 

where r is the distance (always positive) from the source point to the field point, and e_r is a unit vector directed from the source point toward the field point.  Then

 

E(r) = kq/r² e_r

 

Here you can see where the sign of q is crucial.  If q is positive, the field lines are directed radially away from the source point, parallel to e_r.  But if q is negative, the field lines are directed radially towards the source point, opposite the direction of e_r.

Answer 2

Arturo's answer is very complete regarding the case of using polar (or spherical, in three dimensions) coordinates.  In that case, the sign of the charge indicates the direction of the field in the radial dimension.

 

However, in the case of Cartesian coordinates (which are more commonly used at the fundamental level), the situation gets a little trickier.

 

For example, imagine a positive charge at the origin.  The electric field points away from it in every direction.  Along the positive x axis, this means the field points in the positive x direction.  However, along the negatvie x axis, the field points in the negative direction.  This despite the fact that putting a positive charge into the expression yields a positive number in both cases.  The same is true of the y axis.

 

The problem is that the sign indicating direction of the vector, and the sign of charge, do not mean the same thing.  A negative electric field vector means the electric field points in the negative direction.  But the negative sign on the charge has nothing to do with direction, in general.  It is merely meant to distinguish one type of charge from the other.  From the very early study of electric forces and fields, it was known there were two distinct kinds of charges that behaved differently, and had to be distinguished.  The tradition became to call them positive and negative (or plus and minus) charges.  But there is nothing sacrosanct about that.  They could have been called anything to distinguish the pair:  1 and 2, A and B, Peanut Butter and Jelly.  Anything that would make one type of charge distinct from the other.  These labels were not meant to indicate direction, in general.  They can, when certain coordinates are used, but, in general, they do not.

 

For that reason, in Coulomb's law and electric field expressions, I usually expunge the signs of the charges and use them to get the magnitude.  Then I use my knowledge of the sign of the charge, and where I am in space, to get the direction of the force or field.

Now, when it comes to the scalar quantities of electric potential and work, the sign of the charge and the sign of the quantity are much more closely aligned, but that is a different story

 

That is just my preference, but I find it works well in most cases.  Great question!