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# 1.2: Electric Field

#### Coulomb Field

While one can describe the details of forces between charges mathematically, it still is very unsatisfying – how do the charges affect each other from a distance? This question troubled physicists for a long time, and the “solution” (really it is just a model that works) is quite ingenious. It goes like this:

The source of the electric (or for that matter, gravitational) force doesn’t know anything about the existence of another charge “out there.” All it knows is its own charge. The source then sends out a “signal” that radiates away from it radially, and this signal carries with it the information of how much charge the source has and how far the signal travels – the signal gets weaker as it gets farther from the source, because it spreads out on the surface of an ever-growing sphere. Now if another charge happens to be in the space near where this source is, it “receives” the signal, and it takes from it the information about the amount of charge of the source, as well as the signal strength itself (which includes the inverse-square-law separation information), and the direction from which the signal is coming. The affected charge puts all this information together with its own charge to determine the electric force it feels.

This “signal” is constantly emitted, so it is always everywhere in the space around the source charge, and it is called the electric field of that source charge.  Since the signal carries information about both a magnitude (source charge and distance) and a direction (coming from the source charge's position), it essentially associates a vector with every point in space. Defining (as usual) the origin to be at the position of the source charge, the electric field vector at a specific point (defined by the position vector $$\overrightarrow r = r\;\widehat r$$) due to the source charge is:

$\overrightarrow E = \dfrac{k\;q_{source}}{r^2}\;\widehat r$

To visualize what the complete field looks like, imagine all of space filled with vectors. For a positive point charge, the vectors all point directly away from it, and the magnitudes of the vectors drop-off in length as they get farther from the source:

Figure 1.1.2 – Electric Field of a Point Charge

Comparing our Coulomb field equation with Equation 1.1.3, we see that indeed all of information needed to compute the electric force, except for the charge that is affected, is contained within the electric field vector. So if we know the electric field vectors everywhere in space (or, more succinctly, we "know the electric field"), then we can compute the force on a point charge placed at any position, simply by multiplying the affected charge by the electric field vector:

$\overrightarrow F_{on\;q} = q\;\overrightarrow E\left(r\right),\;\;\;\;\;\;\text{where }\overrightarrow r= \text{position vector of the charge }q$