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Physics LibreTexts

2: The Electric Force

As before, we can determine the electrical force between two charges in one of two ways: the direct model or the field model. In the direct model, we determine the magnitude of the electric force without any reference to the field: \[\text{Charge 1} \xrightarrow {\text{creates force on}} \text{Charge 2}\] From Newton’s third law, we know that Charge 2 simultaneously exerts a force of the same magnitude on Charge 1.  In the direct model the two charges are treated the same way.

Given a source charge \(Q\) and test charge \(q\), separated by a distance \(r\) between the centers of charge, the direction and magnitude of the electric force between them is given by: \[\mathbf{F}_{\text{electric, } Q \text{ on } q} = \begin{cases} \text{Magnitude} & = kQq/r^2 \\ \text{Direction} & = \text{attractive if } q \text{ and } Q \text{ have the opposite sign, repulsive if } q \text{ and } Q \text{ have the same sign} \end{cases}\]  The constant \(k\) has the value \(9 \times10^9 \text{ Nm}^2/\text{C}^2\).  It converts everything to the proper units of Newtons.

Compare the direct model for electrical forces to the direct model for gravitational forces. Note that in both cases, the strength of the force depends on the inverse square of the distance between the objects (\(r^2\)). That is, if we double the distance between two charged objects, both the gravitational force and the electrical force between them will be one-fourth the previous values. Also note that mass and charge enter the equation in an identical fashion.

If you are asked to draw a force diagram for an electrical setup, note that you determine the directions and the magnitudes of all relevant forces in separate steps. To determine the direction of the electric force, consider the term \(Qq\) in our equation.  If \(Q\) and \(q\) have the same sign, this term is positive; if they have different signs, this term is negative.  Given what you know about electrical charge, you can deduce that the positive sign indicates a repulsive force and the negative sign indicates an attractive force.  Of course calculating the direction isn't that important; the "opposites attract, same-signs repel" convention can be adopted more easily and is always effective.