# Scalar Form

We saw in the previous chapter that the electric force increases with the strength of the electric charges, and decreases as the charges get farther apart. If we want to be more precise and calculate the exact electric force, we can use Coulomb's Law:

$$\displaystyle\abs{\vec F}=k\frac{\abs{q_sq_t}}{d^2}$$ Coulomb's Law

Strictly speaking, this equation gives the force on one point charge due to another point charge. Point charges are objects with charge and maybe mass, but no size. This might make this equation seem rather pointless (ahem), but in practice it serves as a reasonable approximation for the force between any two charged objects, so long as the distance between them is large compared to their size. In later chapters we'll learn how to calculate the force when the objects cannot be approximated as points. To describe the variables in this equation:

• The variable d is the distance between the two charges, in meters. The force obeys what is known as an inverse-square law: if you double the distance between two charges, the force on each drops to one-quarter of its original value. Inverse-square laws appear all over the place in physics (gravity is one, as is the amount of light a detector receives as you move it away from a light source), and is due to the fact that our universe has three spatial dimensions.
• The variable q is used to represent the charge of an object: q > 0 for positive charges and q < 0 for negative ones. In the SI system of units, charge is measured in coulombs. The subscripts s and t refer to the two point charges; I'll explain their significance in a bit. The charges are in absolute value bars because we are calculating the magnitude of the force, which is by definition a nonnegative quantity.
• The variable k is a fundamental constant, sometimes called Coulomb's constant. It is equal to
$$k=9\ten9\u{Nm^2/C^2}$$ Coulomb's constant
Suppose you hold a 2C charge in one hand and a 1C charge in the other hand, so that they are 1 meter apart.
What force does the 2C charge feel? ----- The 2C charge is repelled by the 1C charge with a force of
$$F=\left(9\ten9\u{Nm^2\over C^2}\right)\frac{\abs{(2\u{C})(1\u{C})}}{(1\u{m})^2} =18\ten9\u{N}$$
or 18 billion newtons. For comparison, the force of gravity on a human being is roughly a thousand newtons.
What force does the 1C charge feel? ----- The two charges are pushing on each other, so according to Newton's Third Law they will feel forces of equal magnitude and opposite direction. Thus the 1C charge also feels 18 billion newtons of force.

From the example above, we see that if you were holding a 1 coulomb charge in each hand, you would definitely notice it! A coulomb is a large charge: most charges in everyday life are measured in microcoulombs (1 µC = 10-6 C), nanocoulombs (1 nC=10-9 C), or even smaller. At the microscopic level, particles tend to have charges which are some small multiple of the charge of a proton e—which, as we mentioned in a previous chapter, is

$$e=1.6\ten{-19}\u{C}$$ charge of a proton

The formula above gives us the magnitude of the force, but force is a vector so it must have a direction as well. In simple situations we can determine the direction from the rule "like charges repel, opposite charges attract". However, if we want the force in component form then it would be useful to have a version of Coulomb's Law for vectors.