Physics with Calculus/Electromagnetism/Electric Charge and Coulomb's Law

Electricity

The Electric Force

Using the concepts of atoms and fields, we can begin our discussion of electric forces.

The first important idea is that of electric charge. An object can be said to have an electric charge if it has more protons than electrons, or vice-versa. If you rub a balloon on your hair, both the balloon and your hair will become charged, resulting in the balloon lifting up your hair in defiance of gravity.

As we already discussed, there are two types of fundamental charged particles: the proton and the electron, with positive and negative charges, respectively. When you rub the balloon against your hair, one becomes positively charged and the other becomes negatively charged, as electrons are moved from one object to another. If you charge a glass rod by rubbing it with a piece of silk, you'll see that the silk is attracted to the glass rod. This is because electrons have been stripped off the glass and transferred to the silk, resulting in opposite charges. Furthermore, if you bring two similarly prepared glass rods in close proximity, you'll feel a repulsive force, as both objects have the same, net positive charge. From these observations, we can conclude that identical charges will repel each other and opposite charges will attract each other.

There are only these two types of charge. We have never found a particle with a charge that is attracted or repelled by both positive and negative charges.

This motion of charge from one object to another can also occur within a single object, where the motion of charge is called electrical conduction. An object that allows charges to move within it easily and without much resistance is called a conductor, whereas an object that resists the motion of charges is called an insulator. In general, metals tend to be excellent conductors, which is why if you take apart a piece of electrical equipment, you'll see that the wires are all made of copper. You'll also see that they are coated with rubber, which is an excellent insulator, in order to make sure that the charges are moving in the proper direction, and that someone can touch the wires without receiving a shock.

There is another method to charge an object without direct contact, called charging by induction. This requires the concept of grounding. If a conductor is connected to the earth by another conductor, it is said to be grounded. We can treat the earth as an infinite reservoir of charges, that is, we can give or take as many electrons as we want and the earth will not acquire any charge, as there are so many electrons in it. Consider an uncharged conducting sphere that is surrounded by an insulator so that it cannot be grounded. A negatively charged rod is brought near the sphere. The repulsive force on the electrons in the sphere will push them away from the side nearest the rod, leaving a net positive charge in that area. If we then ground the sphere, some of the electrons will leave the sphere, due to the repulsive force from the rod. If we then remove the ground, the sphere is left with an excess of positive charge. This positive charge will then redistribute uniformly over the sphere. Note that the negatively charged rod never actually touches the sphere, and therefore, loses none of its charge. Charging an object by induction requires no physical contact with the inducing object. If we then bring this positively charged sphere near, say, a neutral wall, we will find that our sphere will stick to the wall. This is because the net positive charge will cause the electrons to move towards the sphere, resulting in a net negative charge in a small region on the wall.

We can mathematically determine how strong the force of attraction or repulsion between two charged objects will be, and as it turns out, the electric force is proportional to the inverse square of the distance between the charges. If we have two charges, then the magnitude of the force between them is given by

$F_{e}=k_{e}{\frac {q_{1}q_{2}}{r^{2}}}$ (1)

where $k_{e}$ (the Coulomb constant) is equal to $8.9876\times 10^{9}N\cdot m^{2}\cdot C^{-2}$, r is the distance between the objects in meters, and $q_{1}$ and $q_{2}$ are the charges of the objects as measured in coulombs. This relation will then give the force and its direction in newtons.

Note: The Coulomb constant can also be written as

$k_{e}={\frac {1}{4\pi \epsilon _{0}}}$

where $\epsilon _{0}$ is the permittivity of vacuum, with a value of $8.8542\times 10^{-12}C^{2}\cdot N^{-1}\cdot m^{-1}$. This notation is mainly for historic reasons and, as we will see, because the $4\pi$ will cancel out somewhere else. Why $4\pi$? Because $4\pi r^{2}$ is the surface area of a sphere, and it will be useful to enclose a point charge in an imaginary sphere.

If we want the direction of the force in addition to its magnitude, Coulomb's Law becomes

$\mathbf {F} _{12}=k_{e}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} _{12}$ (2)

where $\mathbf {F} _{12}$ is the force exerted by $q_{1}$ on $q_{2}$ and $\mathbf {\hat {r}}$ is a unit vector directed from $q_{1}$ towards $q_{2}$.

Then, by Newton's Third Law, we can see that $\mathbf {F} _{21}$, or the force exerted by $q_{2}$ on $q_{1}$ is equal to $-\mathbf {F} _{12}$.

Electrostatics: when an electric charge is confined on an isolated object. Charge carriers: electron, proton, neutron Electron: 9.10938188 × 10-31 kilograms Proton: 1.67262158 × 10-27 kilograms Neutron: 1,6749 x 10^(-27) kg e=elementary charge...not electron e=1.60x10-19Coulombs

Coulomb (C is the unit to measure charge.)

The charge magnitude (q) is an integer (N) multiple of the charge (e). q=Ne or N=q/e

I. Coulomb's Law of Electrostatic force Fe=Kq1q2/r2

if Fe is positive the force is attractive, if Fe is negative, the force is repulsive 1μC=10−6C

2/22/06 Acceleration of Charged Particle FE=ma FE=Eq a=(qE)/m

Uniform (constant) acceleration requires a uniform field