Electric Forces and Electric Fields

Electricity

The electrical nature of matter is inherent in atomic structure.

Proton's charge: $+1e$ = $+1.602*10^{-19}\ C$ (Coulombs)

Electron's charge: $-1e$ = $-1.602*10^{-19}\ C$

Neutron's charge: none

Removing an electron from a neutral atom (where number of protons = number of electrons) yields a positively charged ion, called electrification. This results in a transfer of charge, not a creation of charge.

Charge can neither be created or destroyed.

Law of Conservation of Electrical Charge

Charged Objects and the Electrical Force

Objects with excess charge will exert forces on each other.

Electrical Force (like gravitational force)

Opposite charges attract, like charges repel. There are two types - attractive and repulsive.

Newton's 2nd Law: $F=ma$ , $\therefore$ Electric charges can accelerate forces.

Conductors & Insulators

Conductors and insulators are determined by atomic structure. The valence (outermost) electrons are more weakly bound to the nucleus. They can "break free" and move through the material. These are called conduction electrons.

(negative charged object: $-5 C$ ) --> ~conductor~ --> (positively charged obj: $+13C$ )

Electrons will continue to flow until the charge on each object is equal.

$-5C+13C=8C$ , so each object must end up with a charge of $+4C$ .

Generally, good thermal conductors are good electrical conductors.

Conductors - copper, silver, salt water, gold, iron Insulators - rubber, glass, wood, plastic

Charging an Object

Two ways to charge -
1. Charging by contact: touch a positively charged object with a negatively charged object, or vice versa
2. Charging by induction: bring a negatively charged object close to a positively charged object, or vice versa.
(does not work for insulators, instead forms dipoles) - called Polarization
Dipoles: two equal but opposite charges separated by some distance $r$ .
produces static cling by forming a slight positive charge

Coulomb's Law

Coulomb's Law: $F=k\dfrac{q_1*q_2}{r^2}$ (watch the magnitude! don't negate) Point charges: charges that are much smaller than the distance separating them.

What if there's 3 charges? -- Then the net force on one charge is the vector sum of the other two forces.

Electric Field

A charge creates an electric field that fills all space. Any other charge in that field will feel a force. - Similarly, in a gravitational field, the earth fills all space, and the moon feels the effect of this field. - Stationary charges create electric fields that fill all space. - Other (non-stationary) charges will feel forces in these electric fields.

$\vec{E}=\dfrac{\vec{F}}{q_0}$

i.e. the electric field is force per unit charge: $\left[ \dfrac{N}{C} \right]$

If we place a charge $q$ in an electric field, it will feel a force given by $\vec{F}=q\vec{E}$ . $E$ is created by other charges, not q.

Electric Field of a Point Charge

Electric field of a point charge: $E=k\dfrac{Q}{r^2}$

What is the magnitude of the acceleration of a particle of mass $m$ and charge $+Q$ placed between the plates of a parallel plate capacitor of charge density $\delta$ ?

$a=Q\delta/\epsilon_0m$

Electric Field Lines

Electric fields lines always start at and are directed away from positive charges and always end at and are directed toward negative charges.

Lines start from positive charges and end on negative charges.
The electric field vector at some point in space is always tangent to the field line at that point.
The more lines you have per unit volume (density), the stronger the field.
The number of field lines is proportional to the magnitude of the charge.

Parallel Plate Capacitor

Consider a metallic plate with a total charge $+q$ distributed uniformly over its surface. If the face of the plate has a surface area $A$ , then $\sigma=\dfrac{q}{A}$ where $s$ is the surface charge density.

Now let's take two identical plates, one with total charge $+q$ and one with total charge $-q$ . So the magnitude of $s$ is $q/A$ for each plate. In between the plates, there's a uniform electric field that points from the positive plate to the negative one. This is a parallel plate capacitor.

Electric field between the plates: $E=\dfrac{q}{\epsilon_0A}=\dfrac{\sigma}{\epsilon_0}$ .

The field doesn't depend on the distance from the charged plates. It's uniform, but only near the middle of the plates. This is a fringe field.

Electric Field Inside a Conductor

Let's place a bunch of electrons at the center of a solid conducting sphere. Repulsion spreads electrons out! So, they move to the surface.

At equilibrium under electrostatic conditions, any excess charge resides on the surface of the conductor.

The interior of the conductor still has electrons, but they are compensated exactly by the positive changes in the interior, so the interior is electrically neutral. Since the charges are static, the force on them is zero, therefore $E=0$ .

This is a static condition - once the surface charges are induced, they don't move.

This electric field has to be perpendicular to the surface of a conductor.

Electric Flux and Gauss' Law

Flux: measure of how much field passes perpendicularly through a surface

$\Phi_E=E_{\tau}A=(E\cos\phi)A$

i.e. [(electric field)(area)] = $\left[ \dfrac{N*m^2}{C} \right]$

We can now use the concept of electric flux and Gauss' Law to determine the electric field. Let's start with a positive point charge: $E=k\dfrac{Q}{r^2}$ .

To use Gauss' Law, you have to pick a surface. For example, for my point charge $+Q$ , I'll enclose it in a spherical radius $r$ . This is my surface. - Gaussian surface: surface to which I will apply Gauss' Law - Ideally, it'll have high symmetry, like a sphere or cylinder, so it's easy to calculate the area of the surfaces.

Gauss' Law: The electric flux passing through a Gaussian surface is equal to the charge enclosed by the surface divided by $\epsilon_0$ .

$\Phi_E=\dfrac{Q_{\text{enc}}}{\epsilon_0} \rightarrow \boxed{\sum(E\cos\phi)\Delta A=\dfrac{Q_{\text{enc}}}{\epsilon_0}}$

(The summation here is to sum over the different areas you have in your Gaussian surface.)