Ch.1 Electric Charges and Force

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Charge

• two kinds: positive and negative
• charging: transfer of electrons from one object to another
  • rubbing plastic rod with wool or glass rod with silk
• discharing: remove charge
• like repel, opposite attract
• neutral attacks to both
• force is long-range, with more charge or smaller distance increasing force
• can be transferred
• conserved
• conductors: charge move easily
  • freely flowing sea of electrons; creating array of positive ion cores
  • added charge instantaneously spreads
• insulators: charge is immobile
  • electrons tightly bound to nuclei
  • rubbing transfers charge, but do not move
• inherent property of matter, like mass
• electrons and protons have opposite charges with exactly the same magnitude
  • fundamental unit of charge $e$
  • protons: $+e$
  • electrons: $-e$
• neutral objects have no net charge
• charge quantization: amount of charge is discrete
• ionization: removing electrons; creates a positive ion
• adding electrons creates negative ion
• law of conservation of charge: charge is not created nor destroyed only transferred
• current: flow of charge through material
• charge carriers: charges that physically move
  • are electrons in metals, ions in ionic solutions
• electrostatic equilibrium: charges at rest; no net force on electrons
  • excess electrons in an isolated conductor push each other to the surface
• grounded: the earth is an enormous conductor; objects touching the earth get discharged
• charge polarization: slight separation of positive and negative charges in a neutral object
  • repulsive force between electrons and attractive force between polarizing rod balance in less than $10^{-15}\text{ m}$
  • attractive force of electrons on one side are slightly greater than the repulsive force of ions on the other, creating polarization force
    • in an insulator, every atom within the object is polarized, creating an electric dipole and a net force
• charging by induction: polarization to move electrons, touch a conductor to spread charge, remove conductor, remove polarizing rod

Coulomb's Law

• electric force follows inverse-square law
• Two charged particles with charge $q_1$ and $q_2$ a distance of $r$ apart exert a force of magnitude
  $F_{1\text{ on }2}=F_{2\text{ on }1}=K\frac{|q_1||q_2|}{r^2}$

• Coulomb's Law describes the force between point charges
• Coulomb: the SI unit of charge
  • $e=1.60\cdot10^{-19}\text{ C}$
  • electrostatic constant: $K=8.99\cdot10^9\text{ N m}^2/\text{C}^2\approx9.0\cdot10^9\text{ N m}^2/\text{C}^2$
• permittivity constant$\epsilon_0=\frac{1}{4\pi K}$
  • makes formula $F=\frac{1}{4\pi K}\frac{|q_1||q_2|}{r^2}$
• only applies to point charges; can be approximated if object is much smaller than distance

Electric Fields

If one particle moves, does the other particle respond to the change in force immediately?
It doesn't, especially considering if the particles were thousands of lightyears apart.
To determine this, Faraday invented fields
From a Newtonian perspective, particle A directly responds to particle B.
From Faraday's view, particle A affects the space around it and particle B responds to the change in space.
field: in math, a function that assigns a vector to every point in space; in physics, the idea that a physical entity exists everywhere in space

• graivtational field formed around masses
• electric field formed around charges

Faraday Model

• electric field exerts electric force on charged particles
  • ie charged particles interact via electric fields

Postulates:

• source charges: alters space around them to create an electric field $\vec{E}$ at all points in space
• separate charge $q$in the field experiences a force $\vec{F}=q\vec{E}$ exerted by the field.
  • force on a positive charge is in the direction of $\vec{E}$; negative, opposite
    • i.e. exactly as $\vec{F}=q\vec{E}$ suggests

If a probe charge$q$ is placed on the field at $(x,y,z)$ and force $\vec{F}_{\text{on }q}$ is measured, we can find the electric field at $(x,y,z)$ by
$\vec{E}(x,y,z)=\frac{\vec{F}_{\text{on }q}\text{ at }(x,y,z)}{q}$

• The units for electric field is $\text{N}/\text{C}$ (Newtons per Coulomb)
• The magnitude $E$ is the electric field strength
• The field is dependent only on the source charge; the force exerted is proportional to the probe charge, so the force is also dependent on the probe charge

Field Model

• electric force is exerted by electric field
• electric field is created by other charges, the source charges
  • electric field is a vector
  • the field exists at all points in space
  • a charge does not feel its own field
• if an electric field at a point in space is $\vec{E}$, a particle with charge $q$ experiences electric force $\vec{F}_{\text{on }q}=q\vec{E}$
• force on positive charge is in the direction of $\vec{E}$
• force on negative charge is opposite the direction of $\vec{E}$

Electric Field of a Point Charge

We use $q'$ to probe the electric field of $q$.
Define $\vec{r}$ to be the unit vector from the point of origin pointing toward the point of interest; i.e. the direction vector
By Coulomb's Law, we calculate the force on $q'$ as
$\vec{F}_{\text{on }q'}=\frac{1}{4\pi\epsilon_0}\frac{qq'}{r^2}\vec{r}$
It is customary to use $1/4\pi\epsilon_0$ instead of $K$ for field calculations. Then by the electric field equation, we can find
$\vec{E}=\frac{\vec{F}_{\text{on }q'}}{q'}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\vec{r}$
Calculating the field at several places lets us draw a field diagram.