# Chapter 18: Electric Forces and Fields

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Chapter 18: Electric Forces and Fields
Charges The electric force The electric field Electric flux and Gauss’s Law

Charges Thales of Miletus, ~ 600 B.C.: a piece of amber, rubbed against fur, attracted bits of straw “elektron” – Greek for “amber”

Charges electric charge: an intrinsic property of matter
two kinds: positive and negative net charge: more of one kind than the other neutral: equal amounts of both kinds

Charges charge is quantized: comes in integer multiples of a fundamental (“elementary”) charge SI unit of charge: the coulomb symbol: C Size of elementary charge: 1.60×10-19 C Elementary charge: often written as “e”

Charges Charge is a conserved quantity.
If a system is isolated, its net charge is constant. Charges exert forces on each other, without touching. Attraction if charges are unlike (opposite sign) Repulsion if charges are like (same sign)

Charges Motion of charges Conductors: Insulators:
Charges can move freely on the surface or through the material – loosely bound valence electrons Typically: metals Insulators: Little movement of charge on or through the material Electrons are tightly bound Typically: rubber, plastic, glass, etc.

Charges Separation of charges
Sometimes possible by mechanical work (friction) Example: friction between hard rubber and fur or hair electrons leave the fur and go to the rubber rubber acquires a net negative charge fur acquires a net positive charge net charge of total system remains zero

Charges Transfer of charge By contact By induction
Objects touch – net charge moves from one to the other By induction Charged object brought near to another object Like charges driven from second object through path to earth Path to earth taken away Original charged object withdrawn: opposite net charge remains on second object

The Electric Force Studied systematically by Charles-Augustin Coulomb
French natural philosopher,

The Electric Force: Coulomb’s Law
Attractive or repulsive – like or unlike charges Magnitude: Constant of proportionality: magnitudes of charges constant of proportionality distance between charges

The Electric Force: Coulomb’s Law
Coulomb’s Law (electric force) Newton’s Law of Universal Gravitation (gravitational force)

The Electric Field Field: the mapping of a physical quantity onto points in space Example: the earth’s gravitational field maps a force per unit mass (acceleration) onto every point Electric field: maps a force per unit charge onto points in the vicinity of a charge or charge distribution

The Electric Field Place a test charge q0 at a point a distance r from a charge q

The Electric Field Use Coulomb’s Law to calculate the force exerted on the test charge:

The Electric Field Divide the electric force by the magnitude of the test charge:

The Electric Field Take away the test charge and define the quantity E as the ratio F/q0:

The Electric Field We calculated the magnitude of E, in terms of the magnitude of F : Both E and F are vectors. For a positive test charge, E points in the same direction as F. E always has the same direction as the electric force on a positive charge (opposite direction from the force on a negative charge).

The Electric Field The electric field is “set up” in space by a charge or distribution of charges The electric field produces an electric force on a net charge q1 : If more than one charge is present, each charge produces an electric field vector at a given point in space. These vectors add according to the usual vector rules.

The Electric Field Parallel-Plate Capacitor two conducting plates
each has area A each has net charge q (one +, one -) electric field magnitude between plates: (where e0 is the permittivity of free space) field points from + plate to - plate

The Electric Field: Field Lines
Electric Field Lines Directed lines (curves, in general) that start at a positively-charged object and end at a negatively-charged one Field lines are drawn so that the electric field vector is locally tangent to the field line

The Electric Field in Conductors
A net charge in a conducting object will move to the surface and spread out uniformly mutual repulsive forces make the charges “want” to get as far from each other as possible In the steady state, the electric field inside a conducting object is zero because the charges in a conductor are free to move, if there is an electric field, the charges will move to a distribution in which the electric field is reduced to zero

The Electric Field in Conductors
Example: a conducting sphere is placed in a region where there is an electric field Initially, the field is present inside the sphere

The Electric Field in Conductors
The field causes the charges to separate, and the separated charges produce their own field.

The Electric Field in Conductors
The motion continues until the “internal” field is equal and opposite to the “external” one …

The Electric Field in Conductors
… and their sum is zero.

Electric Flux We define a quantity associated with the electric field:
SI unit of electric flux: Nm2/C electric flux area angle between electric field vector and surface normal

Electric Flux Consider a positive charge q … what is the electric field at a spherical surface centered on the charge and a distance r from it?

Electric Flux Rearrange and substitute for the area of a sphere:
Note that the left side is the electric flux through the spherical surface. Since the field vectors are radial, f = 0° everywhere.

Electric Flux: Gauss’ Law
Johann Carl Friedrich Gauss German mathematician – 1855 Mathematics, astronomy, electricity and magnetism

Electric Flux: Gauss’ Law
Our result for the sphere enclosing the charge q : is a statement of Gauss’ Law for a spherical surface, where f is everywhere zero (the electric field vector is everywhere perpendicular to the surface). The sphere is an example of a Gaussian (closed) surface.

Electric Flux: Gauss’ Law
In general, a Gaussian surface is any surface that continuously encloses a volume of space. Such a closed surface wraps continuously around the volume. Think of a water balloon, hanging over your palm, assuming some strange, arbitrary shape.

Electric Flux: Gauss’ Law
Here is an arbitrary Gaussian surface, containing an arbitrarily-distributed net charge Q : This is the general form of Gauss’ Law.

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor charge q, spread uniformly over plate area A Gaussian cylinder radius = r Flux through surfaces 1 and 2 zero

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor Flux through surface 3: Net charge enclosed in cylinder: Flux according to Gauss’ Law:

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor Equate the two expressions for and solve for E : Sometimes is defined as a “charge density”: Then: