INSTRUCTIONAL OBJECTIVES
Physics 2220, Fall, 2013
Chapters 20, 21, 22, 23
Coulomb's Law for two point charges: There are two acceptable forms:
F on 2 due to 1
= kq
1 q
2 r /r
3
and F = kq
1 q
2 e r
/r
2
where r = r
2
r
1
is a vector from charge 1 to charge 2, q
1
and q
9
2
are the values of the charges, e
Nm
2
/C
2 r
(= r /r) is a unit vector in the direction of r , and k is 9x10 .
• Nature of equation: Experimentally derived law with an experimentally determined constant k.
Coulomb's Law for n point charges: To get the net force on one of the n charges add each of the n-1 forces resulting from each of the other charges using Coulomb's Law n-1 times .
Electric field: The definition of electric field is
E at r
= F on a test charge q
/q
To be precise one should take the limit of F/ q as q approaches zero.
• Nature of equation: Definition with the content of Coulomb's law.
• Think of: When you think of electric field think of force.
• Units: Newtons per Coulomb; N/C
Electric field resulting from a single point charge q:
E at r due to charge q
= kq r /r
3
or E = kq e r
/r
2 where r is a vector from q to the field point.
Electric field resulting from an assembly of point charges: Add the electric fields generated by each of the point charges.
Electric field resulting from a continuous distribution of charge :
E ( r f
) = ∫ kρ( r s
) r sf dV/r sf
3 where r s
is a vector pointing from the origin to the element of volume dV (sources), r f
is a vector pointing from the origin to the point in space where you want to find the field, and r sf point to the field point. Note that the amount of charge in volume dV is
= r f
– r s is a vector from the source dq=ρdV. You integrate over all the sources (all space).
Gauss's Law
∫ E•dA =q enclosed
/ε o where the integral is taken over any closed surfaced S, dA is a vector of magnitude dA and direction of the outward normal to the surface and ε o
= 1/4πk and is called the permittivity of free space.
•
Nature of the equation: contains the same information as Coulombs Law.
• When is it true? Always
• When is it useful? When a nice symmetrical surface can be chosen such that for all portions of the surface the integral gives either EA or 0. Useful symmetries are rectangular, cylindrical, and spherical.
•
What determines the closed surface? You do, but it must have proper symmetry to be useful.
Electric Potential :
V at point B with respect to point A
=- ∫ E•dr where the integral is taken by any path from A to B.
Be able to calculate potential due to a uniformly charged sheet and due to a uniformly charged sphere.
Nature of the equation: Definition that contains the content of Coulomb's Law.
Think of: When you think of electric potential think of potential energy or work.
Units: Volts=Joules/Coulomb; V=J/C
Potential values require a reference point.
Electric field from potential: E x
= -dV/dx etc.
Electric Field Lines: An electric field line gives the direction of the electric field. The inverse spacing between lines indicates the field strength – narrow spacing→high field; wide spacing→low field.
Equipotential Surfaces: Surfaces all points of which are at the same electric potential.
Relationship between Field Lines and Equipotential Surfaces: Field lines are perpendicular to the surfaces. Be able to draw field lines and equipotential surfaces for given charge distributions.
Capacitance:
C=│Q/V │ where V is the difference between the voltages on two conductors and Q is the charge on one plate.
• Nature of equation: Definition of capacitance. Contains content of Coulomb's Law

• Units: Farads=Coulombs/Volt; F=C/V
• Use of a capacitor: Store charge and/or energy.
Calculating capacitance: Usually for two conductors with net positive charge on one plate Q. First calculate the electric field E using Gauss's law or Coulombs law. Second, using E calculate the voltage between the two plates.
Third, find the ratio Q/V. That is the capacitance.
Capacitance of a parallel-plate capacitor: C=ε o
A/d, where A is the plate area and d is the spacing. Be able to derive this.
Capacitors in parallel:
V=V
1
=V
2
and Q=Q
1
+Q
2
which, with repeated use of C=Q/V, yields C=C
1
+C
2
.
Capacitors in series:
V=V
1
+V
2
and Q=Q
1
=Q
2
which, with repeated use of C=Q/V, yields 1/C=1/C
Energy Stored in a Capacitor: U=CV
2
/2.
Energy Density in an Electric field in free space: u=ε o
E
2
/2
1
+1/C
2
.
Dielectric Constant κ: κ is the ratio of the measured capacitance of a capacitor having a dielectric between its two plates to that of the same setup without the dielectric. When working in a dielectric replace ε o
by κε o
and capacitance C o
by κC o
.
Conductor: A material such as copper and other metals containing electrons that are free to move about the material. After equilibrium is reached E =0 inside the conductor. Electric fields near a charged conductor are small where the radius of curvature is great and large where the radius is small (sharp point).
Insulator: A material where the atoms or molecules can twist and orient but cannot otherwise move around. E need not be zero.
Important constants: You need not memorize these. They will appear on the formula sheet if needed. c=3x10 k=9x10
8
9
m/s
Nm
2
/C
2 e=1.6x10
-19
C; charge of the electron r e
~3x10
-15
m; classical radius of the electron. Modern theorists often conclude it is zero.
Interesting Numbers: Have some feeling for these.
Dielectric breakdown electric field for air~30,000 V/cm
Voltage and current in lightning: up to 10
9
V and 500 kA
Voltage developed in James' van de Graff generator: ~ 100,000 V
Voltage in typical dental x-ray machine: 60,000 V
Electron charge first measured to within ~20% by: Harvey Fletcher
Nobel Prize for measuring electron charge given to: Robert A. Millikan
Fletcher's contribution to the oil-drop experiment: He modified Millikan's design, used oil instead of water, and worked closely with Millikan as his graduate student to make the final precise measurement. Fletcher wrote most of the first paper reporting the results of the experiment but it contained only Millikan's name as author. Refer to
Wikipedia under "Oil Drop Experiment"
Dilemma: If the electron has internal structure how does it hold together against the enormous repulsive force due to like charges repelling.