Physics 272
February 27: Review
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 401
Coulomb's law
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Strength of electric force is proportional to 1/r2
The Magnitude of the electric force between two point charges is
directly proportional to the product of the charges and inversely
proportional to the square of the distance between them.
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Force magnitude is always positive
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Direction is always along the line of the two charges
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If both charges are positive: repulsive
If both charges are negative: repulsive
If have charges have opposite charge signs: attractive
Phys272 - Spring 14 - von Doetinchem - 402
Superposition of forces
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Total charge is the vector sum of charges:
principle of superposition of forces:
Phys272 - Spring 14 - von Doetinchem - 403
Electric field and electric forces
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The fields are responsible for exerting the electric
force on the other charge
An electric field creates an electric force on a test
charge q0
Phys272 - Spring 14 - von Doetinchem - 404
Electron in a uniform field
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Release of an
electron in a
uniform electric
field
Concepts:
–
electric
field-electric
force relation
–
Force and
acceleration
We know the
field, mass, and
charge
Phys272 - Spring 14 - von Doetinchem - 405
Electron in a uniform field
●
●
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Release of an
electron in a
uniform electric
field
Concepts:
–
electric
field-electric
force relation
–
Force and
acceleration
We know the
field, mass, and
charge
Phys272 - Spring 14 - von Doetinchem - 406
Electric field lines
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A field line illustrates the direction of the electric field at a certain
point
If you draw the tangent to a point on a field line you get the direction
of the field at this point
Spacing of electric field lines is chosen such the density illustrates
the magnitude
Field lines do not intersect
Phys272 - Spring 14 - von Doetinchem - 407
Electric flux
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Analogy: positive charges flow out of the surface and negative charges flow into the surface.
If there is no enclosed charged: electric field flux going into the surface cancels flux out of
the surface
Radius cancels out
→ the flux through any surface enclosing a single point charge is independent of the shape
or size of the surface
Phys272 - Spring 14 - von Doetinchem - 408
General form of Gauß's law
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Total electric field is the vector sum of the electric
fields of the individual charges inside the enclsoed
surface
General form:
The total electric flux through a closed surface is
equal to the total (net) electric charge inside the
surface, divided by ε 0.
Phys272 - Spring 14 - von Doetinchem - 409
Electric potential energy
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Charged particle moving in a field: field exerts work
on particle
Work can be expressed as potential energy:
position of a charge in an electric field
Conservative force → independent of exact path
Potential energy increases if charged particle
moves in opposite direction of electric force
Phys272 - Spring 14 - von Doetinchem - 410
Electric potential
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Describe potential energy on
a “per unit charge” basis
(like the electric field describes
force per unit charge)
Determination of electric field is
often easier by using the potential
Source: http://de.wikipedia.org/wiki/Alessandro_Volta
Alessandro Volta
1745-1825
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Potential energy and potential are scalars
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Potential difference in circuits is often called voltage
Phys272 - Spring 14 - von Doetinchem - 411
Potential gradient and electric field
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Vector electric field can be calculated from scalar electric potential
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Potential gradient points towards the most rapid change in position.
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Absolute value of potential is not important for electric field, only the
local change.
Phys272 - Spring 14 - von Doetinchem - 412
Capacitors and capacitance
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Electric field is
proportional to the
stored charge (the
same is true for the
potential difference)
Capacitance stays
constant:
Capacitance is a measure of the ability of a
capacitor to store energy.
Phys272 - Spring 14 - von Doetinchem - 413
Capacitors in series
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Charges on all plates
have the same
magnitude
Equivalent capacitance
of a series combination
of capacitors is always
less than any individual
capacitance.
Charges on plates are the same, but if the
dimensions are different
→ potential for each capacitor different
Phys272 - Spring 14 - von Doetinchem - 414
Capacitors in parallel
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Charges can reach capacitors independently from the source
Imagine one big capacitor that you split into multiple smaller
capacitors
The parallel combination of capacitors always has a higher
capacitance than the individual capacitances
Charges are generally not the same on each capacitor
Phys272 - Spring 14 - von Doetinchem - 415
Capacitor network
Phys272 - Spring 14 - von Doetinchem - 416
Electric field energy
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Charging a capacitor: work against the electric field
between the plates
Energy is stored in the field in the region between
the plates
Phys272 - Spring 14 - von Doetinchem - 417
Induced charge and polarization
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When dielectric is inserted and charge is kept constant
→ potential difference drops
→ electric field drops
→ surface charge density drops, but not the charge
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Redistribution of charges in dielectric occurs: polarization
Phys272 - Spring 14 - von Doetinchem - 418
Current, drift velocity, current density
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Amount of charge flowing through an area:
Current in a conductor is the product of the density of moving
charged particles, the magnitude of charge of each such
particle, the magnitude of the drift velocity, and the
cross-section area
Phys272 - Spring 14 - von Doetinchem - 419
Resistivity
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Generally current density in
conductor depends on electric field
and the properties of the material as
a function of temperature
Ohm's law:
Source: http://de.wikipedia.org/wiki/Georg_Simon_Ohm
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Resistivity ρ of a material is the ratio of electric field
and current density
The greater the resistivity the greater the field has to be
to achieve the same current density
Phys272 - Spring 14 - von Doetinchem - 420
Resistance
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Current and potential difference are easier to
measure than current density and electric field
As current flow through electric potential difference
→ electric potential energy is dissipated
→ energy heats up material
Phys272 - Spring 14 - von Doetinchem - 421
Resistors in series
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Voltages are directly proportional to resistance and
current
The equivalent resistance of any number of
resistors in series equals the sum of their individual
resistances
Equivalent resistance is greater than any individual
resistance
Phys272 - Spring 14 - von Doetinchem - 422
Resistors in parallel
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Current is proportional to common voltages, but inversely
proportional to the resistance
For any number of resistors in parallel, the reciprocal of
the equivalent resistance equals the sum of the
reciprocals of their individual resistances.
The equivalent resistance is always lower than any
individual resistance.
More current goes through the path of least
of resistance.
Phys272 - Spring 14 - von Doetinchem - 423
Electromotive force and circuits
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Ideal source of emf
brings charge to higher
potential energy level
without increasing the
kinetic energy
Charge is not used up in
a circuit and is not
accumulating in the
circuit elements. Both
sides of the terminal of a
battery have the same
current for an ideal
source of emf.
Phys272 - Spring 14 - von Doetinchem - 424
Energy and power in electric circuits
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How fast is energy delivered or extracted?
If a charge passes through a circuit element: change of
potential energy
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Current stays the same → no gain of kinetic energy
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Power:
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Power for a pure resistance:
Phys272 - Spring 14 - von Doetinchem - 425
Kirchhoff's rules
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Junction: three or more conductors meet
Algebraic sum of currents is zero at any junction:
Conservation of charge
Loop: Algebraic sum of potential differences is zero
in any loop. Electrostatic force is conservative.
Potential energy is the same after going around a
loop
Phys272 - Spring 14 - von Doetinchem - 426
Charging a capacitor
●
Capacitor charges:
–
vbc increases (charge builds up)
–
vab decreases (Kirchhoff's loop rule)
–
Current decreases (Ohm's law)
Phys272 - Spring 14 - von Doetinchem - 427
Charging a capacitor
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Charge at any time t during the charging process:
Phys272 - Spring 14 - von Doetinchem - 428
Charging a capacitor
Phys272 - Spring 14 - von Doetinchem - 429
Magnetic forces on moving charges
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Magnitude of the force is proportional to amount of
charge
Magnitude of the force is proportional to the
magnetic field strength.
Magnitude of the force is proportional to the velocity
–
electric force is always the same: no matter if charge
moves or not!
–
Particle at rest does not feel magnetic force
Force is perpendicular to the velocity and magnetic
field
Phys272 - Spring 14 - von Doetinchem - 430
Helical particle motion in a magnetic field
Phys272 - Spring 14 - von Doetinchem - 431
Magnetic torque
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Greatest torque when magnetic dipole moment and
magnetic field are perpendicular
Torque is zero when magnetic dipole moment and
magnetic field are (anti)parallel
Analogue to electric dipole moment and electric field
Phys272 - Spring 14 - von Doetinchem - 432
Potential energy for a magnetic dipole
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If magnetic dipole changes orientation in magnetic
field
→ the field does work on it
In analogy to the potential energy of an electric dipole
→ potential energy for a magnetic dipole:
Potential energy is zero when magnetic moment is
perpendicular to the field
torque tries to align magnetic moment and magnetic
field
Phys272 - Spring 14 - von Doetinchem - 433
The Hall effect
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Conductor strip perpendicular to a magnetic field
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Magnetic force causes polarization effect in material
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Electric field between lower and upper side of strip builds
up
Phys272 - Spring 14 - von Doetinchem - 434