Physics 272
January 23
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 - 101
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Course website: www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm
Phys272 - Spring 14 - von Doetinchem - 102
Summary
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Electric dipole in an external field
Potential energy of a dipole:
φ=0: pot. energy minimal, stable equilibrium, dipole parallel to field
φ=π/2: pot. energy 0, dipole perpendicular to field
φ=π: pot. energy maximum, dipole antiparallel to field
Phys272 - Spring 14 - von Doetinchem - 103
Summary
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Electric flux: electric fields passing through an area
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Relation between electric field and flux, charge, and enclosing surface
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the total flux through any surface enclosing a charge distribution is
independent of the exact shape or size of the surface
Phys272 - Spring 14 - von Doetinchem - 104
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Gauß's law is an alternative to Coulomb's
law and is a different way to express the
relationship between electric charge and
electric field.
Source: http://en.wikipedia.org/wiki/Carl_Gauss
Gauß's law
Carl Friedrich Gauß
(1777 - 1855)
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Electric flux is independent
of exact radius and only
depends on enclosed charge.
If you increase the size of the sphere, the electric field gets smaller, but
the area increases → electric flux stays constant
Phys272 - Spring 14 - von Doetinchem - 105
Point charge inside a nonspherical surface
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Electric flux is positive (negative) where the electric
field points out (into) of the surface
Electric field lines can begin or end inside a region
of space only when there is charge in that region
Phys272 - Spring 14 - von Doetinchem - 106
General form of Gauß's law
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Surface encloses multiple charges
Total electric field is the vector sum of the electric
fields of the individual charges
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 - 107
Faraday's icepail experiment
http://www.youtube.com/watch?v=GNizWxAD-9M
Phys272 - Spring 14 - von Doetinchem - 108
Testing Gauß's law experimentally
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Faraday's test of Gauß's law test also Coulomb's law with great precision
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Ball inside pail:
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Position of ball does not matter: reading of electroscope does not change
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Charge on inside wall opposite of outside wall
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Charge of pail is the same as the ball
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Take ball out: electroscope reads zero
Ball touches pail:
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Charge transfers to surface of pail
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Take ball out: no charge on ball, but charge on pail surface
Phys272 - Spring 14 - von Doetinchem - 109
Parallelpiped
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Find the net flux and the charge within the
parallelpiped
Phys272 - Spring 14 - von Doetinchem - 112
Parallelpiped
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electric field lines must terminate on charges
→ field is not only due to charges inside, but also due to
outside charges
Phys272 - Spring 14 - von Doetinchem - 113
Electric flux and enclosed charge
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Results only depend
on enclosed charges
not on the shape of
the surface
Alternative
explanation with field
lines:
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A: all field lines are
coming out of surface
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B: all field lines are
inward
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C/D: number of field
lines going in and out
is the same
Phys272 - Spring 14 - von Doetinchem - 116
Applications of Gauß's Law
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If charge distribution is known and has enough symmetry the
integral can be evaluated
If we know the field we can find the charge distribution.
Typical question: what is the electric field caused by a charge
distribution on a conductor.
Excess charge at rest on a solid conductor resides
entirely on the surface, and not in the interior.
Phys272 - Spring 14 - von Doetinchem - 117
How to solve problems with Gauß's law
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What are the known quantities?
Select the right Gaussian surface according to symmetry of
the problem:
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Spherical: concentric spherical surface
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Cylindrical: coaxial cylindrical surface, flat ends perpendicular to
the axis of symmetry
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Planar: cylindrical surface, flat ends to the plane
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Field magnitude at point: surface must include this point
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Smart surface choice should make integration unnecessary
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If electric field is constant and perpendicular to the surface
→ no integration needed
No charge within a Gaussian surface → no total electric flux
through surface: However, field at any given point might
be non-zero
Phys272 - Spring 14 - von Doetinchem - 118
Field of a charged conducting sphere
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Charge moves freely on the
conductor
→ distributed uniformly
→ spherical symmetry
Gaussian spherical surface
inside or outside of sphere
→ calculate electric field on
surface
Electric field must be radial
→ electric field depends only on
distance, same electric field on
the whole surface
Field outside is the same as
for a point charge
Phys272 - Spring 14 - von Doetinchem - 120
Field of a charged insulating sphere
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Electric field is a continuous function of the radius
(in contrast to conductor)
Phys272 - Spring 14 - von Doetinchem - 122
Charges on conductors
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In electrostatics the electric field inside a solid
conductor is 0, excess charges only on surface
Charge in isolated cavity of neutral conductor:
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Excess charge in conductor can only be on the surface
Phys272 - Spring 14 - von Doetinchem - 123
Van de Graaf generator
http://www.youtube.com/watch?v=sy05B32XTYY
Phys272 - Spring 14 - von Doetinchem - 124
Van de Graaf generator
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Can be used to build
up very high charges
on top of shell surface
Phys272 - Spring 14 - von Doetinchem - 125
Faraday cage
http://www.youtube.com/watch?v=WqvImbn9GG4
Phys272 - Spring 14 - von Doetinchem - 126
Faraday cage
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Faraday cages protect against the external
influence of electric fields
Vacuum chamber:
Electric noise from pumps
→ distorts measurement
Sensitive silicon detector
Phys272 - Spring 14 - von Doetinchem - 127
Electric field inside a Hydrogen atom
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Hydrogen atom: proton surrounded by electron (treat both point-like)
Electron moves → charge is smeared out → position probability follows
an exponential function
Phys272 - Spring 14 - von Doetinchem - 129
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Ratio of electric field of
Hydrogen and the
proton alone
E/Eproton
Electric field inside a Hydrogen atom
Study the influence and
relative field strength
electron+proton field
goes much quicker to
zero than than proton
alone
r/a0
Phys272 - Spring 14 - von Doetinchem - 130
Review
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Electric flux is a measure of the “flow” of electric
field through a surface
Gauß's law is the relation between the charge inside
a closed surface and the corresponding electric field
and is equivalent to Coulomb's law. However,
solving problems is easier as symmetries can be
used to simplify problems.
Excess charge resides on conductor surfaces and
the electric field inside a conductor is zero.
Phys272 - Spring 14 - von Doetinchem - 131
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
Use electric potential to describe potential electric energy
→ potential differences are important for understanding of electric circuits
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Work done on a particle to move from a to b:
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change of potential energy for a conservative force (reversible):
Phys272 - Spring 14 - von Doetinchem - 133
Electric potential energy in an uniform field
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Energy conservation:
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Analogous to gravitational force:
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Conservative force → independent of exact path
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Potential energy increases if charged particle moves in
opposite direction of electric force
Phys272 - Spring 14 - von Doetinchem - 134
Electric potential energy in an uniform field
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Potential energy decreases if a charged particle
moves in the direction of the electric field
If the displacement of a positive charge is in the
direction of the electric field the work is positive
Phys272 - Spring 14 - von Doetinchem - 135
Electric potential energy of two point charges
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Point charge in the field of another point charge
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Work only depends on radial displacement along the force
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Does not depend on exact path, can move without loss/increase perpendicular to force
Phys272 - Spring 14 - von Doetinchem - 136
Electric potential energy of two point charges
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Potential energy of two point charges
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Potential energy defined to some reference point
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Shared property of both charges
Phys272 - Spring 14 - von Doetinchem - 137
Electric potential energy with several point charges
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Electric field is the vector sum and the total potential
energy is the algebraic sum (potential energy is
NOT a vector):
For every electric field due to a static charge
distribution, the force exerted by that field is
conservative.
Phys272 - Spring 14 - von Doetinchem - 138