Download Physics 241 Recitation

PHYS 241 Recitation
Kevin Ralphs
Week 3
Overview
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HW Questions
A Bit of History
Flux
Gauss’s Law
Electrostatics
Conductors vs Insulators
A Bit of History
• History
– The 18th century was very productive for the
development of fluid mechanics
– This lead physicists to use the language of fluid
mechanics to describe other physical phenomena
• Mixed Results
– Caloric theory of heat failed
– Electrodynamics wildly successful
Flux
• Flux, from the Latin word for “flow,” quantifies
the amount of a substance that flows through a
surface each second
• It makes sense that we could use the velocity of
the substance at each point to calculate the flow
• Obviously we only want the part of the vector
normal to the surface, 𝑣𝑛 , to contribute because
the parallel portion is flowing “along” the surface
• Intuitively then we expect the flux to then be
proportional to both the area of the surface and
the magnitude of 𝑣𝑛
Flux
• For the case of a flat surface and uniform
velocity, it looks like this:
Flux
• For curved surfaces and varying flows, if we chop the
surface up into small enough pieces so that the
surface is flat and the velocity uniform, then we can
use an integral to sum up all the little “pieces” of flux
Φ=
𝑣 ∙ 𝑑𝐴
𝑆
Gauss’s Law
• What does it tell me?
– The electric flux (flow) through a closed surface is
proportional to the enclosed charge
• Why do I care?
– You can use this to determine the magnitude of
the electric field in highly symmetric instances
– Flux through a closed surface and enclosed charge
are easily exchanged
3 Considerations for Gaussian Surfaces
Gauss’s law is true for any imaginary, closed surface and any
charge distribution no matter how bizarre. It may not be
useful, however.
1. The point you are evaluating the electric field at needs to
be on your surface
2. Choose a surface that cuts perpendicularly to the electric
field (i.e. an equipotential surface)
3. Choose a surface where the field is constant on the
surface
*Note this requires an idea of what the field should look like
Common Gauss’s Law Pitfalls
• Your surface must be closed
• The charge you use in the formula is the charge
enclosed by your surface
• The Gaussian surface need not be a physical
surface
• Start from the definition of flux and simplify only
if your surface/field allows it
𝑆
𝑞𝑒𝑛𝑐
𝐸 ∙ 𝑑𝐴 =
𝜀𝑜
Universal
Electrostatics
• It may not have been explicit at this point, but we
have been operating under some assumptions
• We have assumed that all of our charges are
either stationary or in a state of dynamic
equilibrium
• We do this because it simplifies the electric fields
we are dealing with and eliminates the presence
of magnetic fields
• This has some consequences for conductors
Conductors vs Insulators
• Conductors
– All charge resides on the surface, spread out to
reduce the energy of the configuration
– The electric field inside is zero
– The potential on a conductor is constant (i.e. the
conductor is an equipotential)
– The electric field near the surface is perpendicular
to the surface
Note: These are all logically equivalent statements,
but only apply in the electrostatic approximation
Conductors vs Insulators
• Insulators
– Charge may reside anywhere within the volume or
on the surface and it will not move
– Electric fields are often non-zero inside so the
potential is changing throughout
– Electric fields can make any angle with the surface