Download PHYS 221 Recitation

PHYS 221 Recitation
Kevin Ralphs
Week 2
Overview
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HW Questions
Gauss’s Law
Conductors vs Insulators
Work-Energy Theorem with Electric Forces
Potential
HW Questions
Ask Away….
Flux/Gauss’s Law
• 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
Φ ∝ 𝑣𝑛 𝐴 = 𝑣 ⋅ 𝐴
Flux
• For the case of a flat surface and uniform
velocity, it looks like this (pretend the electric
field vector is a velocity):
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=
𝜌𝑣 ∙ 𝑑 𝐴
𝑆
𝑣: velocity field, 𝜌: weighting factor – for fluids
this is usually the density of the fluid, for electricity
we will take it to be 1
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?
𝑞𝑒𝑛𝑐
Φ𝐸 =
𝜀𝑜
Situational:
Closed Surface
– You can use this to determine the magnitude of the
electric field in highly symmetric instances; The
symmetries of the charge distribution are reflected in the
field they create
– Flux through a closed surface and enclosed charge are
easily exchanged
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
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
Potential Energy
• In a closed system with no dissipative forces
Δ𝑃𝐸𝑒𝑙𝑒𝑐 + 𝑊 = 0
• The work done is due to the electric force so
𝑊 = 𝐹Δ𝑥 = 𝑞𝐸Δ𝑥
This formula assumes the following:
• 𝐹 is constant in both magnitude and direction
• The displacement Δ𝑥 is parallel to 𝐹
• WARNING: Since charge can be negative, 𝐸 and 𝐹 might point in
opposite directions (this is called antiparallel) which would change the
sign of W
• This can be combined with the work-energy theorem to
obtain the velocity a charged particle has after moving
through an electric field
Potential
• What does it tell me?
– The change in potential energy per unit charge an
object has when moved between two points
Δ𝑃𝐸𝑒𝑙𝑒𝑐
Δ𝑉 ≡
= −𝐸Δ𝑥
𝑞
• Why do I care?
– The energy in a system is preserved unless there is
some kind of dissipative force
– So the potential allows you to use all the conservation
of energy tools from previous courses (i.e. quick path
to getting the velocity of a particle after it has moved
through a potential difference)
Potential
• Word of caution:
– Potential is not the same as potential energy, but they are
intimately related
– Electrostatic potential energy is not the same as potential
energy of a particle. The former is the work to construct
the entire configuration, while the later is the work
required to bring that one particle in from infinity
– There is no physical meaning to a potential, only difference
in potential matter. This means that you can assign any
point as a reference point for the potential
– The potential must be continuous
Tying it Together
Multiply by q
Vectors
Electric
Field
Electric
Force
Multiply by -Δx
Scalars
Multiply by -Δx
Change in
Potential
Change in
PE
Multiply by q
Analogies with Gravity
• Electricity and magnetism can feel very abstract because we
don’t usually recognize how much we interact with these forces
• There are many similarities between gravitational and electric
forces
• The major difference is that the electric force can be repulsive
• Gravity even has a version of Gauss’s law
Charge
Force
Electricity
q
𝑄𝑞
𝑘 2𝑟
𝑟
Gravity
m
𝐺
𝑀𝑚
𝑟
𝑟2
Field
𝐸=𝑘
𝑔=𝐺
𝑄
𝐹
𝑟
=
𝑟2
𝑞
PE
𝑞Δ𝑉
𝑀𝑚
𝐹 𝑚 𝑔Δ𝑦
𝑟
=
𝑟2
𝑚