PHYS 221 Recitation

Kevin RalphsWeek 2

Overview

• 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, , 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

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

: velocity field, : weighting factor – for fluidsthis is usually the density of the fluid, for electricitywe 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?– 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

Situational:Closed Surface

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 surfaceNote: 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

• The work done is due to the electric force so

This formula assumes the following:• is constant in both magnitude and direction• The displacement is parallel to • WARNING: Since charge can be negative, and 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

• 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

Electric Field

Change in Potential

Change in PE

Electric Force

Multiply by q

Multiply by q

Multiply by -Δx Multiply by -Δx

Vectors

Scalars

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 Field PE

Electricity q

Gravity m