PHYS 241 Exam 1 Review

Kevin Ralphs

Overview

• General Exam Strategies• Concepts• Practice Problems

General Exam Strategies

• Don’t panic!!!• If you are stuck, move on to a different

problem to build confidence and momentum• Begin by drawing free body diagrams• “Play” around with the problem• Take fifteen to twenty minutes before the

exam to relax… no studying.• Look for symmetries

Concepts

• Electrostatics• Coulomb’s Law• Principle of Superposition• Electric Field• Continuous Charge Distributions• Conductors vs. Insulators• Gauss’s Law• Potential• Capacitance

Electrostatics

• Our study of electric fields so far has been based on a few assumptions

• These assumptions collectively are known as the electrostatic approximation

• Basically we assume that our systems have to come to a dynamic equilibrium before we do our calculations

• We will be ignoring transitory behavior or steady state behaviors (no currents or magnetic fields)

Coulomb’s Law

• What does it tell me?– It tells you the force between two charged particles

• Why do I care?– Forces describe the acceleration a body undergoes– The actual path the body takes in time can be found

from the acceleration in two ways1. Use integration to get the particle’s velocity as a

function of time, then integrate again to gets its position2. Kinematic equations (the result when method 1. is

applied in the case of constant acceleration)

Coulomb’s Law

• Forces have magnitude and direction so Coulomb’s law tells you both of these– Magnitude: – Direction: Along the line connecting the two

bodies. It is repulsive in the case of like charges, attractive for opposite charges

Principle of Superposition

• What does it tell me?– The electric force between two bodies only

depends on the information about those two bodies

• Why do I care?– Essentially, all other charges can be ignored, the

result obtained in pieces and then summed… this is much simpler

Electric Field

• What does it tell me?– A vector proportional to the force a positive test

charge would experience at a point in space• Why do I care?– Calculating the force a particular charge feels

doesn’t directly tell you how other charges would behave

– The electric field gives you a solution that applies to any charge, so it reduces your work

Electric Field

• Electric field due to a point charge at distance r with charge q

• Principle of superposition still applies– You can sum individual fields due to discrete

charges– You can integrate continuous charge distributions

where the charge becomes and the field becomes

Continuous Charge Distributions

• Motivation for the equation:

– Very far from a charge distribution, it looks like a point charge

– So if we “chop” up the distribution into small enough pieces, each one will have a field contribution we can calculate

– The principle of superposition then allows the integrand to approach the true field

Continuous Charge Distributions

• General procedure to setup the integrals– Prepare your integral– Change integral to integrate over where the charge

lies (aka parameterization)– Identify elements of the integrand that depend on

the integrating variable– Determine explicit relationships with the integrating

variable– Integrate

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

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 allows it

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

• Why do I care? (cont.)– If you have the potential defined over a small

area, the potential function encodes the information about the electric field in the derivative

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

Capacitance

• What does it tell me?– The charge that accumulates on two conductors is

proportional to the voltage between them• Why do I care?– Capacitors are vital components in electronics– They can be used to temporarily store charge and

energy, and play an even more important role when we move to alternating current systems

– Camera flashes, touch screen devices, modern keyboards all exploit capacitance

Capacitance

• In circuits– In well-behaved configurations, capacitors may

be combined into a single equivalent capacitor– Parallel

* This is like increasing the area of the plates *– Series

* This is like increasing the separation distance *

Capacitance

• Dielectric– Put simply, a dielectric is a material (an insulator)

that weakens the electric field around it– This allows more charge to be placed on the

plates for the same voltage (i.e. capacitance is increased)

– The permittivity of a dielectric tells you how it affects the capacitance

– The ratio of the permittivity of a dielectric and the permittivity of free space is the dielectric constant

Capacitance

• Capacitors are in equilibrium…– Series: when they have the same charge– Parallel: when they have the same voltage

Practice Problems

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Practice Problem

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Practice Problem