PHYS 241 Recitation

Kevin RalphsWeek 1

Overview

• Introduction• Preliminaries• Tips for Studying Physics• Working with Vectors• Coulomb’s Law• Principle of Superposition• Electric Fields

Introduction

• I’m Kevin Ralphs, your TA– Email: kralphs@purdue.edu– Web site: web.ics.purdue.edu/~kralphs– Office: PHYS 137– Office Hours: By appointment

Preliminaries

• Recitations will largely focus on a conceptual approach to the material

• Memorization of equations and constants will not help you on the exam

• Understanding the concepts often allows you to almost instantly answer a question

Tips for Studying Physics• Universal vs Situational Formulae

– Very few of the formulas you will encounter are always applicable– It is imperative to know the situation for which a formula is valid

• Limited Amount of Symbols– We will use a mix of Latin and Greek letters as symbols– Unfortunately we will have more than 49 different quantities which

means we will sometimes reuse symbols– Just because 2 formulae have the same symbols does not mean they

can be used together• Identify the Type of Problem

– Particle Motion– Raw Calculation– Before and After

Working with Vectors

• Vectors are commonly defined as having direction and magnitude, but this is false– This is a geometric description, but vectors are

defined by their algebraic properties– They can be given a notion of direction and

magnitude, however• In reality, vectors are members of a set called a

vector space that has two binary operations, addition and scalar multiplication, that combine linearly

Working with Vectors

• Addition: If , , and are vectors then,– is a vector (closure)– (commutativity)– (associativity)– There exists a special vector, , such that for any

vector, (identity)– Every vector, , has a special vector associated with

it such that . We call this vector (additive inverse)

Working with Vectors

• Scalar Multiplication: If is a vector and α, β are real numbers, then– is a vector (closure)– (commutativity)– (associativity or compatability)– (multiplicative identity)

Working with Vectors

• Combine Linearly (Distributivity): If and are vectors and α, β are real numbers, then

Working with Vectors• There are many examples of vector spaces:

polynomials, real valued functions, periodic functions when they have thecorrect operations defined

• We will primarily use a vector space called the Cartesian plane or its 3D analog. It is the set of ordered pairs (triples) of real numbers with appropriate operations defined for addition and multiplication

– Addition: – Scalar Multiplication: – Identity:

• It’s a good exercise to verify all the properties listed before hold

Working with Vectors

• Direction and Magnitude– Everything we’ve talked about so far are algebraic properties of

vector spaces– It would be nice to develop tools that allow us to think about

vectors geometrically– But really if you think about it, our intuition already informs us

that there is a natural way to talk about directions and lengths• is obviously a vector of length in the x-direction and is obviously a

vector of length in the y-direction• The x-axis and y-axis have an angle of 90°, so we can think of as the

hypotenuse of a triangle with sides and • So the length of any vector should just be from the Pythagorean

theorem

Working with Vectors

• Direction and Magnitude (cont.)– We can formalize this with an operation called the dot

product

• Magnitudes: • Directions: , where θ is the angle between the two vectors

– You can check that this agrees completely with our intuition from the previous slide.

– The dot product tells you both how parallel two vectors are and how much of one vector lies along another

– Note also that which proves that scalar multiplication effectively “stretches” a vector

Working with Vectors

• Direction and Magnitude (cont.)– A vector with unit magnitude is called a unit vectors and has

a “hat” over it: .– You can get a unit vector in any direction you wish in a

couple different ways• Take a vector that already points in that direction and divide by

the magnitude: • If you know the angle the vector makes with the positive x-axis,

then

– It is convenient to write a vector in the form

– This makes the direction and magnitude explicit

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?– The force a positive test charge q 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

Universal

Electric Field

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

, θ: angle the direction makes with the positive x-axis, k: Coulomb’s Consant

• 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

Situational

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

Electric Field

• Word of caution when summing or integrating:– ALWAYS verify what changes from charge to

charge in the discrete case or over the integrating variable in the continuous case

– Constants may be removed from the sum or integral

– Remember that you are integrating vectors, so the direction unit vector often depends on the integrating variable