PHYS 241 Recitation Kevin Ralphs Week 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 π’, π£, and π€ are vectors then, – π£ + π€ is a vector (closure) – π£ + π€ = π€ + π£ (commutativity) – π’ + π£ + π€ = π’ + π£ + π€ (associativity) – There exists a special vector, 0, such that for any vector, π£ + 0 = π£ (identity) – Every vector, π£, has a special vector associated with it such that π£ + π€ = 0. We call this vector π€ = −π£ (additive inverse) Working with Vectors • Scalar Multiplication: If π£ is a vector and α, β are real numbers, then – πΌπ£ is a vector (closure) – πΌπ£ = ππ£ (commutativity) – πΌ π½ π£ = πΌπ½ π£ (associativity or compatability) – 1π = π (multiplicative identity) Working with Vectors • Combine Linearly (Distributivity): If π£ and π€ are vectors and α, β are real numbers, then – πΌ + π½ π£ = πΌπ£ + π½π£ – πΌ π£ + π€ = πΌπ£ + πΌπ€ Working with Vectors • There are many examples of vector spaces: polynomials, real valued functions, periodic functions when they have the correct 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: π₯π , π¦π + π₯π , π¦π = π₯π + π₯π , π¦π + π¦π – Scalar Multiplication: πΌ π₯π , π¦π = πΌπ₯π , πΌπ¦π – Identity: 0 = 0,0 • 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 • (πΌ, 0) is obviously a vector of length πΌ in the x-direction and (0, π½) is obviously a vector of length π½ in the y-direction • The x-axis and y-axis have an angle of 90°, so we can think of πΌ, π½ as the hypotenuse of a triangle with sides (πΌ, 0) and (0, π½) • So the length of any vector should just be πΌ 2 + π½2 from the Pythagorean theorem Working with Vectors • Direction and Magnitude (cont.) – We can formalize this with an operation called the dot product π ⋅ π = π₯π π₯π + π¦π π¦π • Magnitudes: π = π ⋅ π = • Directions: cos π = vectors π⋅π π π π₯π 2 + π¦π 2 , where θ is the angle between the two – 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 πΌ π = πΌ π 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: π. – 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 π by the magnitude: π = π • If you know the angle the vector makes with the positive xaxis, then π = cos π , sin π – It is convenient to write a vector in the form π= π π – 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 ways 1. Use integration to get the particle’s velocity as a function of time, then integrate again to gets its position 2. 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: πΉ = π π1 π2 π12 2 – 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 π πΉπ = πΉ1 + πΉ2 + β― + πΉπ π=1 Electric Field • What does it tell me? – The force a positive test charge q would experience at a point in space πΉ Universal πΈ ≡ lim ⇒ πΉ = ππΈ π→0 π • 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 π Situational πΈ = π 2π π π = cos π , sin π , 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 ππ and the field becomes ππΈ Electric Field • Electric field due to a point charge at distance r with charge q π πΈ = π 2π π • Principle of superposition still applies – You can sum individual fields due to discrete charges – You can integrate continuous charge distributions where the charge becomes ππ and the field becomes ππΈ 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 π often depends on the integrating variable