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Transcript
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: [email protected] – 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 𝑢, 𝑣, and 𝑤 are vectors then, – 𝑣 + 𝑤 is a vector (closure) – 𝑣 + 𝑤 = 𝑤 + 𝑣 (commutativity) – 𝑢 + 𝑣 + 𝑤 = 𝑢 + 𝑣 + 𝑤 (associativity) – There exists a special vector, 0, such that for any vector, 𝑣 + 0 = 𝑣 (identity) – Every vector, 𝑣, has a special vector associated with it such that 𝑣 + 𝑤 = 0. We call this vector 𝑤 = −𝑣 (additive inverse) Working with Vectors • Scalar Multiplication: If 𝑣 is a vector and α, β are real numbers, then – 𝛼𝑣 is a vector (closure) – 𝛼𝑣 = 𝑎𝑣 (commutativity) – 𝛼 𝛽 𝑣 = 𝛼𝛽 𝑣 (associativity or compatability) – 1𝑎 = 𝑎 (multiplicative identity) Working with Vectors • Combine Linearly (Distributivity): If 𝑣 and 𝑤 are vectors and α, β are real numbers, then – 𝛼 + 𝛽 𝑣 = 𝛼𝑣 + 𝛽𝑣 – 𝛼 𝑣 + 𝑤 = 𝛼𝑣 + 𝛼𝑤 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: 𝑥𝑎 , 𝑦𝑎 + 𝑥𝑏 , 𝑦𝑏 = 𝑥𝑎 + 𝑥𝑏 , 𝑦𝑎 + 𝑦𝑏 – Scalar Multiplication: 𝛼 𝑥𝑎 , 𝑦𝑎 = 𝛼𝑥𝑎 , 𝛼𝑦𝑎 – 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 • (𝛼, 0) is obviously a vector of length 𝛼 in the x-direction and (0, 𝛽) is obviously a vector of length 𝛽 in the y-direction • The x-axis and y-axis have an angle of 90°, so we can think of 𝛼, 𝛽 as the hypotenuse of a triangle with sides (𝛼, 0) and (0, 𝛽) • So the length of any vector should just be 𝛼 2 + 𝛽2 from the Pythagorean theorem Working with Vectors • Direction and Magnitude (cont.) – We can formalize this with an operation called the dot product 𝑎 ⋅ 𝑏 = 𝑥𝑎 𝑥𝑏 + 𝑦𝑎 𝑦𝑏 • Magnitudes: 𝑎 = 𝑎 ⋅ 𝑎 = • Directions: cos 𝜃 = vectors 𝑎⋅𝑏 𝑎 𝑏 𝑥𝑎 2 + 𝑦𝑎 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 𝛼 𝑎 = 𝛼 𝑎 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: 𝑎. – 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 𝑎 by the magnitude: 𝑎 = 𝑎 • If you know the angle the vector makes with the positive xaxis, then 𝑎 = cos 𝜃 , sin 𝜃 – It is convenient to write a vector in the form 𝑎= 𝑎 𝑎 – 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: 𝐹 = 𝑘 𝑞1 𝑞2 𝑟12 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 𝑛 𝐹𝑖 = 𝐹1 + 𝐹2 + ⋯ + 𝐹𝑛 𝑖=1 Electric Field • What does it tell me? – The force a positive test charge q would experience at a point in space 𝐹 Universal 𝐸 ≡ lim ⇒ 𝐹 = 𝑞𝐸 𝑞→0 𝑞 • 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 𝑞 Situational 𝐸 = 𝑘 2𝑟 𝑟 𝑟 = cos 𝜃 , sin 𝜃 , 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 𝑑𝑞 and the field becomes 𝑑𝐸 Electric Field • Electric field due to a point charge at distance r with charge q 𝑞 𝐸 = 𝑘 2𝑟 𝑟 • Principle of superposition still applies – You can sum individual fields due to discrete charges – You can integrate continuous charge distributions where the charge becomes 𝑑𝑞 and the field becomes 𝑑𝐸 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 𝑟 often depends on the integrating variable