Transcript PHYS 241 Recitation

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