Download PHYS 241 Exam Review

PHYS 241 Exam 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.
Concepts
• Electricity
–
–
–
–
Gradient
Potential Energy
Potential
Capacitance
• Circuits
– Current
– Resistance/Resistivity
– Kirchoff’s Rules
• Magnetism
– Magnetic Fields
– Magnetostatics
– Electrodynamics
Gradient
• The gradient is a vector operator that gives two pieces of
information about a scalar function
1. Direction of steepest ascent
2. How much the function is changing in that direction
𝜕
𝜕
𝜕
Situational: Cartesian
𝛻=
𝑥+
𝑦+ 𝑧
Coordinates
𝜕𝑥
𝜕𝑦
𝜕𝑧
•
•
It transforms a scalar function into a vector field where
every vector is perpendicular to the function’s isosurfaces
Every smooth scalar function has an associated vector
field, not every vector field has an associated scalar
function, and there are an infinite number of scalar
functions that get mapped to the same vector field
Potential Energy
• In a closed system with no dissipative forces
Δ𝑃𝐸𝑒𝑙𝑒𝑐 + 𝑊 = 0
Situational
• The work done is due to the electric force so
𝑊=
𝐹 ⋅ 𝑑𝑠 =
𝑞𝐸 ⋅ 𝑑𝑠
Universal
WARNING: Since charge can be negative, 𝐸 and 𝐹 might point in
opposite directions (this is called antiparallel) which would change
the sign of W
• This can be combined with the work-energy theorem
to obtain the velocity a charged particle has after
moving through an electric field
Potential
• What does it tell me?
– The change in potential energy per unit charge an object
has when moved between two points
Δ𝑃𝐸𝑒𝑙𝑒𝑐
Δ𝑉 ≡
= − 𝐸 ⋅ 𝑑𝑠 Situational: 𝛻 × 𝐸 = 0
𝑞
• 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
𝐸 = −𝛻𝑉
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝐸𝑥 = −
; 𝐸𝑦 = −
; 𝐸𝑧 = −
𝜕𝑥
𝜕𝑦
𝜕𝑧
Potential
• For charge distributions obeying Coulomb’s
law we get the following:
𝑞𝑖
𝑉=
𝑘
𝑟𝑖
𝑖
Situational
𝑑𝑞
𝑉= 𝑘
𝑟
𝑞
Potential
• We recover the electric field
from the potential using the
gradient
𝐸 = −𝛻𝑉
• The isolines (or isosurfaces)
of the potential are called
equipotentials
• So the electric field is
perpendicular to the
equipotential lines
(surfaces)
• This means also that electric
flow lines are perpendicular
to equipotential lines
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
– The potential must be continuous
Tying it Together
Multiply by q
Electric
Field
Vectors
−
Scalars
𝐸 ⋅ 𝑑𝑠
−𝛻𝑉 𝑟
Electric
Force
−
𝐹 ⋅ 𝑑𝑠
−𝛻𝑈 𝑟
Potential
Energy
Potential
Multiply by q
Analogies with Gravity
• Electricity and magnetism can feel very abstract because we
don’t usually recognize how much we interact with these forces
• There are many similarities between gravitational and electric
forces
• The major difference is that the electric force can be repulsive
• Gravity even has a version of Gauss’s law
Charge
Force
Field
PE
Electricity
q
𝑄𝑞
𝑘 2𝑟
𝑟
𝐸=𝑘
𝑄
𝐹
𝑟
=
𝑟2
𝑞
𝑞Δ𝑉
Gravity
m
𝐺
𝑀𝑚
𝑟
𝑟2
𝑔=𝐺
𝑀
𝐹
𝑟
=
𝑟2
𝑚
𝑚 𝑔Δ𝑦
Capacitance
• What does it tell me?
– The charge that accumulates on two conductors is
proportional to the voltage between them
𝑄 = 𝐶∆𝑉
Q: charge on the capacitor’s plates, C: capacitance,
ΔV: potential difference across the capacitor
• 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
• What does capacitance depend on?
– Geometry of the plates
– Material between the plates
– For parallel plates: 𝐶 =
𝜀𝐴
𝑑
C: capacitance, ε: permittivity of the material between the plates, A: area of
the plates (may or may not be square), d: distance between the plates
• Unit of capacitance is the Farad
– To demystify this, units are (meters*permittivity)
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
𝜀
Situational: Assumes steady fields
𝜅≡
𝜀𝑜
κ: dielectric constant, ε: permittivity of a material, εo : permittivity of free space,
𝐶𝑑
𝜅=
𝐶𝑣𝑎𝑐
Situational:
Assumes uniform
dielectric
Cd : capacitance with a uniform dielectric, Cvac: capacitance in the vacuum
Capacitance
• The permittivity of free space has no physical
meaning
• It merely changes physical quantities into their
appropriate SI units
Physical Units
SI Units
Length
Farads
Length/Charge
Volts
Length^2/Charge^2
Newtons
Length/Charge^2
Joules
Capacitance
• A charged capacitor has potential energy from
the work done to push the charge onto the
plates
𝑄
𝑈=
𝑄
𝑑𝑄′∆𝑉 =
0
0
𝑄′
𝑄2
𝑑𝑄′ =
𝐶
2𝐶
– Note: This means that inserting a dielectric into a
capacitor while it is disconnected from a voltage
source will lower the potential energy (in fact, it
will be sucked in)
Capacitance
• In circuits
– In well-behaved configurations, capacitors may be
combined into a single equivalent capacitor
– Parallel
𝐶𝑒𝑞 =
𝐶𝑖
* Always bigger than the smallest capacitance *
– Series
𝐶𝑒𝑞 =
1
𝐶𝑖
−1
* Always smaller than the smallest capacitance *
Capacitance
• Capacitors are in equilibrium
– Series: when they have the same charge
– Parallel: when they have the same voltage
Current
• What does it tell me?
– The amount of charge flowing through a boundary
– The unit of measure is the ampere: 1 𝐴 = 1 𝐶 𝑠
The word “flow” implies there should be an equation
similar to flux that describes this
𝑑𝑄
𝐼=
=
𝐽 ⋅ 𝑛𝑑𝐴
𝑑𝑡
𝑆
𝐽 = 𝜌𝑞 𝑣𝑑 = 𝑞𝑛𝑣𝑑
𝐽 is the current density 𝑣𝑑 is the drift velocity. It is the average velocity of the charge
carriers, 𝜌𝑞 is the charge density, 𝑛 is the number density (# of charge carriers/unit
volume)
Resistance
• What does it tell me?
– The ratio of the potential drop in the “direction” of the current and the
current in a segment and is measured in ohms (Ω)
– Essentially it is telling you how tough it is to push charge through an
object
• Why do I care?
– All things have resistance so it is critical to understand how it affects
electric current
– The resistor is another one of our linear electronic components
– The potential difference across a resistor is given by Ohm’s Law
+
− 𝐼𝑅
ΔV: potential difference across a resistor, I: current passing through the resistor,
R: resistor’s resistance, the sign depends on the direction of the current
Δ𝑉 =
Resistivity
• The resistivity (𝜌) tells us how easy it is to
push charge through a material, regardless of
its dimensions
𝐿
𝑅= 𝜌
𝐴
R: resistance, ρ: resistivity, L: length of resistor,
A: cross-sectional area of resistor (assumes A is constant along the resistor’s length)
• It has temperature dependence
∆𝜌 = 𝜌 − 𝜌𝑜 = 𝜌𝑜 𝛼 𝑇 − 𝑇𝑜 = 𝜌𝑜 𝛼∆𝑇
ρ: resistivity at temperature T, ρo: Resistivity at To =20°C
Kirchoff’s Rules
• Loop Rule
– Based on conservation of energy
Δ𝑉𝑖 = 0
𝐿𝑜𝑜𝑝
𝐸 ⋅ 𝑑𝑙 = 0
• Node Rule
– Based on conservation of charge
𝐼𝑗 = 0
Kirchoff’s Rules
General Procedure:
– Choose loops so that every branch is covered by at
least one loop
– Choose current directions in each branch – this
does not have to correspond to the direction of
your loop
– Write down each loop/node equation and solve
using method of your choice. You need as many
independent equations as you have currents to
solve.
Kirchoff’s Rules
• The most common errors in applying Kirchoff’s rules are
sign errors
Voltage Source
Resistor
Current
(Black arrows denote a positive change in voltage; red negative)
Capacitors and Inductors
• Capacitors and inductors act like mirrors of
one another
Proportionality
Energy
Charging
Discharging
Voltage
Capacitor
Inductor
𝑄 = 𝐶𝑉
Φ𝑀 = 𝐿𝐼
𝑈=
1 2
𝐶𝑉
2
𝑈=
𝑡
𝑄 = 𝑄𝑜 (1 − 𝑒 −
𝑡
𝐼 = 𝐼𝑜 𝑒 − 𝜏
𝑡
𝜏)
𝑄 = 𝑄𝑜 𝑒 − 𝜏
𝑡
𝐼 = 𝐼𝑜 (1 − 𝑒 − 𝜏 )
𝑄
𝑉=
𝐶
1 2
𝐿𝐼
2
𝑡
𝐼 = 𝐼𝑜 (1 − 𝑒 −
𝑡
𝐼 = 𝐼𝑜 𝑒 −
𝑑𝐼
𝑉=𝐿
𝑑𝑡
𝜏
𝜏)
Kirchoff’s Rules
• Loop Rule
– Based on conservation of energy
Δ𝑉𝑖 = 0
𝐿𝑜𝑜𝑝
𝐸 ⋅ 𝑑𝑙 = 0
• Node Rule
– Based on conservation of charge
𝐼𝑗 = 0
Kirchoff’s Rules
General Procedure:
– Choose loops so that every branch is covered by at
least one loop
– Choose current directions in each branch – this
does not have to correspond to the direction of
you loop
– Write down each loop and node equation and
solve using method of your choice. You need as
many independent equations as you have currents
to solve.
Kirchoff’s Rules
• The most common errors in applying Kirchoff’s rules
are sign errors
Voltage Source
Resistor
Current
Right-Hand Rule and the Cross Product
• Cross product is perpendicular
to BOTH of the vectors in the
product
• You sweep your hand from the
first vector to the second through
the smallest angle between
• Measures how perpendicular
two vectors are
•
𝑎×𝑏 =
𝑎
𝑏 𝑆𝑖𝑛(𝜃)
Right-Hand Rule and the Cross Product
• It’s also possible to do these
calculation algebraically, if you are
given the vectors by knowing some
basic properties of the cross-product
(no right-hand rule needed!!!)
– Anti-Commutative: 𝑣 × 𝑤 = − 𝑤 × 𝑣
– Bilinear:
𝛼𝑣 × 𝛽𝑤 = 𝛼𝛽(𝑣 × 𝑤)
𝑣 + 𝑤 × 𝑢 = (𝑣 × 𝑢)+(𝑤 × 𝑢)
– Cyclical in our Cartesian basis vectors:
𝑥 × 𝑦 = 𝑧, 𝑦 × 𝑧 = 𝑥,
𝑧×𝑥 =𝑦
Magnetic Fields
• Lorentz Force
– What does it tell me?
• The force a charged particle experiences in an
electromagnetic field
𝐹 =𝑞 𝐸+ 𝑣×𝐵
• For a wire this becomes
𝐹 = 𝐼𝑙 × 𝐵
Magnetic Fields
• Lorentz Force (cont.)
– Why should 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.
2.
Use integration to get the particle’s velocity as a function of
time, then integrate again to gets its position
Kinematic equations (the result when method 1. is applied
in the case of constant acceleration)
• This along with Maxwell’s equations describe all
electromagnetic phenomena
Magnetostatics
• Electrostatics vs Magnetostatics
– When we were talking about electrical phenomenon
earlier in the course, we assumed we were at an
equilibrium so no charges were moving
– For our study of magnetism we will assume that our
current is steady (or at least not varying rapidly) and
that we are not too far away from our magnetic field
source
– Note that the principle of superposition is valid in
both of these approximations
Magnetic Moment
• What does it tell me?
– How a current loop or magnet responds to an external
magnetic field
• Why should I care?
– This drastically simplifies your calculations
– You end up treating it like an electric dipole
Wire
𝜇 = 𝐼𝐴𝑛
Torque
𝐼
Magnetic
Moment
𝑟 × 𝑑𝑟 × 𝐵
𝜇×𝐵
𝑤𝑖𝑟𝑒
Potential Energy
𝜏 𝑑𝜃
−𝜇 ⋅ 𝐵
Biot-Savart Law
• What does it tell me?
– The magnetic field produced by a current in the
magnetostatic approximation
• Why should I care?
– This is a fundamental physical principle derived
from experimental data
𝜇0
𝐼 𝑑𝑙 × 𝑟
𝐵=
4𝜋 𝑤𝑖𝑟𝑒 𝑟 2
Biot-Savart Law
• When running a Biot-Savart Law integral, it
often becomes crucial to draw a picture to
make sure you get the cross product correct
• FYI: If the magnetostatic approximation fails
you would have to use the equation below!
Gauss’s Law for Magnetism
• What does it tell me?
– The net magnetic flux through a closed surface is
zero
𝐵 ⋅ 𝑑𝐴 = 0
𝑆
– If you recall our discussion about electric flux, the
net flux of a field through a closed surface is
proportional to the total sources and sinks that
are within the volume bounded by the surface
– This means that there are NO magnetic charges
Gauss’s Law for Magnetism
• Why should I care?
– Gauss’s law gives you important information
about the shape of magnetic field lines
– Essentially, magnetic lines of flux are loops and
they never converge on or diverge from a point
Note: when there are no currents flowing, we can
use the concept of magnetic “charge” to solve
problems, but this is a theoretical tool only
Ampere’s Law
• What does it tell me?
– A closed path integral of the magnetic field is
proportional to the current that flows through the
loop
𝐵 ⋅ 𝑑 𝑙 = 𝜇𝑜 𝐼𝑒𝑛𝑐
𝐶
• Why should I care?
– You can always use it to calculate the current within a
region and when there is a HIGH of degree symmetry
you can figure out the magnetic field
Ampere’s Law
• Although this isn’t called Gauss’s law, this idea
functions much like Gauss’s law for electric fields.
• This means that all the details about Gauss’s law
apply here
– You must use a closed loop
– The current is that which is enclosed by the loop: this
plays the analog as the source of a magnetic field
– A line integral is a sum: Just because it evaluates to
zero, does not mean that the magnetic field is zero
– You must already know something about the magnetic
field prior to applying Ampere’s Law
Practice Problems
Practice Problem
Practice Problem
Practice Problem
Practice Problem
Practice Problem
Practice Problems
Practice Problems
Practice Problems
Quiz Questions
Remember: the negative charge
Flips the direction of the force
Relative to the cross product
Quiz Questions
Quiz Questions
Practice Problems
Practice Problems
Quiz Question
Quiz Question
Quiz Question