phys 241 exam review kevin ralphs. overview general exam strategies concepts practice problems
TRANSCRIPT
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PHYS 241 Exam Review
Kevin Ralphs
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Overview
• General Exam Strategies• Concepts• Practice Problems
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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.
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Concepts• Electricity
– Gradient– Potential Energy– Potential– Capacitance
• Circuits– Current– Resistance/Resistivity– Kirchoff’s Rules
• Magnetism– Magnetic Fields– Magnetostatics– Electrodynamics
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Gradient
• The gradient is a vector operator that gives two pieces of information about a scalar function
1. Direction of steepest ascent2. How much the function is changing in that direction
• 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
Situational: Cartesian Coordinates
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Potential Energy
• In a closed system with no dissipative forces
• The work done is due to the electric force so
WARNING: Since charge can be negative, and 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
Universal
Situational
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Potential
• What does it tell me?– The change in potential energy per unit charge an object
has when moved between two points
• 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)
Situational:
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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
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Potential
• For charge distributions obeying Coulomb’s law we get the following:
Situational
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Potential• We recover the electric field
from the potential using the gradient
• 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
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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
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Tying it Together
Electric Field
Potential Potential Energy
Electric Force
Multiply by q
Multiply by q
Vectors
Scalars
−∫ �⃑�⋅ 𝑑 �⃑� − �⃑�𝑉 (𝑟 ) − �⃑�𝑈 (𝑟 )−∫ �⃑� ⋅𝑑 �⃑�
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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
Gravity m
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Capacitance
• What does it tell me?– The charge that accumulates on two conductors is proportional to
the voltage between them
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
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Capacitance
• What does capacitance depend on?– Geometry of the plates – Material between the plates– For parallel plates:
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)
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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
κ: dielectric constant, ε: permittivity of a material, εo : permittivity of free space,
Cd : capacitance with a uniform dielectric, Cvac: capacitance in the vacuum
Situational:
Situational:
Assumes steady fields
Assumes uniform dielectric
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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
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Capacitance
• A charged capacitor has potential energy from the work done to push the charge onto the plates
– 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)
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Capacitance
• In circuits– In well-behaved configurations, capacitors may
be combined into a single equivalent capacitor– Parallel
* Always bigger than the smallest capacitance *– Series
* Always smaller than the smallest capacitance *
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Capacitance
• Capacitors are in equilibrium– Series: when they have the same charge– Parallel: when they have the same voltage
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Current
• What does it tell me?– The amount of charge flowing through a boundary– The unit of measure is the ampere:
The word “flow” implies there should be an equation similar to flux that describes this
is the current density is the drift velocity. It is the average velocity of the charge carriers, is the charge density, is the number density (# of charge carriers/unit
volume)
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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 ()
– 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
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
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Resistivity
• The resistivity () tells us how easy it is to push charge through a material, regardless of its dimensions
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: resistivity at temperature T, ρo: Resistivity at To =20°C
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Kirchoff’s Rules
• Loop Rule– Based on conservation of energy
• Node Rule– Based on conservation of charge
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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.
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Kirchoff’s Rules
• The most common errors in applying Kirchoff’s rules are sign errors
Voltage Source
Resistor
(Black arrows denote a positive change in voltage; red negative)
Current
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Capacitors and Inductors
• Capacitors and inductors act like mirrors of one another
Capacitor Inductor
Proportionality
Energy
Charging
Discharging
Voltage
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Kirchoff’s Rules
• Loop Rule– Based on conservation of energy
• Node Rule– Based on conservation of charge
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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.
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Kirchoff’s Rules
• The most common errors in applying Kirchoff’s rules are sign errors
Voltage Source
Resistor
Current
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Right-Hand Rule and the Cross Product
• Cross product is perpendicularto BOTH of the vectors in theproduct
• You sweep your hand from thefirst vector to the second throughthe smallest angle between
• Measures how perpendiculartwo vectors are
•
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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: – Bilinear:
)
– Cyclical in our Cartesian basis vectors:
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Magnetic Fields
• Lorentz Force– What does it tell me?• The force a charged particle experiences in an
electromagnetic field
• For a wire this becomes
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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 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)
• This along with Maxwell’s equations describe all electromagnetic phenomena
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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
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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 Magnetic Moment
Torque
Potential Energy�⃑�=𝐼𝐴 �̂�
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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
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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!
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Gauss’s Law for Magnetism
• What does it tell me?– The net magnetic flux through a closed surface is
zero
– 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
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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
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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
• 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
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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
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Practice Problems
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Practice Problem
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Practice Problem
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Practice Problem
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Practice Problem
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Practice Problem
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Practice Problems
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Practice Problems
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Practice Problems
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Quiz Questions
Remember: the negative chargeFlips the direction of the forceRelative to the cross product
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Quiz Questions
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Quiz Questions
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Practice Problems
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Practice Problems
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Quiz Question
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Quiz Question
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Quiz Question