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.
• Look for symmetries
Concepts
• Circuits
– Current
– Resistance/Resistivity
– Capacitors and Inductors
– Kirchoff’s Rules
• Magnetism
– Magnetic Fields
– Magnetostatics
– Electrodynamics
Current
• What does it tell me?
– The amount of charge flowing through a boundary
– 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
Resistivity
• The resistivity tells us how easy it is to push
charge through a material, regardless of its
dimensions
𝐿
𝑅= 𝜌
𝐴
• Resistivity depends on temperature and the
temperature coefficient gives us this
relationship measured relative to 20℃
𝜌𝑜 𝛼Δ𝑇 = Δ𝜌
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
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)
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
Electromotive Force (EMF)
• What does it tell me?
– The change in potential energy per unit charge an object
has when moved along a path
Δ𝑈
ℰ≡
𝑞
– It can also refer to the voltage measured across two
terminals
• Why do I care?
– So far we have considered conservative electric fields
which have scalar potentials
– For non-conservative fields, the change in potential energy
becomes path dependent and EMF is accounting for that
Electromotive Force (EMF)
• Why do I care?
– If a particle is free to move around in space, this is
not all that helpful, but when they are constrained
to move on a specified path (like an electronic
circuit), it becomes well-defined.
Note:
1. This is not a force, it has units of volts
2. This is not a potential, the path taken matters
very much
Motional EMF
• When a conductor moves through a magnetic
field, it acquires an EMF (this is more along
the lines of the two terminal definition)
• This happens because a Lorentz force from the
magnetic field shuffles charges to opposite
ends of the conductor
• This sets up a voltage like a parallel plate
capacitor bringing the charges into an
equilibrium
Motional EMF
Farraday’s Law
• Two earlier approximation schemes
– Electrostatics
• Stationary charges
• Conducting charges at equilibrium
𝐸 ⋅ 𝑑𝑙 = 0
𝛻 × 𝐸=0
– Magnetostatics
• Steady Currents
Farraday’s Law
• In electrodynamics we allow single charges to
move
• This causes time varying magnetic fields
bringing Farraday’s law into effect
𝜕𝐵
𝛻 × 𝐸=−
𝜕𝑡
𝑑
𝐸 ⋅ 𝑑𝑙 = −
𝐵 ⋅ 𝑛𝑑𝐴 = Ɛ
𝑑𝑡
Farraday’s Law
• What does it tell me?
– A changing magnetic field creates a non-conservative
electric field
– Anything that affects that flux integral induces an EMF
in a loop
• Why should I care?
– Without this law, you could not see, there would be
no cell phones or radio: electromagnetic waves exist
because of this
– Inductors and transformers exploit this phenomenon
Lenz’s Law
• What does it tell me?
– When the flux through a loop changes, a current is
produced that fights this change
• Why should I care?
– This principle is how you determine the direction
of an induced current
Lenz’s Law
• If you are having problems with this, you are not alone
– People spend thousands of hours researching this (no
kidding)
• The idea is to find the direction of the induced
magnetic field and use the right hand rule to find the
current
• To find the direction of the induced field
– Note the direction of the original field through the loop
– Determine whether this field is getting stronger or weaker
– The direction of the induced field will maintain the status
quo
Practice Problem
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Answer: E
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