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

Resistivity

• The resistivity tells us how easy it is to push charge through a material, regardless of its dimensions

• Resistivity depends on temperature and the temperature coefficient gives us this relationship measured relative to

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

Capacitors and Inductors

• Capacitors and inductors act like mirrors of one another

Capacitor Inductor

Proportionality

Energy

Charging

Discharging

Voltage

Kirchoff’s Rules

• Loop Rule– Based on conservation of energy

• Node Rule– Based on conservation of charge

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 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

•

Magnetic Fields

• Lorentz Force– What does it tell me?• The force a charged particle experiences in an

electromagnetic field

• For a wire this becomes

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

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 Magnetic Moment

Torque

Potential Energy�⃑�=𝐼𝐴 �̂�

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

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

– 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

• 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

– 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 volts2. 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

– 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

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

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Answer: E

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems