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 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

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

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:

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

Potential

• For charge distributions obeying Coulomb’s law we get the following:

Situational

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

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

Electric Field

Potential Potential Energy

Electric Force

Multiply by q

Multiply by q

Vectors

Scalars

−∫ �⃑�⋅ 𝑑 �⃑� − �⃑�𝑉 (𝑟 ) − �⃑�𝑈 (𝑟 )−∫ �⃑� ⋅𝑑 �⃑�

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

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

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)

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

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

– 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

* Always bigger than the smallest capacitance *– Series

* 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:

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)

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

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

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 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

(Black arrows denote a positive change in voltage; red negative)

Current

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

•

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:

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

Practice Problems

Practice Problem

Practice Problem

Practice Problem

Practice Problem

Practice Problem

Practice Problems

Practice Problems

Practice Problems

Quiz Questions

Remember: the negative chargeFlips the direction of the forceRelative to the cross product

Quiz Questions

Quiz Questions

Practice Problems

Practice Problems

Quiz Question

Quiz Question

Quiz Question