Lecture 1 Physics 2

Coulombs law Matter is composed of charged objects affected by

electric (& magnetic) forces Charge is quantized and comes in two varieties

dubbed positive and negative Different charges attract, like repel The Coulomb force varies as The Coulomb force obeys superposition Quantitative experiments establish the magnitude

of the Coulomb force

x t

Electric Field expresses the consequence of charge reports force on test charge particle: Point charge q : Superposition:

Replace sum by integral for continuous distribution

“Field point”

Electric Flux and Gauss law Represent by field lines

From positive to negativeDensity proportional to is tangent to field lines

Define flux of vector field through surface

Non-zero flux of vector field through closed surface “springs from” or “ends in” the interior

Gauss law relates flux of electric field through any closed surface with enclosed electric charge

Gauss law: Applications For simple charge distributions:

Symmetry determines field configuration

Gauss law determines the magnitude

Conductor: Charge moves in response to within conductor in equilibrium Net charge density only at surface of conductor Field normal to surface of conductor:

Electric Potential Energy Work by Coulomb force is path independent

(conservative force) Work done by Coulomb force

is my work to affect change

My work to assemble charge configuration:

ri rf

q1 q2q2

Electric Potential is work to place unit charge at

From potential to electric field:

From potential to electric field:

Procedures for calculating

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Capacitance Charge displaced between disconnected

conductors is proportional to potential difference

The constant of proportionality, , characterizes the conducting structure

To calculate capacitance:

a) Mentally displace charge from plate 1 to 2

b) Determine the resulting

c) Using convenient path calculate:

Capacitors: connected & enhanced Polarizable medium in capacitor reduces

Energy stored in capacitor

Alternate route to C: calculate energy, U, then

Connecting capacitors:

Series: smallerParallel: larger

So energy density

So capacitance increases

Current and Resistance Resistivity and conductivity are materials properties:

Resistance is property of materials and geometry:

Terminal characteristics for resistor:

Series connection of resistors:

Parallel connection of resistors:

Power delivered by battery:

Power dissipated in resistor:

Circuit Analysis Kirchoff’s laws

Analysis of multi-loop circuitSimplify circuit

Define suitable variables (currents or potentials)

Write down Kirchoff #1

Write down Kirchoff #2

Solve the linear equations

Charge conservation

Energy conservation

Dynamic Circuits: RC RC circuit: systematic approach:

Write Kirchoff’s equations considering initial conditions

Use relationship between current and charge:

Solve linear differential equation

Match solution to initial conditions

RC circuit: quick solution

R is resistance the capacitor, C, “looks into”

is value immediately following step disturbance

is value far later where all capacitors open circuit

Recall energy is stored in capacitor:

Effects of the magnetic B field

B-field exerts force on moving charged particle

“never works”: Spiraling charged particles: Crossed E-B fields:

Rutherfords discovery of electron:Hall effect (discovered here):

Force on current carrying wire: Torque on current-coil: where Energy of magnetic dipole in B-field:

Where B-fields come from Force between parallel current carrying wires:

This implies current carrying wire generates field

For general current distribution Biot-Savart:

Amperes law Amperes law relates a closed loop integral to the

enclosed current:

Field in solenoid with winding density, n, current, i

• Any current distribution• Any closed path

Motional EMF & Faraday’s law

Moving with respect to B-field:

Motional EMF:

Generalize the result through definition of B-flux

Equivalent formulation of Faraday’s law

Lenz: Induced current counteracts change in flux

Inductors and R-L Circuits Faraday’s law implies a change in current is met

by an opposing electric potential

For long solenoid

Time constant for LR circuit:

Energy stored in inductor:

Magnetic energy density:

Mutual inductance:

+ -

Circuits that Oscillate (L-C & R-L-C) LC circuit supports oscillations with

resonance angular frequency

Energy oscillates between electric and magnetic

forms in capacitor and inductor respectively

RLC circuit can support damped oscillations with

frequency and time cnst:

To derive the differential equation write loop

equation for q(t)

Complex numbers greatly simplify solutions

AC Circuits: Analysis with phasors

Circuits consisting of linear elements (R, L, C)

respond harmonically when driven harmonically

Generalized relationship between AC current and

AC voltage : Vm=ImZ here Z is called “Impedance”

Voltage and current are not generally in phase:Resistor: voltage and current in phase

Inductor: current lags voltage

Capacitor: voltage lags current

Use “Phasors” to analyze RLC circuit

Analysis of resonance with Phasors

Generalized relationship between AC current and

AC voltage : Vm=ImZ here Z is called “Impedance”

Voltage and current are not generally in phase:Resistor: voltage and current in phase

Inductor: current lags voltage by

Capacitor: voltage lags current by

Use “Phasors” to analyze RLC circuit

Average power dissipated:

Ideal Transformer (AC!):

Maxwell’s term & waves Maxwell fixes Ampere:

Maxwell equations imply wave equation:

Solutions: travelling waves with speed

EM-waves exist in vacuum accounting for micro-waves to visible light to gamma rays and beyond

Transverse polarized: propagation along A fixed ratio of amplitudes:

1775-1836 1831-1879

AmpereMaxwell

Polarization, Reflection & Refraction Light slows in matter: where

Matching E and B across interface implies:In-plane components of wave vectors match:

This implies incident and reflected angles match and Snell’s law:

Also obtain normal reflected intensity: Total internal reflection above Polarizing Brewster angle:

Medium n1

Medium n2 > n1

Imaging with lenses & mirrors Common formulae for mirrors and thin lenses:

Sign conventions:

Spherical refraction:

Thin lens in air:

Mirror: