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
Sne
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Bru
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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: