Lecture 1-1 Coulomb’s Law Charges with the same sign repel each other, and charges with opposite signs attract each other. The electrostatic force between two particles is proportional to the amount of electric charge that each possesses and is inversely proportional to the distance between the two squared. q1q1 q2q2 r 1,2 where   is called the permittivity constant. Coulomb constant: 1,2 by 1 on 2

Lecture 1-2 Electric Field Define electric field, which is independent of the test charge, q 2, and depends only on position in space: Electric Field due to a Point Charge Q

Lecture 1-3 Dynamics of a Charged Mass in Electric Field For -Q<0 in uniform E downward: Oscilloscope Ink-Jet Printing Oil drop experiment -Q v y = at = qE/m t v x >>0

Lecture 1-4 Electric Field from Coulomb’s Law Bunch of Charges (volume charge) (surface charge) (line charge) Continuous Charge Distribution dq P Summation over discrete charges Integral over continuous charge distribution P k

Lecture 1-5 Gauss’s Law: Quantitative Statement The net electric flux through any closed surface equals the net charge enclosed by that surface divided by ε 0. How do we use this equation?? The above equation is TRUE always but it doesn’t look easy to use. BUT - It is very useful in finding E when the physical situation exhibits a lot of SYMMETRY.

Lecture 1-6 Charges and fields of a conductor In electrostatic equilibrium, free charges inside a conductor do not move. Thus, E = 0 everywhere in the interior of a conductor. Since E = 0 inside, there are no net charges anywhere in the interior. Net charges can only be on the surface(s). The electric field must be perpendicular to the surface just outside a conductor, since, otherwise, there would be currents flowing along the surface.

Lecture 1-7 Electric Potential Energy of a Charge in Electric Field Coulomb force is conservative => Work done by the Coulomb force is path independent. Can associate potential energy to charge q 0 at any point r in space. It’s energy! A scalar measured in J (Joules)

Lecture 1-8 Electric Potential So U(r)/q 0 is independent of q 0, allowing us to introduce electric potential V independent of q 0. [Electric potential] = [energy]/[charge] SI units: J/C = V (volts) U(r) of a test charge q 0 in electric field generated by other source charges is proportional to q 0. taking the same reference point Potential energy difference when 1 C of charge is moved between points of potential difference 1 V Scalar!

E from V Expressed as a vector, E is the negative gradient of V We can obtain the electric field E from the potential V by inverting the integral that computes V from E :

Lecture 1-10 Electric Potential Energy and Electric Potential positive charge High U (potential energy) Low U negative charge High U Low UHigh V (potential) Low V Electric field direction High V Low V Electric field direction

Lecture 1-11 Two Ways to Calculate Potential Integrate - E from the reference point at (∞) to the point (P) of observation: Integrate dV (contribution to V(r) from each infinitesimal source charge dq) over all source charges:  A line integral (which could be tricky to do)  If E is known and simple and a simple path can be used, it may be reduced to a simple, ordinary 1D integral. q1q1 q2q2 q3q3 q4q4 P Q P r P Q

Lecture 1-12 Capacitance Capacitor plates hold charge Q The two conductors hold charge +Q and –Q, respectively. The capacitance C of a capacitor is a measure of how much charge Q it can store for a given potential difference ΔV between the plates. Expect Capacitance is an intrinsic property of the capacitor. farad (Often we use V to mean ΔV.)

Lecture 1-13 Steps to calculate capacitance C 1.Put charges Q and -Q on the two plates, respectively. 2.Calculate the electric field E between the plates due to the charges Q and -Q, e.g., by using Gauss’s law. 3.Calculate the potential difference V between the plates due to the electric field E by 4.Calculate the capacitance of the capacitor by dividing the charge by the potential difference, i.e., C = Q/V.

Lecture 1-14 Energy of a charged capacitor How much energy is stored in a charged capacitor? Calculate the work required (usually provided by a battery) to charge a capacitor to Q Total work is then Calculate incremental work dW needed to move charge dq from negative plate to the positive plate at voltage V.

Lecture 1-15 Dielectrics between Capacitor Plates + Q- Q free charges neutral -q +q Electric field E between plates can be calculated from Q – q. Polarization Charges ± q

Lecture 1-16 Capacitors in Parallel Equivalent Capacitor: V is common where

Lecture 1-17 Capacitors in Series Equivalent Capacitor: where q is common

Lecture 1-18 Electric Current Current = charges in motion Magnitude rate at which net positive charges move across a cross sectional surface Units: [I] = C/s = A (ampere) Current is a scalar, signed quantity, whose sign corresponds to the direction of motion of net positive charges by convention J = current density (vector) in A/m²

Lecture 1-19 Ohm’s Law Current-Potential (I-V) characteristic of a device may or may not obey Ohm’s Law: or V = IR with R constant Resistance (ohms) gas in fluorescent tube tungsten wire diode

Lecture 1-20 Energy in Electric Circuits So, Power dissipation = rate of decrease of U = Steady current means a constant amount of charge ΔQ flows past any given cross section during time Δt, where I= ΔQ / Δt. Energy lost by ΔQ is => heat V

Lecture 1-21 Resistors in Parallel Devices in parallel has the same potential drop Generally,

Lecture 1-22 Kirchhoff’s Rules Kirchhoff’s Rule 1: Loop Rule  When any closed loop is traversed completely in a circuit, the algebraic sum of the changes in potential is equal to zero. Kirchhoff’s Rule 2: Junction Rule  The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction.  Conservation of charge  In and Out branches  Assign I i to each branch  Coulomb force is conservative

Lecture 1-23 Galvanometer Inside Ammeter and Voltmeter Ammeter: an instrument used to measure currents Voltmeter: an instrument used to measure potential differences galvanometer shunt resistor galvanometer Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance R g is small for not disturbing what is being measured.

Lecture 1-24 Galvanometer Inside Ammeter and Voltmeter Ammeter: an instrument used to measure currents Voltmeter: an instrument used to measure potential differences galvanometer shunt resistor galvanometer Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance R g is small for not disturbing what is being measured.

Lecture 1-25 Discharging a Capacitor in RC Circuits 2. Loop Rule: 3.Convert to a differential equation 4.Solve it! 1. Switch closed at t=0. Initially C is fully charged with Q 0 I

Lecture 1-26 Charging a Capacitor in RC Circuits 1.Switch closed at t=0 C initially uncharged, thus zero voltage across C. 2. Loop Rule: 3. Convert to a differential equation 4.Solve it! (τ=RC is the time constant again)

Lecture 1-27 Magnetic Field B Magnetic force acting on a moving charge q depends on q, v. This defines B. direction by Right Hand Rule. B is a vector field 1 T = 10 4 gauss (earth magnetic field at surface is about 0.5 gauss) Vary q and v in the presence of a given magnetic field and measure magnetic force F on the charge. Find: F varies sinusoidally as direction of v is changed (q>0) If q<0

Lecture 1-28 Magnetic Force on a Current Consider a current-carrying wire in the presence of a magnetic field B. There will be a force on each of the charges moving in the wire. What will be the total force dF on a length dl of the wire? Suppose current is made up of n charges/volume each carrying charge q < 0 and moving with velocity v through a wire of cross-section A. Force on each charge = Total force = Current = For a straight length of wire L carrying a current I, the force on it is: A

Lecture 1-29 Both B and E present when balanced velocity selector No deflection when E=3 kV/m, B=1.4 G

Lecture 1-30 Magnetic Torque on a Current Loop If B field is parallel to plane of loop, the net torque on loop is 0. Definition of torque: abut a chosen point so that n is twisted to align with B If B is not zero, there is net torque. magnetic moment direction B

Lecture 1-31 Potential Energy of Dipole Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. Define a potential energy U (with zero at position of max torque) corresponding to this work.  Therefore,  B x .

Lecture 1-32 Sources of Magnetic Fields Permeability constant Moving point charge: Bits of current: I Biot-Savart Law also The magnetic field “circulates” around the wire.

Lecture 1-33 Gauss’s Law for Magnetism sources No sources Gauss’s Law Gauss’s Law for Magnetism

Lecture 1-34 Ampere’s Law in Magnetostatics The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, Choosing a direction of integration. A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration. Biot-Savart’s Law can be used to derive another relation: Ampere’s Law

Lecture 1-35 Potential Energy of Dipole Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. Define a potential energy U (with zero at position of max torque) corresponding to this work.  Therefore,  B x . U = +μ B cos θ

Lecture 1-36 Faraday’s Law of Induction The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time. wher e Minus sign indicates the sense of EMF: Lenz’s Law Decide on which way n goes Fixes sign of Δ ϕ B RHR determines the positive direction for EMF N N

Lecture 1-37 Ways to Change Magnetic Flux Changing the magnitude of the field within a conducting loop (or coil). Changing the area of the loop (or coil) that lies within the magnetic field. Changing the relative orientation of the field and the loop. motor generator

Lecture 1-38 Motional EMF of Sliding Conductor  Lenz’s Law gives direction Induced EMF: counter-clockwise  Faraday’s Law  This EMF induces current I  Magnetic force F M acts on this I  F M decelerates the bar

Lecture 1-39 Self-Inductance As current i through coil increases, magnetic flux through itself increases. This in turn induces counter EMF in the coil itself When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease. Self-induction (if flux linked) Faraday’s Law: (henry)

Lecture 1-40 Energy Stored By Inductor 1.Switch on at t=0 2. Loop Rule: 3. Multiply through by I As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + - Rate at which battery is supplying energy Rate at which energy is dissipated by the resistor Rate at which energy is stored in inductor L

Lecture 1-41 RL Circuits – Starting Current 2. Loop Rule: 3. Solve this differential equation τ=L/R is the inductive time constant 1.Switch to e at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + -

Lecture 1-42 Alternating Current (AC) = Electric current that changes direction periodically ac generator is a device which creates an ac emf/current. ac motor = ac generator run in reverse A sinusoidally oscillating EMF is induced in a loop of wire that rotates in a uniform magnetic field. where

Lecture 1-43 Resistive Load Start by considering simple circuits with one element (R, C, or L) in addition to the driving emf. Pick a resistor R first. v R (t) and I(t) in phase Kirchhoff’s Loop Rule: I peak + -- I(t)

Lecture 1-44 Capacitive Load I(t) leads v(t) by 90 o (1/4 cycle) Loop Rule: Power: +--

Lecture 1-45 Inductive Load v L (t) leads I(t) by 90 o (1/4 cycle) Kirchhoff’s Loop Rule: Power: +--

Lecture 1-46 (Ideal) LC Circuit From Kirchhoff’s Loop Rule From Energy Conservation same Natural Frequency harmonic oscillator with angular frequency

Lecture 1-47 ϕ Impedance in Driven Series RLC Circuit impedance, Z

Lecture 1-48 Resonance For given ε peak, R, L, and C, the current amplitude I peak will be at the maximum when the impedance Z is at the minimum. Resonance angular frequency: This is called resonance. i.e., load purely resistiveε and I in phase

Lecture 1-49 Transformer AC voltage can be stepped up or down by using a transformer. AC current in the primary coil creates a time- varying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit. Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load) step-up step-down With resistive load R in secondary, current I 2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must somehow be exactly cancelled because V 1 is a defined voltage source. This occurs by another current I 1 which is induced on the primary side due to I 2.

Lecture 1-50 Maxwell’s Equations The equations are often written in slightly different (and more convenient) forms when dielectric and/or magnetic materials are present. Basis for electromagnetic waves!

Lecture 1-51 Electromagnetic Wave Propagation in Free Space So, again we have a traveling electromagnetic wave speed of light in vacuum Speed of light in vacuum is currently defined rather than measured (thus defining meter and also the vacuum permittivity). Ampere’s Law Faraday’s Law Wave Equation

Lecture 1-52 Maxwell’s Rainbow Light is an Electromagnetic Wave

Lecture 1-53 ©2008 by W.H. Freeman and Company

Lecture 1-54 n 1 sin θ 1 = n 2 sin θ 2 Snell’s Law of Refraction

Lecture 1-55 Summary: Laws of Reflection and Refraction Law of Reflection A reflected ray lies in the plane of incidence The angle of reflection is equal to the angle of incidence Law of refraction A refracted ray lies in the plane of incidence The angle of refraction is related to the angle of incidence by Snell’s Law Medium 1 Medium 2 Where λ is the wavelength in vacuum

Lecture 1-56 Total Internal Reflection In general, if sin θ 1 > (n 2 / n 1 ), we have NO refracted ray; we have TOTAL INTERNAL REFLECTION. All light can be reflected, none refracting, when light travels from a medium of higher to lower indices of refraction. e.g., glass (n=1.5) to air (n=1.0) But θ 2 cannot be greater than 90 O ! Critical angle above which this occurs. medium 2 medium 1

Lecture 1-57 Polarization of Electromagnetic Waves Polarization is a measure of the degree to which the electric field (or the magnetic field) of an electromagnetic wave oscillates preferentially along a particular direction. linearly polarized unpolarized partially polarized Looking at E head-on Linear combination of many linearly polarized rays of random orientations components equal y- and z- amplitudes unequal y- and z- amplitudes

Lecture 1-58 ©2008 by W.H. Freeman and Company

Lecture 1-59 Focal Point of a Spherical Mirror When parallel rays incident upon a spherical mirror, the reflected rays or the extensions of the reflected rays all converge toward a common point, the focal point of the mirror. Distance f is the focal length. Real focal point: the point to which the reflected rays themselves pass through. This is relevant for concave mirrors. Virtual focal point: the point to which the extensions of the reflected rays pass through. This is relevant for convex mirrors. f f Rays can be traversed in reverse. Thus, rays which (would) pass through F and strike the mirror will emerge parallel to the central axis. concave mirror: convex mirror

Lecture 1-60 Mirror Equation and Magnification s is positive if the object is in front of the mirror (real object) s is negative if it is in back of the mirror (virtual object) s’ is positive if the image is in front of the mirror (real image) s’ is negative if it is in back of the mirror (virtual image) m is positive if image and object have the same orientation (upright) m is negative if they have opposite orientation (inverted) f and r are positive if center of curvature in front of mirror (concave) f and r are negative if it is in back of the mirror (convex) (f = r/2)

Lecture 1-61 Locating Images Real images form on the side of a mirror where the objects are, and virtual images form on the opposite side. only using the parallel, focal, and/or radial rays.

Lecture 1-62 Thin Lenses nomenclature A lens is a piece of transparent material with two refracting surfaces whose central axes coincide. A lens is thin if its thickness is small compared to all other lengths (s, s’, radii of curvature). Net convex – thicker in the middle Parallel rays converge to real focus. f > 0 Net concave – thinner in the middle Parallel rays diverge from virtual focus. f < 0 f > 0 Convergent lens r 1 >0 r 2 <0 Divergent lens f < 0 r 1 <0 r 2 >0

Lecture 1-63 Signs in the Lens Equation for Thin Lenses p is positive for real object q is positive for real image q is negative for virtual image m is positive if image is upright m is negative if image is inverted f is positive if converging lens f is negative if diverging lens p is negative for virtual object

Lecture 1-64 Geometric Optics vs Wave Optics Geometric optics is a limit of the general optics where wave effects such as interference and diffraction are negligible.  Geometric optics applies when objects and apertures involved are much larger than the wavelength of light.  In geometric optics, the propagation of light can be analyzed using rays alone. Wave optics (sometimes also called physical optics) - wave effects play important roles.  Wave optics applies when objects and apertures are comparable to or smaller than the wavelength of light.  In wave optics, we must use the concepts relevant to waves such as phases, coherence, and interference.

Lecture 1-65 Thin-Film Interference-Cont’d Path length difference: (Assume near-normal incidence.) destructive constructive where ray-one got a phase change of 180 o due to reflection from air to glass. the phase difference due to path length is: then total phase difference:  =  ’+180.

Lecture 1-66 Interference Fringes For D >> d, the difference in path lengths between the two waves is A bright fringe is produced if the path lengths differ by an integer number of wavelengths, A dark fringe is produced if the path lengths differ by an odd multiple of half a wavelength,

Lecture 1-67 Intensity of Interference Fringes-Cont’d For Young’s double-slit experiment, the phase difference is

Lecture 1-68 Dark and Bright Fringes of Single-Slit Diffraction

Lecture 1-69 Intensity Distribution 1 maxima: central maximum because minima: or

Lecture 1-70 Intensity Distribution from Realistic Double- Slit Diffraction double-slit intensity replace by single-slit intensity envelope How many maxima will fit between central max and first envelope min?:

Lecture 1-71 Diffraction Gratings Devices that have a great number of slits or rulings to produce an interference pattern with narrow fringes. Types of gratings: transmission gratings reflection gratings One of the most useful optical tools. Used to analyze wavelengths. up to thousands per mm of rulings D Maxima are produced when every pair of adjacent wavelets interfere constructively, i.e., m th order maximum