PHYS 241 Final Exam Review
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PHYS 241 Final 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 • “Play” around with the problem • Take fifteen to twenty minutes before the exam to relax… no studying. • Dimensional analysis is a good tool, but can give false results Concepts • Inductance • AC Circuits – – – – RMS Reactance Impedance Phasors • Displacement Current • Electromagnetic Waves (Light) – Wave/Particle Duality – Poynting Vector Concepts • Optics – Refraction • Index of Refraction • Snell’s Law – Total Internal Reflection – Malus’s Law – Mirrors – Lenses – Diffraction Inductance • What does it tell me? – The flux through a loop is proportional to the currents on conductors in the vicinity (including itself) Φπ = πΏπΌ + ππ πΌπ – This is a direct consequence of the principle of superposition and magnetic fields being proportional to the currents that create them Inductance • Why should I care? – This is the sister component to the capacitor making it one of the most fundamental electronic components Capacitor Inductor Depends on geometry and material between the plates Depends on geometry and material in intervening space Proportionality between charge and voltage Proportionality between flux and current Stores energy in an electric field Stores energy in a magnetic field Causes current to lag voltage Causes current to lead voltage π πΆ Current starts at maximum and drops to zero ππΌ ππ‘ Current starts at zero and increases to maximum π= π=πΏ Alternating Current (AC) - RMS • What does it tell me? – RMS is a type of averaging – First square the wave form, then we average and take the square root • Why should I care? – This allows us to keep a form of the Joule heating law πππ£π = πΌπππ 2 π AC - Reactance • What does it tell me? – Capacitors and inductors resist changes in the state of the circuit – Reactance is a measure of this • Why should I care? – Calculating the voltages on capacitors and inductors in an AC circuit can be complicated – Reactance give you a direct link between the average voltage across these components and the RMS current in an Ohm’s law type format – It also shows how the frequency of the applied voltage affects the system 1 ππΏ = π πΏ ππΆ = ππΆ AC - Impedance • What does it tell me? – It represents the relationship (magnitude and phase difference) between the applied voltage and the current • Why should I care? – Impedance provides a compact way to carry a lot of information about your circuit AC - Impedance • Since the impedance carries phase information, it is a complex number 1 π ππΏ = π π πΏ ππΆ = =− πππΆ ππΆ 1 π = π + ππΏ + ππΆ = π + π(ππΏ − ) ππΆ • The circuit is at resonance when the impedance is a real number – This corresponds to maximum power transfer to the resistors AC - Phasors • A phasor is a graphical representation of the relationship between voltage and current in a system • This exploits the power of complex numbers as both vectors and rotations • The phasor rotates through the complex plane and the real projections of the phasor give the measured value • See Demonstration Displacement Current • What does it tell me? – A changing electric field produces a magnetic field as if there was a current flowing that is proportional to the change in flux ππΈ π» × π΅ = ππ π½ + ππ ππ‘ πΦπ π΅ ⋅ π π = ππ πΌ + πΌπ· = ππ πΌ + ππ ππ‘ Displacement Current • Why do I care? – The correction completes Ampere’s law bringing it in agreement with the Biot-Savart Law – Like Faraday’s law, this allows for the propagation of electromagnetic waves Poynting Vector • What does it tell me? – Energy and momentum can be carried away by electromagnetic waves 1 π= πΈ ×π΅ ππ π − π’ππ = π ⋅ πππ΄ + π½ ⋅ πΈππ ππ‘ π ππ π Change in internal energy Energy flowing out Work done inside Poynting Vector • Why do I care? – – – – It is a conservation law Newton’s third law fails without it Hints at the need for special relativity The intensity (power) of light is defined as the time average of the vectors magnitude πΌ≡ π – Radiation pressure is related to the intensity πΌ ππ ππ ≡ π Index of Refraction • What does it tell me? – The ratio of the speed of a wave in a reference medium (we choose the vacuum) and another medium π 1 π≡ = π£ ππ ππ • Why should I care? – The index of refraction influences nearly all optical phenomena in some way • Depends on electrical and magnetic properties of the medium – sensitive to frequency (i.e. π: π βΌ β) Snell’s Law • What does it tell me? – The relationship between the indices of refraction and the angles of refraction and reflection π1 sin π1 = π2 sin π2 • Why should I care? – This concept is the “building block” for more advanced concepts such as thin film diffraction • Remember that ALL angles are measured from the NORMAL of the surface Malus’s Law • What does it tell me? – How the intensity of polarized light is affected by a polarizer πΌ = πΌπ cos2 π – Your book defines theta to be the angle between the transmission axes of two polarizers – Alternatively, it is the angle between the plane of polarization and the transmission axis of the polarizer – If the light is unpolarized, the intensity if halved Assumptions/Conventions • Wavelength of light is much shorter than the length scale of the geometry – Treat light as rays; i.e. no bending • Small angle deviations from the optical axis – Spherical surface is nearly parabolic sin π ≈ π • The biggest challenge in applying the formulae is following the correct sign convention Mirrors Rules for Ray Diagrams Parallel Rays Reflected through focal point Focal Rays Reflected parallel to optical axis Radial Rays Reflected back on itself Sign Convention s is positive if object is on the incident-light side s’ is positive if the image is on the reflected-light side R is positive if the mirror is concave Lenses Rules for Ray Diagrams Parallel Rays Refracted through focal point Focal Rays Refracted parallel to optical axis Central Rays No deflection when refracted Sign Convention s is positive if object is on the incident-light side s’ is positive if the image is on the refracted-light side r is positive if center of curvature is on the refracted-light side Diffraction • What does it tell me? – How a wave behaves near objects – Only an appreciable affect when the length scale of the wave and the geometry are similar • Visible light: 400nm – 700nm • Sound waves: 17mm – 17m Interference • What does it tell me? – How waves mix together – Based on the principal of superposition π΄ cos 2ππ1 + π΄ cos 2ππ2 = 2π΄ cos 2π π1 − π2 π1 + π2 cos 2π 2 2 – Always occurs, but is especially noticeable when the waves are coherent Interference • There are two main sources of interference that we will consider – Path length difference 2ππ βπ = π – Reflected waves can pick up a phase shift when going into a medium with a higher index of refraction π1 − π2 π΄π = π΄πΌ π1 + π2 Main Strategy • For any kind of diffraction, the game is always about counting up phase shifts; these can be expressed in terms of angles or wavelengths – Angles • Constructive Interference: Even multiples of π • Destructive Interference: Odd multiples of π – Wavelengths • Constructive Interference: Integer multiples of λ • Destructive Interference: Odd half-integer multiples of λ Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem Practice Problem
