Formula Sheet for Exam #2

Formula Sheet for Exam #2 Reading and thoroughly familiarizing yourself with this formula sheet is an important part of, but it is not a substitute for, proper exam preparation. The latter requires, among other things, that you have re-worked all assigned homework problem sets (PS) and the in-class quizzes, studied the posted PS solutions, and worked and studied the assigned conceptual practice (CP) problems, as well as (optionally) some practice test (PT) problems, as posted on the LON-CAPA homework and on the PHYS1112 examples and homework web pages. You should consult the syllabus, and in particular review the Class Schedule on the last syllabus page (posted on the PYS1112 course web site), to find out which topics you should cover in preparing for this exam. Wave Optics, Interference, Diffraction (1) Periodic Wave Condition: v = λf = λ τ (2) Index of Refraction for electromagnetic waves, definition: n = c v = λ vacuum λ with λ vacuum c/f = cτ. (3) Definition of Path Length Difference for two-source, double-slit or adjacent slits in multi-slit/diffraction grating: l l 2 l 1 (4) Path Length Difference vs. Angle: l is given approximately in terms of observation angle θ measured from central axis: l = d sin θ if d L where L =distance from slits or sources to observation screen, d =spacing of adjacent sources, slits or lines in double-slit, multi-slit or diffraction grating. (5) Constructive Interference Condition ( intensity maxima, principal maxima, bright fringes) for two-source, double-slit, multi-slit or diffraction grating: l = mλ or d sin θ = mλ (if d L); with m = 0, ±1, ±2,... where m is the order of the (principal) maximum. (6) Destructive Interference Condition 1 ( intensity minima, dark fringes) for twosource or double-slit experiment: l = ( m + 1 2 ) ( 1) ( 1) 1 λ or d sin θ = m + λ (if d L); with m + = ± 2 2 2, ±3 2,... 1

(7) Destructive Interference Condition 2 ( intensity minima, dark fringes) for singleslit diffration with W =slit width and W L: W sin θ = mλ with m = ±1, ±2,... (but m 0). (8) Destructive Interference Condition 3 for first intensity minimum of circular aperture diffration with W =aperture diameter and W L: W sin θ = 1.22λ (for 1st circular intensity minimum). Charge, Force, Electric Field, Flux, Gauss s Law (1) Coulomb s Law for force F F on two point charges Q 1 and Q 2 at distance r: with k = 8.99 10 9 Nm 2 /C 2. F = k Q 1 Q 2 r 2 (2) Definition of Electric Field E in terms of electric force F exerted on test charge q (with E generated by other charges): E F q ; hence F = q E. Note E-field is independent of the test charge q used to detect the electric force F : E depends only on the other charges which generate E. (3) Electric Field E E by Point Charge Q, at an observation point P with distance r from Q: E = k Q r 2 with E pointing radially away from positive charge, Q > 0; or radially towards negative charge, Q < 0. (4) Electric Field E E by Uniform Surface Charge Density σ Q/A on a single planar surface of area A with total charge Q, in close proximity to the surface: E = σ 2ɛ o = Q 2ɛ o A with E = const (uniform E field) where ɛ o 1/(4πk) = 8.85 10 12 C 2 /Nm 2 ; and E normal to the surface, pointing away from positively charged (Q > 0) or towards negatively charged (Q < 0) surface. (5) Electric field E E in Planar Capacitor: between closely spaced, parallel, planar plates of charges Q and Q, and opposing surface areas A, without dielectric (κ = 1): E = Q ɛ o A with E = const (uniform E field) 2

with E normal to the plate surfaces, pointing from positive towards negative plate. (6) Superposition Principle of Electric Field: If an electric field E is being generated by multiple charged objects (Q 1, Q 2,...), then E at any observation point P is the vector sum (resultant vector) of the electric field contributions E 1, E 2,... that would be generated by each of the charged objects in isolation at that point P : E = E 1 + E 2 +... (7) Electric Flux Φ of constant E-field through planar surface of area A with the surface normal at angle θ from E, with 0 o θ 180 o : Φ = EA cos θ (8) Gauss s Law, relating total electric flux Φ(S) of the E-field through a closed surface S (with outward-directed surface normal) to the total charge Q(S) enclosed inside S: Φ(S) = 1 ɛ o Q(S) Electric Potential, Potential Energy, Capacitance, Electric Energy Storage (1) Definition of Electric Potential V and electric potential difference V (also known as voltage drop ), in terms of potential energy U and potential energy difference U, respectively, for a test charge q moving or being moved through E-field: V = U/q, V = U/q ; hence U = qv, U = q V. Note that V or V is a property of the E-field, depends only on the E-field, and is therefore independent of the test charge q. (2) Electric Potential Difference in a Uniform Electric Field ( E = const): V = E s cos θ where V V B V A is the electric potential difference between points B and A; the vector s points from A to B with length s s ; and θ is the angle between s and E with 0 o θ 180 o. Hence, V < 0 when moving from A to B in direction of E (θ < 90 o ); and V > 0 when moving from A to B against direction of E (θ > 90 o ). (3) Electric Potential for Point Charge Electric Field (with E = k Q /r 2 ): V = k Q r where r is the distance from point charge Q to observation point. 3

(4) Superposition Principle of Electric Potential: If an electric field E is being generated by multiple charged objects (Q 1, Q 2,...), then its electric potential V at any observation point P is the scalar sum (sum of numbers) of the electric potential contributions V 1, V 2,... that would be generated by each of the charged objects in isolation at that point P ; and likewise for the electric potential difference V V B V A between any points A and B: V = V 1 + V 2 +... or V = V 1 + V 2 +... (5) Definition of Capacitance: For two oppositely charged metallic objects a and b, with Q stored on a and Q stored on b, their electric potential difference V V b V a is proportional to the charge Q. The capacitance of the two metallic objects is then defined as: C Q V, hence Q = CV or V = Q C (6) Voltage and Capacitance of a Planar Capacitor: For two oppositely charged, parallel planar metallic plates, each of opposing surface area A, closely spaced with distance d, the voltage V and capacitance C are V = Ed = Q d κɛ o A ; C Q V = κɛ A o d κ C o where κ is the dielectric constant of the dielectric (insulating) material between the plates and κ = 1 for vacuum or air, and C o ɛ o A/d is the capacitance without dielectric. (7) Electric Field Energy Storage in a Capacitor: The energy U E required to build up a charge Q and a voltage V = Q/C in a capacitor is stored as electric field energy between the capacitor plates and it is given by U E = 1 2C Q2 = 1 2 CV 2 Electric Field Energy Density. Energy per volume, u E, stored in an electric field is given in terms of the field strength E u E = ɛ o 2 E2 Mechanics Memories: Velocity, Acceleration, Force, Energy, Power (1) Velocity (2) Acceleration v = r t a = v t r if constant; else v = lim t 0 t v if constant; else a = lim t 0 t 4

(3) Constant-Acceleration Linear Motion: for r r f r i and v v f v i r = 1 2 ( v i + v f ) t ; r = v i t + 1 2 a t2 ; v = a t. (4) Constant-Speed Circular Motion: for motion at constant speed v v around a circular trajectory of radius r. The velocity vector v is always tangential to trajectory and perpendicular to acceleration vector a: v a. The acceleration vector a always points towards the center of the circular trajectory. Period T and frequency f of revolution, angular velocity ω, and orbital speed v: T = 1 f = 2πr v = 2π ω ω = 2πf = 2π T = v r v = ωr = 2πfr = 2πr T Circular centripetal acceleration: a = v2 r = ω2 r Orbital angle φ and arc of circumference s covered during time interval t: φ = ω t = v t r (5) Newton s 2nd Law: = s r m a = F s = v t = ω r t = φ r (6) Kinetic Knergy (KE), Work, Work-KE-Theorem: K=kinetic energy of object of mass m moving at speed v; W =work done by force F on an object moving/moved with displacement r, with r pointing at an angle θ from F and 0 o θ 180 o ; K = K f K i = change of kinetic energy due to work done by total force F : K = 1 2 m v2, W = F r cos θ, K = W. (7) Energy Conservation Law for K K f K i and U U f U i : K i + U i = K f + U f or K + U = 0 (8) Mechanical Power: P =rate of work done by force F on an object moving at speed v, with v pointing at an angle θ from F and 0 o θ 180 o : P = F v cos θ. 5

Algebra and Trigonometry az 2 + bz + c = 0 z = b ± b 2 4ac 2a sin θ = opp hyp, adj cos θ = hyp, opp tan θ = adj = sin θ cos θ For very small angles θ (with θ 90 o ): sin 2 θ + cos 2 θ = 1 sin θ = tan θ = θ (in radians) Numerical Data Acceleration of gravity (on Earth): g = 9.81m/s 2 Speed of light in vacuum: c = 3.00 10 8 m/s Coulomb s constant: k = 8.99 10 9 Nm 2 /C 2 Biot-Savart s constant: k m µo 4π = 1 10 7 Tm/A (exact) Permittivity of vacuum: ɛ o 1/(4πk) = 8.85 10 12 C 2 /Nm 2 Permeability of Vacuum: µ o 4πk m = 4π 10 7 Tm/A (exact) Elementary charge: e = 1.60 10 19 C Electron mass: Proton mass: m e = 9.11 10 31 kg m p = 1.67 10 27 kg Other numerical inputs will be provided with each problem statement. SI numerical prefixes: y = yocto =10 24, z = zepto =10 21, a = atto =10 18, f = femto =10 15, p = pico =10 12, n = nano =10 9, µ= micro =10 6, m = milli =10 3, c = centi =10 2, d = deci =10 1, da = deca =10 +1, h = hecto =10 +2, k = kilo =10 +3, M = Mega =10 +6, G = Giga =10 +9, T = Tera =10 +12, P = Peta =10 +15, E = Exa =10 +18, Z = Zetta =10 +21, Y = Yotta =10 +24. 6

Addendum to Formula Sheet Exam #2 for PHYS1212 and PHYS1252 Point Charge Forces and Electric Fields in Vectorial Form (1) Coulomb s Law for vector of force, F, exerted by point charge Q on point charge q: F = k q Q r 2 ˆr, where r = r, ˆr = 1 r r, and k = 8.99 10 9 Nm 2 /C 2. The r is the distance vector drawn from Q to q, r is the distance from Q to q, and ˆr is the unit vector pointing from Q to q. Vector Superposition Principle for Electric Forces. The net vector of force, F, exerted by multiple point charges Q 1, Q 2,... on a point charge q is the vector sum of forces F 1, F 2,..., exerted on q by Q 1 only, by Q 2 only,..., respectively: F = F 1 + F 2 +... = k q Q 1 r 2 1 ˆr 1 + k q Q 2 r 2 2 ˆr 2 +... where r 1 = r 1, r 2 = r 2,..., ˆr 1 = 1 r 1, ˆr 2 = 1 2,... r 1 r 2 r and k = 8.99 10 9 Nm 2 /C 2. The r 1, r 2,... are the distance vectors drawn from Q 1 to q, from Q 2 to q,...; r 1, r 2,... are the distances from Q 1, Q 2,... to q; and ˆr 1, ˆr 2,... are the unit vectors pointing from Q 1, Q 2,... to q, respectively. (2) Electric Field Vector, E, Generated by Point Charge Q, at observation point P : E = k Q r 2 ˆr, where r = r, ˆr = 1 r r, and k = 8.99 10 9 Nm 2 /C 2. The r is the distance vector drawn from Q to observation point P ; r is the distance from Q to P, and ˆr is the unit vector pointing from Q to P. Vector Superposition Principle for Electric Fields. The net electric field vector, E, generated by multiple point charges Q 1, Q 2,... at observation point P is the vector sum of fields E 1, E 2,..., generated at P by Q 1 only, by Q 2 only,..., respectively: E = E 1 + E 2 +... = k Q 1 r 2 1 ˆr 1 + k Q 2 r 2 2 ˆr 2 +... where r 1 = r 1, r 2 = r 2,..., ˆr 1 = 1 r 1, ˆr 2 = 1 2,... r 1 r 2 r and k = 8.99 10 9 Nm 2 /C 2. The r 1, r 2,... are the distance vectors drawn from Q 1 to P, from Q 2 to P,...; r 1, r 2,... are the distances from Q 1, Q 2,... to P ; and ˆr 1, ˆr 2,... are the unit vectors pointing from Q 1, Q 2,... to P, respectively. (3) Force Exerted on Test Charge q by Electric Field E, generated by other charges, Q 1, Q 2,...: F = q E 7