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Unformatted text preview: Quick overview of course topics + look at what’s on formula sheet Dag Gillberg [J] [J]==[N [N m] m]dA [F] [F]==[C/V] [C/V]dA [Hz] [Hz]==[s[s 11]] [C] [V] [⌦] [C] = = [A [A s]s] [V] = = [J/C] [J/C] [⌦] = = [V/A] [V/A] = dxdy (cartesian) Shell Theorem: A shell of uniform charge 22 22 [Wb] = [T m [H] = [T m /A] [T] = [N/(A m)] [Wb] = [T m [H] = [T m /A] [T] = [N/(A = rdrd✓ eV (polar) 19 Stuff inattracts red is not explained JJ eV = = 1.602 1.602⇥ ⇥10 10 19 1) or repels an external charge as a circle: s10= [m] 99 2⇡R 33 GG -- giga M k kilo = 10 giga == 10 M -- mega mega = = 10 1066 k kilo = 10 shell’s charge were at its centre;1212 on the formula sheet 2 2 66 33 99 m -- milli 10 µµ -- micro = nn -- nano pp -- pico mChapters milli ==A 10= ⇡R21-23 micro[m = 10 10 nano = = 10 10 pico = = 10 2) exerts no net electrostatic force on a 2 2 sphere: A = 4⇡R [m ] interior. Mathematical Conservation Mathematical Formulae Formulae and Identities Identities Conservation of of E Emech in isolated isolated system system (Wext 4 3 and 3 mech in ext = 0): ⇡R [m re: (1(1++x)x)nn ==V = 22 3+n(n 1 + nx + 1)x E 00 1 ||q2 | 1 + nx n(n 1)x /2!+ Emech = K K+ + U U= =k|q mech = • Coulomb’s law 33 /2!+ Coulomb’s Law: F = [N] RR ff 2 r ~ ~ n(n n(n 1)(n 1)(n 2)x 2)x /3! /3!+ +pp... ... (if (if |x| |x| < < 1) 1) W ss [J] Wfforce = ii F F ·· d~ d~ [J] orce = ~ ~ tionax (uncorrelated errors) b± bb 4ac 4ac b± Electric Field:theorem: E = FK /q==0W ax22++•bx bx+ + c = 0 has roots x = c = 0 has roots x = Work-KE theorem: K Wnet Work-KE net[N/C or V/m 2a 2a Charge is quantized and conserved q @f ~~ ~ sin(↵±±2 ))= =sin sin↵↵cos cos ± ±cos cos↵↵sin sin sin(↵ ~ = |E ~ Power: P = dW/dt = F · ~ v [W] Power: P = dW/dt = F · ~ v [W] point charge: | E| = k and E + ( @y ) + ... 2 y r to move a mass cos(↵•±± ))= =cos cos↵↵cos cos ⌥ ⌥sin sin↵↵sin sin cos(↵ ! A force is conservative if the work ! A force is conservative if the work Conductors vs non-conductors, ground ~ = /(2 1 issheet: |E| Di↵erential length: ds = = dx dx (cartesian) (cartesian) Di↵erential length: ds or non-conducting charge between between two two points points is path path independent. or charge cz + ... ds = = rd✓ rd✓ (polar) (polar) ds ~ = /(✏0 ) 2 2 2 • Electric Field and Potential Electric Field and Potential Electric fields conducting 1 sheet: | E| ) +Di↵erential (b ) + (c ) + ... y Di↵erential area: z dA = = dxdy dxdy (cartesian) (cartesian) area: dA Shell Theorem: Theorem: A A shell shell of of uniform uniform charge: charge: Shell dA = rdrd✓ (polar) Electric dipole: dA = rdrd✓ (polar) . 1) attracts attracts or or repels repels an an external external charge charge as if all of the 1) • Due to charged particle 
 Circumference of a circle: s = 2⇡R [m] Circumference of a circle: s = 2⇡R [m] m y 2 p z 2 twocharge opposite shell’s charge were at at charges its centre; centre; (±q) separated shell’s were its x 2 22 22 ) Area + ( ofofyaa(point ) + ( charge) ) + ... Area circle: A = ⇡R [m circle: A = ⇡R [m z 2) exerts exerts no no net net electrostatic electrostatic force force on on a charge in its 2) dipole moment: |~ p | = qd directed fro Surface area area of of aa sphere: sphere: A A= = 4⇡R 4⇡R22 [m22]] Surface [m interior. interior. ⇡R33 [m33]] Volume sphere: = 4343⇡R [m Volume aa sphere: VV = • ofof Dipoles ~ ~ k|q ||q ||q || k|q ~ ⌧ = p ~ ⇥ E U = p ~ · E uations Coulomb’s Law: F = [N] Coulomb’s Law: F = Hrr [N] qenc ~~ = ~~/q Error Propagation (uncorrelated (uncorrelated errors) Error Propagation errors) dv ~ ~ Electric Field: E E == F /q00 E [N/C or= Electric Field: F [N/C or V/m] q Gauss’ Law: · d A cceleration: a = = constant •q @f Charged objects: point, sheet,
 e ✏0 @f @f dt qq @f 2 2 2 2 ~ ~ ~ ~ ~ point charge: | E| = k and E = = ( ) + ( ) + ... point charge: | E| = k and E = | E|ˆ r = ( ) + ( ) + ... ff @x xx @y yy rr @x @y vcz = vsphere, Electric Potential:1 = |U/q wire, disk, f+ i + at shell, cylinder ~~ = non-conducting 1V sheet: |E| E| =0 /(2✏0[V] non-conducting sheet: 0) ax+ + by + + ... IfIf ff ==ax by + cz ... p p 2 22 ~~ = 2 2 2 22 + v = v +(c(c2a s...... Uconducting is the electrostatic potential energ conducting 1 sheet: sheet: ||E| E| = /(✏ /(✏00) 1 = (a (a xx)) f+ +(b (b yy))i + + ff = zz)) + • Electric m pp flux and Gauss’1law m Electric dipole: dipole: Electric Axnnyq yq ... IfIf ff ==Ax zz ... sf =mmsi + viptp + 2 at2 potential di↵erence: V = by Vfd; Vi = two opposite charges (±q) separated two opposite charges (±q) separated nn 2 2 2 2 2 2 2 =Using +(( yvy )) + +(( zz )) + +... ... ff (( xx ))symmetries! + ff = q • dipole charge: moment: |~ |~ == qd directed directed from q to +q dipole moment: ppV || = qd acceleration: |~a| = r point k r ~ ~ ~ = p~p~ ⇥ ⇥E E U @V = p~p~ · E ~⌧~⌧ = U = Kinematics Equations Equations Kinematics ~ H H E from V : E = s q dv dv ~ ~ ~~ = Gauss’ Law: = E · A = [email protected] Gauss’ Law: ee = E · ddA Uniform linear linear acceleration: acceleration: = dtdt = = constant constant aa = rgyUniform ✏✏ q1 q2 System 2 charges: U12 [V] = V1 q2 = k = vvii + +at at vvff2= Electricof Potential: = U/q U/q [V] Electric Potential: VV = 1 22 11 22 22 22 xx yy zz enc enc 00 If f = ax + pby + cz + ... (a x )2 + (b y )2 + (c f = m p If f = Axn yq z ... f =f ( nxx )2 + ( m y 2 y ) Chapters 24-25 Kinematics Equations z) 2 + ... + ( pz z )2 + ... a = dv dt = constant vf = vi + at 2 2 v = v + 2a s i f Electric potential sf = si + vi t + 12 at2 v2 Uniform circular acceleration: |~ a | = • Equipotential surfaces r Uniform linear acceleration: • • • Work and Energy Electric potential energy Spring-block: F = k s, Usp = 12 k( s)2 > 0 ) energy transferred to object by force • WFrom system of charged particles K = K f Ki K = 12 mv 2 Ug = mgh Ignoring dissipative energy losses: Calculating Esys = K + U = Wext or Ef = Ei + Wext U = Uf Uifield = Wffrom [J] • Electric orce potential
 If external agent does work against force: U = Wext • ~ = /(2✏0 ) non-conducting 1 sheet: |E| ~ = /(✏0 ) conducting 1 sheet: |E| Electric dipole: two opposite charges (±q) separated by d; dipole moment: |~ p| = qd directed from q to +q ~ ~ ~⌧ = p~ ⇥ E U = p~ · E H ~ · dA ~ = qenc Gauss’ Law: e = E ✏0 Electric Potential: V = U/q0 [V] U is the electrostatic potential energy Rf ~ · d~s potential di↵erence: V = Vf Vi = E i point charge: V = k rq ~ from V : Es = @V E @s System of 2 charges: U12 = V1 q2 = k q1rq2 System of several charges: U = U12 + U13 + U23 + ... Capacitance: C = q/ VC [F] ✏0 A parallel-plate capacitor: C = d P Potential from electric field in series: 1/Ceq = P i (1/Ci ) in parallel: Ceq = i Ci Capacitance UC = 12 C( VC )2 uE = 12 ✏0 E 2 • Depend only on geometry! Capacitor with dielectrics: ✏0 ! ✏ = ✏0 dielectric constant: = ✏/✏0 • Ceq in parallel and series Current: i = dq/dt [A] • Energy stored in capacitor
 Current density: J~ [A/m2 ] (stored in the electric field of capacitor) ~ = i/A in direction of E ~ |J| J = ne evd • Capacitor with dielectric ne is # of conduction electrons/m3 vd is drift speed of electrons Resistance: R = VP [⌦] R /i Chapters 26-27 • • Current & current density • Resistance and resistivity • Ohm’s law & power Circuits • Kirchhoff’s laws • loop = voltage • junction = current • The RC circuit • charging • discharging • time constant Capacitance: C = q/ VC [F] ✏0 A parallel-plate capacitor: C = d P in series: 1/Ceq = P i (1/Ci ) in parallel: Ceq = i Ci UC = 12 C( VC )2 uE = 12 ✏0 E 2 Capacitor with dielectrics: ✏0 ! ✏ = ✏0 dielectric constant: = ✏/✏0 Current: i = dq/dt [A] Current density: J~ [A/m2 ] ~ = i/A in direction of E ~ |J| J = ne evd ne is # of conduction electrons/m3 vd is drift speed of electrons Resistance: R = VP [⌦] R /i in series: Req = i RP i in parallel: 1/Req = i (1/Ri ) Ohm’s Law ) R independent of VR R = ⇢L/A resistivity: ⇢ [⌦m] conductivity: = ⇢ 1 [⌦ 1 m 1 ] current density: J = E Power: P = iV (general case) [W] if Ohm’s law holds: P = i2 R = V 2 /R emf: E = dW [V] dQ Kirchho↵ ’s Rules:P voltage (loop): Vi = 0 i P current (junction): i ii = 0 RC circuit: ⌧ = RC [s] discharging: q(t) = qmax e t/⌧ charging: q(t) = qmax (1 e t/⌧ ) qmax = CE Magnetic Field: Chapters 28-29 • • Magnetic fields • Force on moving charge • The Hall effect • Charged particle moves
 in circle! What radius? • Force on current carrying wire • Torque on current loop Magnetic field due to currents • Magnetic flux • Ampere’s law • Force between wires • Solenoid & toroid if Ohm’s law holds: P = i R = V /R emf: E = dW [V] dQ Kirchho↵ ’s Rules:P voltage (loop): Vi = 0 i P current (junction): i ii = 0 RC circuit: ⌧ = RC [s] discharging: q(t) = qmax e t/⌧ charging: q(t) = qmax (1 e t/⌧ ) qmax = CE Magnetic Field: ~ = µ0 i long straight wire: B 2⇡ d (direction tangent to circle, RHR) solenoid: Bsolenoid = µ0 ni where n = N/l ~ = µ0 id~s2⇥ˆr Biot-Savart Law: dB 4⇡ r H ~ · d~s = µ0 ienc + µ0 ✏0 d e Ampere’s Law: B dt ~ Force from B: ~ moving charge: F~on q = q~v ⇥ B ~ wire current: F~wire = i~l ⇥ B li1 i2 2 parallel wires: Fk wires = µ02⇡d (k attract, anti-k repel) Magnetic dipoles: loop’s magnetic dipole moment: µ ~ = N iA (direction from RHR of I) ~ loop = µ0 2~µ3 magnetic field on axis: B 4⇡ z ~ torque on current loop: ~⌧ = µ ~ ⇥B ~ potential energy: U = µ ~ ·B Magnetic flux through loop: R ~ ~ [Wb] m = N loop B · dA Chapters 30-31 • Faraday’s and Lenz’ laws • • Capacitance: C = q/ VC [F] parallel-plate capacitor: C = ✏0dA P in series: 1/Ceq = P i (1/Ci ) in parallel: Ceq = i Ci UC = 12 C( VC )2 uE = 12 ✏0 E 2 Capacitor with dielectrics: ✏0 ! ✏ = ✏0 dielectric constant: = ✏/✏0 Current: i = dq/dt [A] Current density: J~ [A/m2 ] ~ = i/A in direction of E ~ |J| J = ne evd ne is # of conduction electrons/m3 vd is drift speed of electrons Resistance: R = VP [⌦] R /i in series: Req = i RP i in parallel: 1/Req = i (1/Ri ) Ohm’s Law ) R independent of VR R = ⇢L/A resistivity: ⇢ [⌦m] conductivity: = ⇢ 1 [⌦ 1 m 1 ] current density: J = E Inductor and inductance • RL circuit • Energy stored in magnetic field EM oscillations and AC • LC circuit • Damped oscillations (RLC) • Forced oscillation Power: P = iV (general case) [W] if Ohm’s law holds: P = i2 R = V 2 /R emf: E = dW [V] dQ Kirchho↵ ’s Rules:P voltage (loop): Vi = 0 i P current (junction): i ii = 0 RC circuit: ⌧ = RC [s] discharging: q(t) = qmax e t/⌧ charging: q(t) = qmax (1 e t/⌧ ) qmax = CE • Three simple circuits • Power in AC circuits, power factor • Transformers Magnetic Field: ~ = long straight wire: B µ0 i Faraday’s Law: E = d dtm or |E| = | d dtm | direction of induced current such that ~ will oppose the charge in m . induced B Induced electric field: H ~ · d~s = d m E= E dt Inductance: L [H] = [Wb/A] solenoid: L = N i m = µ0 n2 `A di di Ecoil = L| dt | VL = L dt direction from Lenz’s Law Uinductor = 12 Li2 uB = 2µ1 0 B 2 L LR circuit: ⌧ = R [s] rising current: i(t) = imax (1 e t/⌧ ) falling current: i(t) = imax e t/⌧ imax = E/R Oscillatory Motion: x(t) = A cos(!t + 0 ) Frequency: f = 1/T [Hz] period: T = 1/f [s] angular frequency: ! = 2⇡f [rad/s] LC Circuit: q(t) = q0 cos(!t + 0 ) 1 2 Analogy ) q spring-block system (U = 2 kx ) != k m for LC circuits: ! = q 1 LC AC Circuit: p Impedance: Z = R2 + (XL XC )2 capacitive reactance: XC = 1/(!C) [⌦] inductive reactance: XL = !L [⌦] E(t) = Emax sin(!t) i(t) = imax sin(!t ) where imax = Emax /Z Phase: tan p = (XL XC )/R p Irms = imax / p2 Vrms = Vmax / 2 2 Erms = Emax / 2 Pave = Irms R Kirchho↵ ’s Rules:P voltage (loop): i PVi = 0 current (junction): i ii = 0 RC circuit: ⌧ = RC [s] discharging: q(t) = qmax e t/⌧ charging: q(t) = qmax (1 e t/⌧ ) Chapter 32-33 • Magnetic Field: ~ = µ0 i long straight wire: B 2⇡ d (direction tangent to circle, RHR) solenoid: Bsolenoid = µ0 ni where n = N/l ~ = µ0 id~s2⇥ˆr Biot-Savart Law: dB 4⇡ r H ~ Ampere’s Law: B · d~s = µ0 ienc + µ0 ✏0 ddte ~ Force from B: ~ moving charge: F~on q = q~v ⇥ B ~ wire current: F~wire = i~l ⇥ B li1 i2 2 parallel wires: Fk wires = µ02⇡d (k attract, anti-k repel) Magnetic dipoles: loop’s magnetic dipole moment: µ ~ = N iA (direction from RHR of I) ~ loop = µ0 2~µ3 magnetic field on axis: B 4⇡ z ~ torque on current loop: ~⌧ = µ ~ ⇥B ~ potential energy: U = µ ~ ·B Magnetic flux through loop: R ~ · dA ~ [Wb] = N B m loop Maxwell’s equations • • • • qmax = CE Gauss’ law for magnetic fields Induced magnetic fields Displacement current Electromagnetic waves • • • • Impedance: Z = R2 + (XL XC )2 capacitive reactance: XC = 1/(!C) [⌦] inductive reactance: XL = !L [⌦] E(t) = Emax sin(!t) i(t) = imax sin(!t ) where imax = Emax /Z Phase: tan p = (XL XC )/R p Irms = imax / p2 Vrms = Vmax / 2 2 Erms = Emax / 2 Pave = Irms R Travelling waves: D(x, t) = A sin(kx !t + 0 ) v= f k = 2⇡/ ! = vk Electromagnetic waves: E = Emax sin(kx !t) B = Bmax sin(kx !t) p ~ B ~ c = 1/ ✏0 µ0 E = cB E? ~= 1E ~ ⇥B ~ Poynting vector: S [W/m2 ] µ0 2 I = Save = Erms /(cµ0 ) Optics: Index of refraction: n = c/v Snell’s Law: n1 sin ✓1 = n2 sin ✓2 Direction of E and B wave components, and the wave Direction of EM wave Power and intensity & the Poynting vector Point source, isotropic, plane wave ...
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