Fields, sources, forces,

Transcription

1 Phys 208 Summary Fields, sources, forces, etc Applications Materials Math techniques

2 Phys 208 Summary Fields, sources, forces, etc E and charge B and current Fields and forces Charge conservation Potentials and energy Ampere and Faraday Equations of E&M Reference frames Radiation Materials Conductors and insulators Sparks Current and resistance Electric and magnetic properties Atomic models Applications DC circuits AC circuits Motors/generators Transformers Antennas Math techniques Coordinate systems Vector techniques Integral techniques Field lines & flux tubes Solid angle Use of symmetry *Differential form of integral eqns 1 st and 2 nd order differential eqns *Intro to complex numbers

3 Fields, sources, forces Fields, sources, forces, etc E and charge B and current Fields and forces Charge conservation Potentials and energy Ampere and Faraday Equations of E&M Reference frames Radiation

4 Elements of E&M Sources Charges, currents Fields E,B Interactions Fields result in force on sources Sources are support for fields Fields support themselves

5 Eqs of E&M Integral Local ò E ÿ A = Q ÿe = 1 r Gauss e 0 e 0 Source eqs ò Bÿ A = 0 ÿb = 0 '' for magnetism ò E ÿ s = - F B t ò Bÿ s = m0 I + m0 e0 F E t Q t =Ÿ r t V = -ò S j ÿ A äe = B t äb= m0 j + m0 e0 E t r t Faraday Ampere - Maxwell Maxwell Secretly linked Caution: If loop moves, include motional EMF. + ÿ j = 0 Continuity F = qhe+ vâbl Force Linked by relativity

6 Forces Coulomb + Lorentz F = qhe+ vâbl Force on wire F = I l ä B Forces/torques on dipoles Ue = -pÿe Ub = -mÿb Fe= - HpÿEL... Fx = p ÿ E x te = päe tb = m äb Fb= - Hm ÿbl... Fx = m ÿ B x

7 Gauss s Law Flux through a closed surface is proportional to the charge interior to that surface. F =ò E ÿ A=4 pke Q Q=Ÿ n V Q=S qi

8 Gauss s law for B NO magnetic charge FB = ò Bÿ A = 0 ñ B = 0

9 Ampere s law B around loop is proportional to current enclosed does not depend on details of surface ò Bÿ s = m0 Ÿ j ÿ A = m0 I äb = m0 j Don t forget Biot-Savart: B = m 0 4 p I s ä r` r 2

10 Ampere-Maxwell ò Bÿ s = m0 I + m0 e0 F E t = m0 I + m0 ID

11 Faraday E ÿ s = - F B t Also: increasing current in 2 results in increase in B, and EMF in 1.

12 charge conservation Q =Ÿ r t t V = - ò S j ÿ A Current item Analogy for E E r - - Gauss Fund thm calc - Diff. form of Gauss

13 Energy Potential energy of charges E=- f f = -Ÿ E ÿ s Energy stored in devices (capacitor/inductor) Field energy U = C Q2 U = 1 2 LI2 ue = 1 e 0 E 2 ub = 1 B m 0

14 Energy, work, power Joule heating P = I V = I 2 R = V2 R Mechanical work F = - U x B does no work on particles W =Ÿ F ÿ s =Ÿ qhhväbl ÿvl t = 0

15 Reference frames E & B depend on observer they are part of one object observer at rest in B, sees moving bar Lorentz force on e - observer at rest on bar sees B and E = v x B

16 EM waves - radiation 2 B t - 2 m0 e0 2 B z = 0 2 E = Ex coshwt - k zl B = By coshwt - k zl EM wave illustration E = cb, c 2 = 1 m 0 e 0 w = kc w = 2 p n, k = 2 p l and ln = c Energy, momentum, radiation pressure S = 1 m 0 HEäBL = 1 m 0 E0 B0 k` u = 1 Je 0 E B 2 N 2 m 0 = P A t = u = S c

17 Energy conservation conserved quantity charge energy density r u current j S diff cont. eqn int cont eqn r t Q t + ÿ j = 0 u t =-Ÿ j ÿ A U t + ÿs = 0 =-Ÿ S ÿ A

18 Materials Materials Conductors and insulators Sparks Current and resistance Electric and magnetic properties Atomic models

19 Conductors and Insulators Insulators Electron positions are stable Small local currents Polarization Conductors Charges free to move Statics: Constant potential Applied field gives currents

20 Currents j = qe ne vd = se= 1 r E

21 Materials

22 Atomic models

23 Applications Applications DC circuits AC circuits Motors/generators Transformers Antennas

24 Circuit elements ideal wires (R = 0) carry current I with no voltage drop node - current divides voltage common capacitor V = Q/C V - ideal voltage supply (ideal battery) resistors V = IR switch open/close inductors V = -L di/dt AC power

25 Kirchoff s rules Good for AC circuits too 1. Charge Conservation: sum of all currents into a node is zero. 2. Energy conservation: sum of voltage changes around circuit is zero.

26 Simple parallel circuits I = I1 + I2 = Q 1 + Q 2 toy

27 Simple series circuit DV = DV1 + DV2

28 AC circuits R wl 1 wc z =@R 2 +HCL - CCL 2 D 1ê2 =AR 2 +IwL - 1 wc M2 E 1ê2 = RA1 +I L Rw M2 Hw 2 - w 0 2 L 2 E 1ê2 f = ArcTan C L -C C R

29 Generators & Motors

30 Transformers DV1 = - N1 F t DV2 = - N2 F t = N 2 DV1 N 1

31 Math techniques Math techniques Coordinate systems Vector techniques Integral techniques Field lines & flux tubes Solid angle Use of symmetry * Differential form of integral eqns 1 st and 2 nd order differential eqns * Intro to complex numbers

32 Coordinate systems rectangular (x,y,z) cylindrical (r, θ, z) spherical (r, θ, φ)

33 Vector techniques Proficiency with vectors E & B are vector fields Forces are vectors Field lines & flux tubes

34 More Vectors Cross products a ÿb = ab cos q c = aäb= ab sin q x ` z = x` ä ỳ HaäbL 1 = a2 b3 - a3 b2 HaäbL 2 = a3 b1 - a1 b3 HaäbL 3 = a1 b2 - a2 b1 Intro to vector calculus*(supplemental) Gradient Divergence Curl E = - F ÿe = r e 0 âe = - B t

35 Superposition Discrete sums Integral distributions

36 Use of symmetry Use symmetry to reduce dimension of integrals. 2D and 3D problems to 1D problems C = Qê V l = Qê l E = 2 ke lê r V = 2 ke l lnhbê al C = Qê V = Q 2 ke Qêl lnhbêal = 2 pe 0 l lnhbêal 1D to algebra

37 Differential equations RC & RL circuits LC & LRC circuits L + C = 0 = L I t + Q C fl L 2 Q t 2 + Q = 0 C Q = Q0 CosHwt + fl, w = è!!!!!!! 1 LC

38 Wave equation

39 Using complex numbers for AC circuits*(supplemental) complex numbers, z = x + i y e iθ = cos(θ) + i sin(θ) z = r e iθ multiplication, division, e iπ = -1 real functions power impedance of simple components the RC filter