51
Concept
Prerequisites:
"Study Physics Guide" for Final ("Laundry List" of important concepts) Todd Sauke (important concepts in bold; vectors also shown in bold) Symbol or Equation
Physics quantities are typ. either scalars or vectors (magnitude & direction) components of vectors add From mechanics, total external force on a body = mass x acceleration Fext = m a (SI newton, "N") Mass (SI kilogram, "kg") resists change in motion (via "momentum", p) p = mv, Fext = dp/dt A mass moving in a circle undergoes centripetal acceleration acentr = v2 / r Conservation of linear momentum: Isolated system (Fext = 0) p=0 ; pf=pi m1vf1 + m2vf2 = m1vi1 + m2vi2 A moving mass has energy of motion, "Kinetic Energy" (SI joule, "J") KE = m v2 (a scalar) A spring being compressed pushes back proportional to compression F=-kx A compressed spring has energy of compression, elastic "Potential Energy" U = k x2 For conservative forces, mechanical energy is conserved E = KE+PE = constant (Wnc =0) Electromagnetics: Electric Charge is the fundamental quantity in Electrostatics Q (SI coulomb, "C") Charge is conserved, quantized, and comes in "positive" and "negative" e = 1.602 x 10-19 C Like charges repel (radially); opposite charges attract; Coulomb's Law F = (1/4 0) q1 q2 / r2 The constant 0 is numerically related (by definition) to the speed of light, c 0 = 107 / (4 c2) = 8.854 x 10-12 All "normal" matter is made up of protons, neutrons and electrons mp = 1.67 x 10-27 kg Protons have +e charge; electrons have e. Their mutual attraction holds me = 9.11 x 10-31 kg everything together. In a conductor, electrons are free to move around. Total force (vector) is the vector sum of individual forces (superposition) F = Fi The Electric field vector is the force per unit charge on a "test charge", q0 E = F0 / q0 F=qE For distributions of charge (eg. , ), vector integrate over the distribution Field lines provide a graphical representation of E (and B) fields An Electric Dipole is a separation of equal magnitude, opposite charges An Electric Dipole, p, in an Electric field, E, experiences a torque An Electric Dipole oriented in an Electric field has potential energy, U Electric Flux; "flow" of E through a surface. (dA is a vector to surface) E = dE = dq/(4 0 r2)
E strong where lines are dense
p = qd (d=separation - +) =pxE = p E sin() U = -p E = -p E cos()
Gauss's Law expresses the fact that the source of (static) flux is charge Charge on a conductor at rest resides on its surface. Also for conductor Einside = 0 (for static case) Use Gauss's Law to determine E field for symmetric charge distributions eg. E = / 20 (for sheet) Gauss's Law easily shows E from a line of charge (instead of nasty integral) E = / (2 0 r) A symmetric distribution will be easier to solve for E using Gauss's Law eg.E=/0 ('tween two cond. plates) Electric force from a static charge distribution is a conservative force Work done on "test charge" is path-independent change in potential energy U=qV U=q0/(40)qi/ri Electric Potential is potential energy per unit charge (SI volt, "V") V=U/q V=1/(40)qi/ri We always speak of "potential difference" (the zero is chosen for convenience) The reverse of this is that E field is the (minus) gradient of the potential Equipotential surfaces are everywhere perpendicular to the E field lines A capacitor (any pair of separated conductors) holds charge per volt Capacitance depends ONLY on geometry (& what's between conductors) When capacitors are connected in parallel, the equivalent capacitance is: When in series, capacitors have an equivalent capacitance given by: It takes work (energy) to charge a capacitor. W = potential energy, U Energy (U) stored in a capacitor "resides" in the electric field If the insulation separating capacitor conductors is dielectric, not vacuum: VaVb=Edl E = - Grad ( V )
= E dA (through surface) = E dA = Qencl / 0
C = Q / Vab (SI farad, "F") C = 0 A / d (parallel plates) Ceq = C1 + C2 + C3 + ... 1 1 1 1 /Ceq= /C1 + /C2 + /C3+... 2 2 U= Q /C = CV = QV u = 0 E2 (u = U-density) just replace 0 with !
Current is the amount of charge flowing through an area per unit time I = dQ/dt (SI ampere, "A") In a conductor (non-static case now!), free charges (typ. electrons) can I = n q vd A (A = area) move (with drift velocity, vd) in response to an Electric field (above: n = charge density) The current per unit area is called current density (a vector!) J = n q vd Even conductors have resistance to the flow of charge (material dependent) = E / J ( = "resistivity") The resistivity of a material is temperature dependent, typ. increasing w/ T (T) = 0 [1 + ( T T0 ) ] A source of electromotive force (emf or E) makes current flow in a circuit E is provider of voltage, V For materials obeying Ohm's Law, current is proportional to voltage V=IR The ratio V/I is called resistance and is related to a material's resistivity R = L / A (SI ohm, "") Current flowing through a resistor is accompanied by a voltage drop label R w/ '+' & '-' in diagrams! An "ideal" source of emf supplies a perfectly constant voltage to the circuit A real source of emf (eg. a battery) has internal resistance, r (voltage drop) Vab = E I r A circuit element with potential difference, Vab, across it and current, I, P=VabI=I2R=V2/R (SI watt, "W") (a resistor always takes energy out) flowing through it is a source or sink of power depending on sign of I When resistors are connected in series, the equivalent resistance is: Req = R1 + R2 + R3 + . . . 1 When in parallel, resistors have an equivalent resistance given by: /Req= 1/R1+1/R2+1/R3+ . . . Series resistors all have the same current; when in parallel, the same voltage Kirchoff's junction rule (based on conservation of charge) I =0 Kirchoff's loop rule (based on conservation of energy) V =0 Use Kirchoff's rules to generate equations ("n equations in n unknowns") An "ideal" ammeter has zero resistance and measures the current through it An "ideal" voltmeter has rin & measures voltage across probed points iin = 0 (admits no current) When a circuit ("RC circuit") has a capacitor being charged or discharged q =CE(1-e-t/RC)= Qf(1-e-t/RC) by a series resistor, the current and charge are not constant. Kirchoff's i = I0 e-t/RC (for charging) loop-rule equation for the circuit results in a differential equation, q = Q0 e-t/RC the solution of which involves decaying exponentials: i = I0 e-t/RC (for discharging) The product of R & C has units of time and is called the "time constant" = RC , (in e-t/) Magnetic interactions are interactions between moving charged particles Magnetic interactions are described by the vector field, B (SI tesla, "T") F = qv x B (right-hand-rule) The magnetic force is always perpendicular to v (and B): no work done Just like for E, field lines provide a graphical representation of B fields B field lines form closed loops Magnetic Flux; "flow" of B through a surface (again, just like with E) mag = B dA Gauss's law for magnetism: there are no "magnetic monopoles" mag=BdA=0 (closed surface) In a uniform B field: charged particle goes in circle (or spiral) of radius, R R = mv / (qB) ; = qB / m Crossed E and B fields: velocity selector (when Fnet(v) = 0, no deflection) vselected = E / B For a current carrying wire in a magnetic field, there is a force on the wire F = dF = I dl x B A current loop of area A & current I in a uniform magnetic field B has Fnet = 0 = n A (for n loops) But it does experience a torque, , in terms of magnetic moment, . = x B = B sin() The work done by the torque can be described as a potential energy U = - B = - B cos() A moving charge q with velocity v creates a magnetic field B that depends on distance as 1/r2, & is perpendicular to both v and the unit vector B = (0 / 4) qv x / r2 from the "source point" (at q) to the "field point" (at B). 0 = 4 x 10-7 1 The constant 0 is defined so that, together with 0, they relate to c c2 = / (0 0) The total B from several moving charges is the vector sum of the fields produced by the individual charges (superposition) B=dB=(0/4) dqv x / r2 The Biot-Savart law gives the previous relations in terms current of in wire B=dB=(0/4) I dl x /r2
From this law, the B field from a long, straight current carrying wire is: and the right-hand-rule gives the direction that B curls around the wire The force per length between two long, parallel current carrying wires is attractive if currents are in the same direction, repulsive if opposite B at distance x along axis of conducting loop (N turns, radius a, current I) Ampere's Law relates the line integral of B around any closed path to the net current through any area bounded (encircled) by the path
We apply Ampere's law to a highly symmetric situation where we can choose the
integration loop (through field point P) to have constant B aligned with the path (or )
B= 0 I / (2 r) (at distance r) F / L = 0 I I' / (2 r) Bx=0 N I a2 / 2 (x2 + a2)3/2
B
Bdl = 0 Ienc (closed loop)
Ienc includes displacement current Choose integration path through P to make the integral easy
The B field inside of a long solenoid with n turns per unit length is: B inside a toroidal solenoid (N turns) at distance r from symmetry axis: B field outside the space enclosed by a tightly wound solenoid is near 0
B = 0 n I (near the center) B = 0 N I / (2 r) B 0 (outside solenoid)
B field inside a long cylindrical conductor of radius R is easy using Ampere's Law B = 0 I r /(2 R2) inside cylinder When magnetic materials are present, there is an effect on the B field just replace 0 with = Km 0!
Changing B flux (B) through a closed loop induces emf (Faraday's Law) E = - dB /dt The minus sign is Lenz's Law; induced current or E opposes the change (lest the universe explode) A "search coil" with N loops of area A, and resistance R, can be used to measure B fields Qtransferred = B A N / R Motional emf is caused by a conductor moving in a magnetic field E = (v x B) dl
B B
Associated with the induced emf is an induced E field (non-conservative) A time-varying electric field generates a "displacement current", id The fundamental relations between electric and magnetic fields have been presented, but are collected together again here to show the unity and symmetry of the relations. They are called Maxwell's Equations and together form a complete basis for all of electromagnetism. For reference, full force equation is repeated here (completes all of E&M) Changing currents in circuits with shared magnetic flux induce emf in one another. Mutual inductance M (SI henry, "H") is geometry dependent. A changing current, i, in any circuit causes a self-induced emf E The self inductance, L (SI henry, "H") is enhanced by number N of loops An inductor is a circuit device intended to have substantial inductance The "" sign (again Lenz's Law) means that an inductor opposes current change An inductor (L) carrying current I has energy associated with the B field If what's inside an inductor is not vacuum, replace 0 with (permeability) When a circuit ("RL circuit") has an inductor connected to an emf and a series resistor, the current is not constant. Kirchoff's loop-rule equation for the circuit gives a diff. eq., with decaying exponential solutions. When a circuit ("LC circuit") has an inductor (L) and capacitor (C), it has electrical oscillations analogous to a mechanical harmonic oscillator L is analogous to mass m; 1/C is like the spring constant k; q is like x; i like v dq dx Many other analogies hold; L i (= L /dt) is like m /dt = mv (momentum) and
E = E dl = - dB /dt
B
id = dE /dt
E dA = Qencl / 0 B dA = 0 Bdl =0 (Ic + 0 dE/dt)enc E dl = - dB /dt
B
F = q (E + v x B ) E2= -Mdi1/dt; E1= -Mdi2/dt eg. M = N2B2/i1 = N1B1/i2 E = -L di/dt L = N B / i
B
U = L I2 is like KE= m v2
F=ma (=m
dv
U = L I2 u = B2/0 (u = U-density) i =E/R(1-e-tR/L)= If(1- e-tR/L) i = I0 e-tR/L (with no emf) = L/R , (in e-t/) i = I cos(t + ) = (1/LC) ( in radians / s)
/dt) is like V=L /dt
di
again lest the universe explode
it requires an electromotive force (voltage) to change it (like force changing p)
For the LC circuit, the energy exchanges back and forth (in the E, B fields) label L and C with '+' & '-' -tR/(2L) When you add resistance R ("LRC circuit") the oscillations (of ') decay in time i = I e cos('t + ) 4L 1 2 2 2 For small enough R, circuit is "underdamped"; for R = /C "critical damping", '=0 ' = ( /LC R /(4L ))
An ac source produces an emf that varies sinusoidally with time. We can Irms = Imax / 2 represent an sinusoidal voltage or current with a phasor vector rotating Vrms = Vmax / 2 counterclockwise with ang. freq. . Instantaneous quantity is x projection i = I cos(applied t) In general, the (sinusoidal) voltage can be "out of phase" with the current v = V cos(applied t + ) The voltage v across a resistor is "in phase" with the i (power is dissipated) VR = I R The v across an inductor "leads" the current i by 90 (no power dissipated) VL = I XL The v across a capacitor "lags" the current i by 90 (no power dissipated) VC = I XC 1 XL and XC are the inductive and capacitive "reactance", respectively XL = L ; XC = / C The reactance is frequency dependent, making high-pass, low-pass circuits In a general ac circuit, the v and i amplitudes are related by "impedance" V= IZ Z= (R2+[XLXC]2) and the phase angle is seen from the phasor diagram to be given by: tan() = (L - 1 / C) / R The average power delivered to an ac circuit depends on the amplitudes Pave = V I cos() of the v and i and on the phase angle (like a dot product of phasors) or Pave = Vrms Irms cos() In the RLC circuit, the i is maximum (Z minimum) at "resonance" o = (1/LC) At the resonant frequency, i and v are "in phase", total reactance = 0 and: Z = R (at resonance) A transformer transforms the voltage and current levels in an ac circuit. The ratio of Vin (V1) to Vout (V2) is = to the turns-ratio (ideal transformer) V2 / V1 = N2 / N1 The current ratio is opposite, so that [power in] = [power out] V1 I1 = V2 I2 Maxwell's equations predict the existence of electromagnetic waves that 1 propagate in vacuum at the speed of light c. That's what light is. c = /(0 0) = 3.00 x 108 m/s E and B are uniform over a plane perpendicular to propagation (plane wave) E = c B Electromagnetic waves are transverse w/ "in phase" sinusoidal field E(x,t ) = Emax cos(kx-t) oscillations. E, B are to each other and the propagation direction. B(x,t ) = Bmax cos(kx-t) "k" is the wave number of the propagating electromagnetic wave k = 2/ Emax = c Bmax
There are two possible polarizations of light for x-propagation (E in y or E in z)
For an electromagnetic wave traveling in a dielectric (eg. glass) v < c The period T, wavelength , and frequency f are related to wave speed v: The energy density of a traveling electromagnetic wave is in E, in B Poynting vector S is energy flow rate (power per unit area) for EM wave The magnitude of the time-averaged value of S is called intensity I of wave Electromagnetic waves also carry momentum, exerting radiation pressure The radiation pressure prad of full intensity sunlight is small but measurable The flow rate of electromagnetic momentum (p) is related to S and c If a perfectly reflecting surface is placed at x=0, the incident and reflected waves add up to form a "standing wave" with nodal planes of E & B Standing E-M waves have stationary sinusoidal oscillations
There is an E-node at the mirror: There can be no tangential E on a conducting surface
v2 = / ( ) = c2 / K Km v = /T = f utot = uE + uB = 2 uE = 0E2 1 S = /0 E x B
I=Sav=EmaxBmax/(20)=Emax2/(20c)
B
1
f = /T
1
prad = I / c ( absorbed) prad = 2 I / c ( reflected) 1 /A dp/dt = S / c = EB /(0c) E-nodes at kx = n B-nodes at kx = (n + )
Ey(x,t)= -2Emaxsin(kx) sin(t) Bz(x,t)= -2Bmaxcos(kx) cos(t)
The E & B variations for a standing E-M wave have E & B "out of phase" Because the E and B are "out of phase", there is no net flow of energy If there is another mirror parallel to the first one, there must be an E-node there too. This optical cavity is the basis of the laser, and can only support the special wavelengths of light that have nodes at both ends. The different colors of light correspond to different wavelengths Visible light is but a small part of the spectrum of electromagnetic waves
version 11-25-2008(5)
I = Sav = 0 (standing wave)
=c/f =2c/ xray<uv<vis<ir<wav<tv