E & M - Basic Physical Concepts
Current and resistance Current: I = = ddtQ = n q vd A Ohm’s law: V = I R, R , E = = ρ J V I E = = ` , J = A , R = ρ` A
Electric force and electric field Electric force between 2 point charges: |q 1 | |q 2 |
|F | F | = k = k r 2 k = 8.987551787 × 109 N m2 /C2 8 .854187817 × 10−12 C2 /N m2 ≤0 = 4 π1 k = 8. q p = −q e = 1.60217 60217733 733 (49) × 10−19 C m p = 1.67262 6726233 (10) × 10−27 kg me = 9.1093 1093897 897 (54) × 10−31 kg
2 2 Power: P = I V = V R = I R ∆ρ Thermal Thermal coefficient of ρ ρ : α = ρ ∆T 0 Motion of free electrons in an ideal conductor:
=
~ ~ = Electric field: E = F q
|Q| ~ = ~ + E ~ + · Point charge: |E | = k = k r 2 , E = E 1 2 + · · · Field patterns: point charge, dipole, k plates, rod,
spheres, cylinders,. cylinders,. . . Charge distributions:
Linear charge density: λ = Area charge density: σA =
∆Q ∆x ∆Q ∆A
∆Qsurf
Surface charge density: σsurf = Volume charge density: ρ =
∆A
∆Q ∆V
m a τ = v d → qmE τ = nJ q → ρ = n q 2 τ V I R Direct current circuits Series: V = I R eq = I R 1 + I R 2 + I R 3 + · + · · ·, I = = I i V V V V Parallel: I = = R = R + R + R + · · ·, V = V i 1 2 3 eq Kirchho ff ’s ’s Rules Steps: in application of Kirchho –Label currents: i1 , i2 , i3 , . . . P P i in = i out –Node equations: P P iR)=0” –Loop equations: “ (±E ) + (∓iR)=0” entering − terminal –Natural: “+” for loop-arrow entering “−” for loop-arrow-parallel to current flow dy RC circuit: if dt + R1C y = 0, y = y = y0 exp(− exp(− RtC )
Electric flux and Gauss’ law ~ · ˆ Flux: ∆Φ = E ∆A⊥ = E · ˆ n∆A Q Gauss law: Outgoing Flux from S, ΦS = enclosed ≤0 Steps: to obtain electric field ~ pattern –Inspect E pattern and construct S H ~ · ~ = Qencl , solve for E ~ –Find Φs = surface E · dA ≤ 0
Spherical: Φs = 4 π r2 E Cylindrical: Φs = 2 π r ` E = E ∆A, 1 side side;; = 2 E ∆A, 2 sides Pill box: Φs = E k ⊥ = σsurf ~ Conductor: E in = 0, E surf = 0, E surf ≤ 0
E − V c − R i = 0, 1c ddtq + R ddti = ci + R ddti = 0 Charging: E − q d q Discharge: 0 = V c − R i = c + R dt , ci + R ddti = 0 Magnetic field and magnetic force µ0 = 4 π × 10−7 T m/A µ i
Wire: B = 2 π0 r
Axis of loop: B =
µ0 a 2 i 2 (a2 +x2 )3/2
~ = i ~ ~ → q ~ ~ Magnetic force: F v×B `×B M ~ × B ~ , Loop-magnet ID: ~ µ = i = i A n ˆ τ τ = i A ~ 2 m v 1 = 2πr Circular motion: F = r = q v B , T = f v ~ ~ ~ Lorentz force: F = q E + + q ~ v×B F d d M M ~ Hall eff ect: ect: V H = q , U = −µ ~ · B
Potential Sources of B and magnetism of matter − 19 µ0 q ~ q ~ v ׈ r Potential Potential energy: ∆U = q ∆V 1 eV ≈ 1.6 × 10 J Biot-Savart Law: ∆B r `׈ ~ = µ0 i ∆~ 4 π r2 , B = 4 π r 2 Positive charge moves from high V V to low V low V µ0 i ∆y a , ∆y = r 2 ∆θ ∆B = θ θ sin , s i n = kQ 2 r a π 4 r Point charge: V = r V = V 1 + V 2 = . . . H ~ Ampere’s law: M = B · d~ s = µ = µ 0 I encircled k q 1 q 2 L Energy of a charge-pair: U = r12 Steps: to obtain magnetic field Potential diff erence: erence: |∆V | V | = | = |E E ∆sk |, ~ pattern and construct loop L –Inspect B R B ~ ~ ∆V = −E · · ∆~ s, V B − V A s ~ . –Find M –Find M and I encl , and solve for B Ø = − A E · · d~ ∆V Ø d V ∂ V V d (E A ) d QA E = = − dr , E x = − ∆x Ø = − ∂ x , etc. Displ. current: I = ≤ d ΦE = ≤ = ~
fix y,z
d
0
dt
0
dt
dt
Magnetism in atom:
Capacitances Q C V Q Q Q Q Series: V = C = C + C + C + · · ·, Q = Q = Qi 1 2 3 eq eq Parallel: Q = C = C eq V = C 1 V + C 2 V + · · ·, V = V i eq V =
≤ A Q Parallel plate-capacitor: C = = V = EQd = 0d 2 R Q 1 2 Energy: U = 0 V dq = = 21 Q C , u = 2 ≤ 0 E 2 1 2 Dielectrics: C = = κ C 0 , U κ = 21κ Q C 0 , uκ = 2 ≤ 0 κ E κ Q Q Spherical capacitor: V = 4 π ≤ r − 4 π ≤ r 1 2 0 0
~ Potential Potential energy: U = − p ~ · E
= i A = 2 em L Orbital motion: µ = i
L = m = m v r = n = n h ¯ , ¯h = 2hπ = 1. 1 .06 × 10−34 J s
h ¯ = 9. µorbit = n µB , µB = 2em 9 .27 × 10−24 J/T ¯, µ = h = µ B Spin: S = spin = µ 2
Magnetism in matter: 0 B = B = B0 + BM = (1 + χ) B0 = (1 + χ) µ0 B µ0 = κ m H Ferromagnetic: χ ¿ 1 Diamagnetic: −1 − 1 ø χ < 0 < 0 C Paramagnetic: 0 < χ ø 1, M = T B
Complete reflection: P = 2cU ,
Faraday’s law d φB ~ · dA ~ , E = −N dt , φB = B
R
~ M ~ = F E q
~ d~ E s, ~ opposes change of Φ Lenz law: Induced B B d (B A⊥ ) d φB d A d B ⊥ = = dt A⊥ + B dt dt dt d 1 R · R θ Moving rods: ddtA = ` v , ddtA = dt 2 d A d ⊥ Rotating loop: = dt ( A cos ω t) dt Cutting B lines → change φ B → E ind → E ind Maxwell equations: ~ · dA ~ = Q , ~ · dA ~ = 0 , E B ≤
R · E =
≥
H H ·
0
~ d~ E s =
−
d φB , dt
¥
H H ·
φ ~ d~ B s = µ 0 [I + ≤0 ddtE ]
Inductance i1 , M = M = N 2 φ21 Mutual: E 2 = −M 21 ddt 21 12 i1 φ N d i d i Self: E = −L dt , L = i , V L = L dt A Long solenoid: L = N B i , B = µ 0 n i 1 L i2 , u = 1 B 2 Energies: U L = 2 B 2 µ0 1 2 U C = 2 C q , uE = 21 ≤ 0 E 2
q L C: V L + V C = 0 ⇒ L ddti = − C q = q 0 cos(ω t + δ ),
ω =
q
1
L C ,
U C + U L = U C max = U L max = U 0
dy = −a y, y = y 0 exp(−a t) dt d V L L R: E = V L + R i, + R LV L = 0, dt E Rt V L = E exp − Rt , i = R 1 − exp L L
Decay Equations:
LRC:
≥ ¥
h
R
Q ≈ Q0 e− 2 L t cos ωd t,
ωd
≥ − ¥i r ≥ ¥ = 1
L C
−
R 2L
2
Underdamped, critically damped & overdamped
A C Circuits Impedance: [Ohm ≡ Ω] Z ≡ R2 + (X L − X C )2
q
1
Inductive X L = ω L, Capactive X C = ω C 1 T f (t) dt Mean value: f ¯(t) = T 0
R
1
1
2 2 [sin ω t]rms = [sin 2 ω t] = [ 12 (1 − cos2 ω t)] = 1 2
√
Electromagnetic waves
P = 2cS
Reflection and Refraction λ2 v2 1 Index of refraction: n n2 = v1 = λ1 Snell’s law: n1 sin θ1 = n 2 sin θ2 Critical angle: n2 > n1 , n2 sin θc = n 1 sin 90◦ Total reflection: θ > θ c
Mirrors and lenses 1 1 1 p + q = f Ray tracing rules: Mirror: At symm pt S , reflected symmetrically through
center of sphere, undeflected. Parallel to axis, converges toward F (or diverges away from F ), f = R 2 . Lens: Through center of lens, undeflected. Parallel to axis, converges toward F (or diverges away from F ) Image: q > 0 (real), q < 0 (virtual) Focal point F : at p = ∞, q = f f = ±| f |, “+” convergent, “ −” divergent 0 q Magnification: M = hh = − p n 2 n2 n1 1 Refraction at spherical surface: n p + q = R R is coordinate of center with origin at S , with
−
S the symmetry point of surface on the axis 1 = n2 − 1 1 1 Lens maker: f n1 R −R
≥
¥≥
1
2
0 q n1 Two media: M = hh = − p n2
¥
Huygen’s principles:
Points in wave front are sources of next wavelets Forward tangent surface is next wave front Interference Maxima φ = 0, 2 π , 4 π, · · ·; Minima φ = π , 3 π , 5 π, · · · φ Double slits: I average = I 0 cos2 2 , φ = k ∆ .
≥¥
y sin θ = ∆ , tan θ = L , d
for small θ , θ ≈ sin θ ≈ tan θ ~ = A ~ + A ~ + A ~ + · · · Phasor diagram: A 1 2 3 Ax = A 1x + A2x + A3x +· · ·, Ay = A 1y + A2y +· · · a
b
c
sin α = sin β = sin γ π First minimum for N slits: φ = 2N
Thin film: φ = k ∆ + |φ1reflected − φ2reflected |,
Properties of em waves:
E = E m cos(k z − ω t), B = E c λ , n = c v = ddtz = ω f = λ = v k T speed of light: c = √ ≤1 µ = 2 .99792458 × 108 m/s 0
0
~ ⊥ E ~ , propagating along: E ~ × B ~ B u = u E + uB , uE = u B ~ ×B ~ ~ = E ¯ = I ¯ = E rms Brms Poynting vector: S , S µ0 µ0 ∆U d z P Intensity: I = A = A ∆z dt = u c ~ · dA ~ = d U + P S Energy conservation: R dt U Complete absorption: Momentum p = c ∆ p 1 ∆U 1 S u Pressure: P = F A = ∆t A = c ∆t A = = c
R
=2t φreflected = π (denser medium); =0 (lighter medium) Diff raction Single slit: I = I 0
2
∑ ∏ sin β2 β
∆
, β = k ∆,
∆
= a sin θ
2
λ Resolution criterion: θ criterion = 1.22 D Grating: Principle maxima ∆ = m λ
Polarization Brewster ( n1 < n2 ): n1 sin θbr = n 2 sin( π 2 − θbr ) Polarizer: E transmit = E 0 cos θ , I = I 0 cos2 θ I = I 0 Unpolarized light: ∆ 2π ∆θ Transmitted Intensity: ∆I 0 = ∆I cos2 θ 2π I 0 = 2I 0π 0 cos2 θ dθ = I 20
R
