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电磁·光学
24
engines
Lorentz force · F = qvB·sinθ
Lorentz force · F = qvB·sinθ
F=qvB·sinθ · 右手定则 · F⊥v 且 ⊥B · 90°最大
The magnetic force on a moving charge — perpendicular to both v and B
Solenoid field · B = μ₀ n I
Solenoid field · B = μ₀ n I
B=μ₀nI · 内部均匀 · 磁感线密而平行
The uniform magnetic field inside a long solenoid
Cyclotron motion · r = mv/(qB)
Cyclotron motion · r = mv/(qB)
r=mv/qB · f=qB/2πm(与v无关) · 圆周轨道
Circular orbit in a uniform magnetic field — frequency independent of speed
Capacitor energy · E = ½CV²
Capacitor energy · E = ½CV²
E=½CV²=½QV · Q=CV · 能量∝V²
Energy stored in a parallel-plate capacitor grows with the square of the voltage
LC oscillation · f = 1/(2π√(LC))
LC oscillation · f = 1/(2π√(LC))
f=1/2π√(LC) · 电场↔磁场能量交换 · 谐振
Energy sloshes between the capacitor's E field and the inductor's B field; the total stays constant
Electromagnetic spectrum · c = λf, E = hf
Electromagnetic spectrum · c = λf, E = hf
c=λf · E=hf · 无线电→伽马 · 可见光彩虹带
One family of waves from radio to gamma; wavelength, frequency and photon energy are locked together
衍射光栅
Diffraction grating
d·sinθ=mλ · 多级 · 色散成谱 · 最高级次
Bright maxima at d·sinθ = mλ · a grating disperses white light into spectra
Thin-film interference
Thin-film interference
2nt=(m+½)λ 亮 · 肥皂泡/油膜色 · 增透膜
Soap bubbles, oil slicks and anti-reflection coatings — colour from 2nt = (m+½)λ
Total internal reflection
Total internal reflection
临界角 sinθc=n₂/n₁ · 玻璃→空气 41.8° · 光纤
Snell, the critical angle θc = arcsin(n₂/n₁), and how optical fibres trap light
Malus's law · polarizers
Malus's law · polarizers
I=I₀cos²θ · 45°→½ · 90°正交消光 · 三偏振片
Transmitted intensity through a polarizer at angle θ: I = I₀·cos²θ
Brewster's angle · total polarization on reflection
Brewster's angle · total polarization on reflection
tanθB=n₂/n₁ · 反射光全偏振 · 空气→玻璃 56.3°
tan θ_B = n₂/n₁ — at θ_B the reflected ray is fully polarized and ⟂ to the refracted ray
Rayleigh scattering · why the sky is blue
Rayleigh scattering · why the sky is blue
强度∝1/λ⁴ · 蓝/红≈4.35× · 天空蓝/日落红
Scattering ∝ 1/λ⁴ — blue scatters ~4.35× more than red, so daytime sky is blue and sunsets red
Gauss's law & flux
Gauss's law & flux
Φ=Q_enc/ε0 · 对称求场 · 点电荷/无限线/无限面
Φ = ∮E·dA = Q_enc / ε₀ · symmetry pins down the field E
RC charging & discharging
RC charging & discharging
V(t)=V0(1−e^(−t/RC)) · τ=RC · 63% · 充/放
V(t) = V₀(1 − e^(−t/RC)) · I(t) = (V₀/R)e^(−t/RC) · τ = RC
Dielectric polarization
Dielectric polarization
κ介电常数 · C=κC0 · 束缚电荷 · E=E0/κ
C = κC₀ · E = E₀/κ · bound charge = (1 − 1/κ)·Q
Biot–Savart law · the magnetic field of a current
Biot–Savart law · the magnetic field of a current
dB=(μ0/4π)I dl×r̂/r² · 直线μ0I/(2πr) · 环心μ0I/(2R)
dB = (μ0/4π)·(I dl × r̂)/r² — closed forms for a straight wire and a loop
Ampère's law · ∮ B·dl = μ0 I_enclosed
Ampère's law · ∮ B·dl = μ0 I_enclosed
∮B·dl=μ0 I_enc · 导线/螺线管B=μ0nI · 环形
One Amperian loop gives B for wires, solenoids and toroids — exactly
Magnetic force on a current-carrying wire
Magnetic force on a current-carrying wire
F=BIL sinθ · 平行导线 μ0 I1 I2/(2πd) · 同向吸引
F = B I L sinθ, and F/L = μ0 I1 I2/(2π d) between two parallel wires
Lenz's law · induced current opposes flux change
Lenz's law · induced current opposes flux change
感应电流反抗磁通变化 · 能量守恒 · Faraday负号
EMF = −dΦ/dt — the induced current fights the change that made it
Motional EMF · a rod sliding on rails
Motional EMF · a rod sliding on rails
棒切割磁感线 EMF=BLv · I=BLv/R · 阻碍力B²L²v/R
EMF = BLv → I = BLv/R → retarding force F = B²L²v/R
Displacement current · Ampère–Maxwell law
Displacement current · Ampère–Maxwell law
I_d=ε0 dΦ_E/dt · 安培-麦克斯韦 · 电容板间B
I_d = ε₀ dΦ_E/dt equals the wire's conduction current I_c = C dV/dt
Poynting vector
Poynting vector
S=(1/μ0)E×B · 能流 · I=½ε0 c E0² · 传播方向
S = (1/μ₀) E×B · the energy flux of an electromagnetic wave
Magnetic dipole torque
Magnetic dipole torque
τ=m×B=mB sinθ · U=−m·B · 线圈/罗盘对齐
τ = m×B = mB sinθ · U = −m·B = −mB cosθ
Hall effect
Hall effect
V_H=IB/(nqt) · 载流子偏转 · 符号揭示电子/空穴
V_H = I B / (n q t) · the sign of V_H reveals the carrier sign