Mechanical Waves

Waves.  The wave speed v=lambda f, lambda – the wavelength, f – the frequency.

Wave functions and wave dynamics.  The displacement of individual particles in the medium
y(x,t)=Acos[omega(frac xv-t)] (or, more intuitively, y(x,t)=Acos[omega(t-frac xv)])
y(x,t)=Acos2pi(frac xlambda-frac tT)
y(x,t)=Acos(kx-omega t), where k=frac{2pi}lambda and omega=2pi f=vk
Wave function: frac{partial^2y(x,t)}{partial x^2}=frac1{v^2}frac{partial^2y(x,t)}{partial t^2}

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Alternating Current

Voltage, current, and phase angle.  In general, the instantaneous voltage v=Vcos(omega t+phi) between two points in an ac circuit is not in phase with the instantaneous current i=Icosomega t passing through those points.

Resistance and reactance.  The voltage across a resistor is in phase with the current, V_R=IR.  The voltage across an inductor leads the the current by fracpi 2, V_L=IX_L, inductive reactance X_L=omega L.  The voltage across a capacitor lags the the current by fracpi 2, V_C=IX_C, capacitive reactance X_C=frac1{omega C}.

Impedance and the L-R-C series circuit.  In general ac circuit, the voltage and current amplitutes are related by the circuit impedance Z, V=IZ.  In an L-R-C series circuit, Z=sqrt{R^2+(omega L-frac1{omega C})^2}, tanphi=frac{omega L-frac1{omega C}}R.

Power in ac circuits.  The average power input to an ac circuit: P_{av}=frac12VIcosphi=V_{mathrm{rms}}I_{mathrm{rms}}cosphi, where phi is the phase angle of the voltage relative to the current.  The factor cosphi is called the power factor of the circuit.

Resonance angular frequencey.  omega_0=frac1{sqrt{LC}}.

Transformers.  frac{V_2}{V_1}=frac{N_2}{N_1}, V_1I_1=V_2I_2.

Mutual Inductance

Mutual inductance.  When a changing current i_1 in one circuit causes a changing magnetic flux in a second circuit, an emf mathcal{E}_2 is induced in the second circuit.  mathcal{E}_2=-Mfrac{di_1}{dt} and mathcal{E}_1=-Mfrac{di_2}{dt}, M=frac{N_2Phi_{B2}}{i_1}=frac{N_1Phi_{B1}}{i_2} – mutual inductance, N_1 – number of turns of coil of the first circuit, Phi_1 – average magnetic flux through each turn of coil 1.

Self-inductance.  A changing current i in any circuit causes a self-induced emf mathcal{E}=-Lfrac{di}{dt}L=frac{NPhi_B}i – depends on the geometry of the circuit and the material surrounding it.

Magnetic field energy.  An inductor with inductance L carrying current I has energy U associated with the inductor’s magnetic field: U=frac12LI^2.  Magnetic energy density: u=frac{B^2}{2mu}.

R-L circuits.  In an R-L circuit the growth and decay of current are exponential with time constant tau=frac LR.

L-C circuits.  An L-C circuit undergoes electrical oscillations with an angular frequency omega=sqrt{frac1{LC}}.

L-R-C circuits.  The frequency of damped oscillations omega'=sqrt{frac1{LC}-frac{R^2}{4L^2}}.

Electromagnetic Induction

  • Faraday’s law.  Induced emf in a closed loop mathcal{E}=-frac{dPhi_B}{dt}, Phi_B – magnetic flux through the loop.
  • Lenz’s law.  An induced current or emf always tends to oppose or cancel out the change that caused it.
  • Motional emf.  mathcal{E}=oint(vec{v}timesvec{B})cdot dvec{l}.
  • Induced electric fields.  When an emf is induced by a changing magnetic flux through a stationary conductor, there is an induced nonconservative electric field vec{E}: ointvec{E}cdot dvec{l}=-frac{dPhi_B}{dt}.
  • Displacement current.  A time-varying electric electric field generates displacement current i_D, which acts as a source of magnetic field in exactly the same way as conduction current: i_D=epsilonfrac{dPhi_E}{dt}.
  • Maxwell’s equations.  The relationships between electric and magnetic fields and their sources:

ointvec{E}cdot dvec{A}=frac{Q_{encl}}{epsilon_0} (Gauss’s law for vec{E} fields)
ointvec{B}cdot dvec{A}=0 (Gauss’s law for vec{B} fields)
ointvec{E}cdot dvec{l}=-frac{dPhi_B}{dt} (Faraday’s law)
ointvec{B}cdot dvec{l}=mu_0(i_C+epsilon_0frac{dPhi_E}{dt})_{encl} (Ampere’s law including displacement current).