## 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: $frac{partial^2y(x,t)}{partial x^2}=frac1{v^2}frac{partial^2y(x,t)}{partial t^2}$

## 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 $omega'=sqrt{frac1{LC}-frac{R^2}{4L^2}}$.

## Oscillation of a Mass-Spring System Compared with Electrical Oscillation in an L-C Circuit

### Mass-Spring System

• Kinetic energy = $frac12mv_x^2$
• Potential energy = $frac12kx^2$
• $frac12mv_x^2+frac12kx^2=frac12kA^2$
• $v_x = pmsqrt{frac km}sqrt{A^2-x^2}$
• $v_x=frac{dx}{dt}$
• $omega=sqrt{frac km}$
• $x=Acos(omega t+phi)$

### Inductor-Capacitor Circuit

• Magnetic energy = $frac12Li^2$
• Electrical energy = $frac{q^2}{2C}$
• $frac12Li^2+frac{q^2}{2C}=frac{Q}{2C}$
• $i = pmsqrt{frac1{LC}}sqrt{Q^2-q^2}$
• $i=frac{dq}{dt}$
• $omega=sqrt{frac1{LC}}$
• $q=Qcos(omega t+phi)$

## 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).