GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.
Fall 2014: General physics II
[TOC]
-
static electricity
- 2 opposite charges attract
- charge conservation
- electrostatic induction : charges reorganized
-
current : change of charge
$I=\frac{\p Q}\dt$ - Rowland experiment
- charge is discrete
- Wimshurt generator
- Kelvin's thunderstorm
- point charge : located on a point without spatial extension
-
charge density
- volume
$\rho(x,y,z)=\rho(r,\theta)$ :$Q=\iiint_t \rho\dt$ - surface
$\sigma(x,y)=\sigma(\theta,\phi)$ :$Q=\iint_S \sigma\dS$ - line
$\lambda(x)$ :$Q=\int_L \lambda\d L$
- volume
-
current density : current per unit area normal to the flow,
$\d I =J\dS_\perp$ or$\v j=\frac{I}{A}=\sigma E$ - volume
$J$ :$I=\int_S \v J·\d \vec S$ - surface
$J_s$ :$I=\int_b J_s\d b$
- volume
- Newton's thid law :
$\v F_{12}=-\v F_{21}$ - Coulomb/Cavendish experiement
-
Coulomb's law :
$\v F =\frac{1}{4\pi\ep_0}\frac{Q_1Q_2}{r^2_{12}}\hat r_{12}$ - electric constant (permittivity of free space) :
$\ep_0 \approx 8.8·10^{-12}\quad Fm^{-1}$ $\frac{1}{4\pi\ep_0}\approx 9·10^9 \quad Nm^2C^{-2}$ $\hat r_{12}=\frac{\v r_{12}}{\norm {r_{12}}}$
- electric constant (permittivity of free space) :
-
superposition principle :
$\F=\sum_n F_n$ - mutual potential energy of charges :
$U=W=\int F\dx=\frac{-Q_1Q_2}{4\pi\ep_0r}$
-
electric field :
$\E=\frac{\F}{Q_t}=\lim_{Q_t\to 0}\frac{\F(Q_t)}{Q_t}\quad NC^{-1},Vm^{-1}$ - infinite thin plane :
$\E=\frac{\sigma}{2\ep_0}$ - rod :
$\E=\frac{\lambda}{2\pi\ep_0 x}$ - in case of
$r< r_0$ : take ratio$r/r_0$ as$r$
- infinite thin plane :
-
electric potential :
$V_{AB}=\frac{W_{AB}}{Q_t}=\int -\E·\dL\quad JC^{-1},V$ - single charge as
$V_\infty=0$ :$V=\frac{Q}{\coul}\frac{1}{r}$ - continuous distribution :
$V=\frac{1}{\coul}\int\frac{\d q}{r}$ - charged disk :
$V=\frac{\sigma}{2\pi\ep_0}(\sqrt{a^2+b^2}-a)$
- single charge as
- electrical engery :
$E=qV$ -
charges in E-fields :
$\F=Q\E=m\v a$ - can accelerate particules
$\v v_i || \E$
- can accelerate particules
- Millikan experiment
- electronvolt :
$1eV=1.6·10^{-19}J$ energy of$e^-$ passing$1V$ potential difference - Ion propulsion
-
change of direction :
$\frac{\sin \alpha_1}{\sin \alpha_2}=\frac{v_2}{v_1}$ -
electric dipole :
$+q$ and$-q$ at distance$a$ -
dipole moment :
$\v p = q\v a$ -
potential :
$V_p=\frac{p\cos\theta}{4\pi\ep_0r^2}=\frac{\v p·\hat r}{4\pi\ep_0r^2}$ $E_\theta=-\frac{1}{r}\frac{\p V}{\p\theta}=\frac{p\sin\theta}{\coul r^3}$ $E_r=-\frac{\p V}{\p r}=\frac{2p\cos\theta}{\coul r^3}$
-
in E-field : dipole will align to E-field
- torque :
$\v\tau =\v p\x\E$ - uniform E-field :
$\sum\F=0$ - potenial energy :
$U=-pE\cos\theta=-\v p·\E$ ($U=0$ for$\theta=\pi/2$ )
- torque :
-
dipole moment :
- multipoles
- monopole :
$V\propto\frac{Q}{r}$ and$E\propto\frac{Q}{r^2}$ - dipole
$\v p$ :$V\propto\frac{p}{r^2}$ and$E\propto\frac{p}{r^3}$ as$\sum Q=0$ - quadrupole
$\v q$ :$V\propto\frac{q}{r^3}$ and$E\propto\frac{q}{r^4}$ as$\sum Q=0$ ,$\sum\v p=0$ - octupole :
$V\propto\frac{1}{r^4}$ and$E\propto\frac{1}{r^5}$ as$\sum Q=0$ ,$\sum\v p=0$ ,$\sum\v q=0$
- monopole :
-
flux :
$\Phi=\int_A\E·\d\v A=\int_A\E_\perp\d\v A$ (positive when outwards normal to the area) -
Gauss's law : flux of E-field over any closed surface is equal to enclosed charge
$\oiint\E·\d\v S=\frac{Q}{\ep_0}$ - no E-field inside conductor, then all charge is located on surface of conductor (i.e. hollow conductor)
$\oiint\E·\d\v S=0$ - charge inside hollow conductor is screened
- conducting sphere or shell (charge on surface)
- inside
$r\le r_0$ :$E=0$ so$\frac{\p V}{\p r}=0$ and$V=\frac{Q}{\coul r_0}$ - outside
$r\ge r_0$ :$E=\frac{Q}{\coul r^2}$ and$V=\frac{Q}{\coul r}$
- inside
- thin plane conductor with surface charge :
$\E=\frac{\sigma}{2\ep_0}$
- no E-field inside conductor, then all charge is located on surface of conductor (i.e. hollow conductor)
- Gauss's divergence theorem :
$\nabla·\E=\frac{\rho}{\ep_0}$ as$\iiint_V \text{div }\F·\d V=\oiint_{\d V}\F·\d\vec S$ - solid angle (3D angle) :
$\Omega=\frac{\v S}{r^2}$ as$\d\Omega=\frac{\dS\cos\theta}{r^2}$ - Rutherford's experiment
-
point effect : Corona discharge due to high field at sharp point (locally
$E=\frac{V}{r_0}$ )- for the same potential :
$\sigma_{big}<\sigma_{small}$ and$E_{big}< E_{small}$
- for the same potential :
-
circuital law (only in electrostatic fields) :
$\oint\E·\d\v L=0$ or$\nabla\x\E=0$ -
Poisson's and Laplace's equation :
$\Delta·V=-\frac{\rho}{\ep_0}$ (if no charge$\Delta·V=0$ )- solution only if boundary conditions are specified
-
capacitance : amout of charge a conductor can hold for a given potential
$C=\frac{Q}{V}\quad CV^{-1},F$ - in vacuum : depend only on geometry
- conducting sphere :
$C=\frac{Q}{V}=\coul r_0$ - ideal capacitor :
$C=\frac{Q}{V_A-V_B}$ where$A$ and$B$ are the same charge (but opposite) - spherical capacitor :
$C=\coul\frac{R_aR_b}{R_a-R_b}$ - parallel plate capacitor :
$C=\ep_0\frac{S}{d}$ - cylindrical capacitor :
$C=\frac{2\pi\ep_0 l}{\ln \frac{R_a}{R_b}}$ - wires and transmission lines
- co-axial :
$C=\frac{2\pi\ep_0}{\ln\frac{b}{a}}$ - twin wire :
$C=\frac{\pi\ep_0}{\ln\frac{d}{r_a}}$
- co-axial :
- combinations of capacitors
- series :
$\frac{1}{C}=\sum_i\frac{1}{C_i}$ - parallel :
$C=\sum_iC_i$
- series :
- polarization of matter : shift of electron distribution, resulting in an induced dipole collection
-
dielectrics : no true insulator exist, matter can be polarized
- dipole moment per volume :
$\v p=\v P\d\tau$ - dipole moment at surface :
$\sigma_p=\v P_\perp$ - amout of charge that has moved through a surface :
$Q_{polarized}=-\oiint\v P·\d\v S$ -
off-topic :
$\vec J_p=\frac{\p\v P}{\dt}$ and$\E=\rho\v J$
- dipole moment per volume :
-
electric susceptibility : how easy it is to polarize a material
$\v P=\ep_0\chi_e\E$ -
D-field : displacement field
$\D=\ep_0\E+\P$ -
$\oiint\D·\d\v S=\sum Q_{free}$ or$\nabla·\D=\rho_{free}$ (only conduction charges are source of D-field) - D-field is what you set, E-field is what you get
-
-
relative permittivity :
$\ep_r=1+\chi_e$ (E-field is reduced by factor$\E=\frac{\E_{ext}}{\ep_r}$ ) -
electric energy :
$E=U_E=W=\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}CV^2=\frac{1}{2}QV=\frac{1}{2}\ep_0\int_\tau \ep_rE^2\d\tau=\frac{1}{2}\int_\tau\D·\E\d\tau$ -
capacitor with dielectric
- isolated (
$Q$ constant) :$V_0=\frac{Q_0}{C_0}$ ,$\frac{C}{C_0}=\ep_r$ - connected (
$V$ constant) :$Q=V_0C=Q_0\ep_r$
- isolated (
-
E- and D-field across interfaces(in a capacitor)
- outside dielectric :
$\E=\E_{ext}=\frac{\sigma_c}{\ep_0}$ and$\D=\ep_0\E=\sigma_c$ - inside dielectric :
$\E=\frac{\E_{ext}}{\ep_r}=\frac{\sigma_c}{\ep_0\ep_r}$ and$\D=\ep_0\ep_r\E=\sigma_c$
- outside dielectric :
-
forces between charged conductors
- constant
$Q$ :$\F=-\nabla U_E$ - constant
$V$ :$\F=+\nabla U_E$
- constant
- piezoelectricity : charge accumulated when stressed
- pyroelectricity : voltage when heated/cooled
- ferroelectricity : spontaneous polarization
- AC/DC networks : alternative current vs direct current
-
resistivity
$\rho$ :$\rho=\frac{1}{\sigma}=\frac{RA}{l}\quad \Omega m$ -
conductivity
$\sigma$ :$\v j=\sigma\E$ -
Ohm's law :
$V=RI$ -
resistance :
$R=\frac{\rho l}{A}$ (dependence on material and on temperature) - Drude model
-
Kirchhoff's laws :
$\sum_i I_i=0$ and$\sum_i \Delta V_i=0$ -
energy dissipation (heat) :
$P=IV=RI^2=\frac{V^2}{R}$ - combination of capacitor and resistor
- delta-star transformation
-
capacitor :
$V_C(t)=V_0(1-e^{-t/\tau})$ and$I(t)=C\frac{\d V_C}{\dt}$ -
measurement schemes
- voltage : across contacts (high internal resistance)
- current : between contacts (low internal resistance)
-
wheatstone bridge :
$R_xR_1=R_2R_3$
- comparison to electrostatics : no monopole, not a central force, only based on B-field
-
current element :
$I\d\v l$ - Oersted experiment
- Ampère's experiment
-
forces on currents :
$\F=\oint_LI\d\v l\x\B$ -
torque :
$\v\tau=I\v A\x\B$ -
Biot-Savart law (magnetic field due to current) :
$\d\B=\frac{\mu_0 I\ds\x\hat r}{4\pi r^2}$ $\B=\oint_L\frac{\mu_0I\dl\x\hat r}{4\pi r^2}=\iiint_V\frac{\mu_0\v j\x\hat r\d\tau}{4\pi r^2}$ - rectangular conducteur :
$\B=\frac{\mu_0I}{2r}$ - thick wire :
$\B=\frac{\mu_0 I}{2\pi r}$ - center of coil :
$\B=\frac{\mu_0 I}{2 r}$
- **permeability **of free space :
$\frac{\mu_0}{4\pi}=10^{-7}$ - Helmholtz coils : constant B-field between two coils
$\frac{\d^2 B_x}{\d x^2}=0$ -
Solenoid :
$B_x=\mu_0nI$ where$n$ is the number of windings per unit length -
force between currents :
$F=\frac{\mu_0I_1I_2l}{2\pi r}$ (attract if same direction) -
Lorentz force :
$\F=q\v v\x\B$ -
B-field due to moving charge :
$\B=\frac{\mu_0q\v v\x\hat r}{4\pi r^2}$ - parallel :
$\F=0$ - perpendicular :
$\F\perp\v v$
- parallel :
-
circular motion :
$a=\frac{v^2}{r}$ resulting$r=\frac{mv}{qB}$ - Bubble chamber, magnetic deflector, mass spectrometer
- charged particule in homogenous B-field : particle will travel on spiral path
- charged particule in non-homogenous B-field : magnetic mirror, Van Allen belt
-
combination of B- and E-fields : E-field determines kinectic energy, B-field determines momentum, together determines velocity
- special case
$F=0$ :$V=\frac{E}{B}$
- special case
-
Hall effect : current through conductor in B-field, measure voltage perpendicular to current
$\F=e\v v_d\x\B$ with speed$v_d=\frac{BI}{net}=\frac{R_H BI}{t}$ where$R_H=\frac{1}{ne}$ -
magnetic dipole :
$\v m =I\v A$ -
$\F =0$ (non-uniform field$\F=(\v m·\nabla)\B$ ) -
torque :
$\v T=I\v A\x\B$ - potential energy :
$U=-\v m·\B$ - angular momentum :
$\v L =m_er^2\omega$ ($\omega=\frac{2\pi}{T}$ ) -
circular current :
$I=\frac{e\omega}{2\pi}$ - gyromagnetic ratio :
$\v m=\gamma\v L$ where$\gamma=\frac{e}{2m_e}$ (electron around atom) - Landé g-factor :
$g=\frac{2m_e}{e}\gamma$ for free electron$g\approx 2$ - Larmor procession :
$\v T\perp\v L$ and$\d\v L =\v T\dt$ - frequency :
$\nu_L=\frac{\omega_L}{2\pi}=\frac{-\gamma}{2\pi}B$ with$\omega=\frac{T}{L\sin\theta}$
- frequency :
-
-
magnetic field :
$B_r=\frac{2\mu_0 m\cos\theta}{4\pi r^3}$ and$B_\theta=\frac{\mu_0m\sin\theta}{4\pi r^3}$ -
magnetic flux :
$\Phi=\int_A\B·\d\v A$ - flux cut in time :
$\d\phi=B\d A=Bvl\dt$ (Solenoid$\Phi=B·nA$ ) - potential energy :
$U=-I\Phi$ - find force :
$F_x=-\frac{\p U}{\p x}=I\frac{\p \Phi}{\p x}$ - find torque :
$T_\theta=-\frac{\p U}{\p \theta}=I\frac{\p \Phi}{\p \theta}$
- flux cut in time :
-
Ampère's circuital law :
$\oint_L\B·\d\v L=\mu_0I$ - B-field due to coil :
$\B=\mu_0 n I$
- B-field due to coil :
-
Gauss's law for B-fields :
$\oiint_S \B·\d\v S=0$ -
magnetic vector potential :
$\B=\nabla\x\v A$
- Faraday's experiments : a voltage or electromotance
$V_{emf}=U_{indu}=-\frac{\d\Phi_m}{\dt}$ is induced when- a rigid stationary circuit across which there is a varying magnetic field
- a rigid circuit moving in a B-field such that the magnetic flux through it changes
- a part of a circuit which moves and cuts magnetics flux
-
motional electromotance : moving conductor in B-field
$\Delta V_{emf}=El=Bvl=\E·\d\v l=\frac{\d\Phi}{\dt}$ - Lenz's law : the direction of any magnetic induction effect is such as to oppose the cause of the effect
-
AC-generator (rotating coil in B-field) :
$N$ turns give$\Delta V_{emf}=-\frac{\d\Phi}{\dt}=NAB\omega\sin\omega t$ -
induced electric field :
$V_{emf}=\oint_L\E·\d\v L\not=0$ and$\nabla\x\E=-\frac{\p\B}{\p t}$ (electric field is no longer conservative in presence of changing magnetic field) -
self-inductance :
$L=\frac{\Phi_{tot}}{I}=\mu_0 n^2 l A$ where$\Phi=LI$ and$\Delta V=-L\frac{\d I}{\dt}$ -
mutual inductance :
$\Phi_2=M_{12}I_1$ and$\Phi_1=M_{21}I_2$ with$M$ determined by geometry (in general same$M$ ) thus$\Delta V_{emf2}=-M\frac{\d I_1}\dt$ - outer/inner solenoid :
$M=\mu_0 n_1 n_2 A$
- outer/inner solenoid :
-
coupled circuits in general : coefficient of coupling
$k=\frac{M}{\sqrt{L_1L_2}}$ - ideal transformer :
$\frac{V_2}{V_1}=-\frac{n_2}{n_1}=-\frac{1}{N}$ and$N=\frac{I_2}{I_1}$
- ideal transformer :
- Tesla coil
- B-field in null in absence of external field (except for ferromagnetic)
-
relative permeability :
$\mu_r=\frac{L_m}{L_0}=\frac{\Phi_m}{\Phi_0}=\frac{B_m}{B_0}$ and$\B=\mu_r\B_0$ -
magnetisation :
$\B=\B_0+\B_M$ and$\d\v m=\v M\d\tau$ -
H-field : the H-field is what you set, the B-field what comes out
$\H=\frac{\B}{\mu_0}-\v M$ (only free currents are sources of H-field)- circuital law :
$\oint_L\H·\d\v l=I$ and$\nabla·\H=\v j$ - magnetic susceptibility :
$\v M=\chi_m\H$ ,$\mu_r=1+\chi_m$ and$\B=\mu_0\mu_r\H$ - hollow solenoid
$\H=nI$ ,$\chi_m=0$ (nothing),$\v M =0$ :$\B=\mu_0 n I$- if non empty :
$\B=\mu_0(1+\chi_m)n I$
- if non empty :
- circuital law :
-
magnetic screening :
$\mu_{r2}B_{\para 1}=\mu_{r1}B_{\para 2}\quad B_{\perp 1}=B_{\perp 2}$ -
magnetic response of types of materials
$F_x\approx\frac{1}{2}\mu_0\chi_m\frac{\p H^2}{\p x}$ - diamagnetisc : no resultant magnetic moment in materials (graphite, water, gold, etc.)
- paramagnetic : resultant atomic magnetic moment but not ordered
- ferromagnetic : metallic, large positive
$\chi_m$ , strong dependence on$H$ and history, become parmagnetic above critical temperature- resultant atomic magnetic moments : parallel order resulting in spontaneous magnetisation
- antiferromagnetic : small positive
$\chi_m$ , dependence on$H$ and history with critical temperature - ferrimagnetic : like ferromagnetic, but non-metallic
- at Curie's temperature : ferro/ferri become para
- spin : intrinsic magnetic moment of elements (and other elementary particles)
- Stern-Gerlach experiment
- if electron paired : no resultant magnetic moment
- if there are unpaired electrons not all spin in cancelled : resultant magnetic moment
-
hysteresis cycle in ferromagnets :
$E=W=\oint B\d H=\oint H\d B$
-
Maxwell's model is for slowly varying field in vacuo
- for rapidly varying field :
$\nabla\x\B=\mu_0\v J+\mu_0\ep_0\frac{\p\E}{\p t}$ or$\oint_L\B·\d\v L=\mu_0I_c+\mu_0\ep_0\frac{\d}{\dt}\iint_S\E·\d\v S$
- for rapidly varying field :
- paradox in Ampère's circuital law for B-field (only for charging the capacitor) : added displacement current :
$\oint_L\B·\d\v L=\mu_0(I_c+I_d)$ -
electromagnetic waves in vacuo (without sources
$Q=0$ and$I=0$ )- energy flux :
$\v S=\frac{1}{\mu_0}\E\x\B$ - energy density :
$E=\frac{\ep_0\E^2}{2}+\frac{\B^2}{2\mu_0}$ - quantity of movement :
$\v p=\ep_0\E\x\B=\frac{1}{c^2}\v S$
- energy flux :
-
wave equations :
$c^2=\frac{1}{\ep_0\mu_0}$ -
electromagnetic spectrum :
$E_{emw}=h\nu=\frac{hc}{\lambda}$ where Planck's constant$h=6.626·10^{-34}\quad Js$ with$\nu=f\lambda$ and$E=cB$ -
monochromatic plane wave : Gauss
$E(x,z,t)$ ,$B(x,y,t)$ , faraday$\frac{\p E}{\p x}=-\frac{\p B}{\p t}$ and ampère$\frac{\p B}{\p x}=-\ep_0\mu_0\frac{\p E}{\p t}$ - polarzitation : oscillation direction of E-field
- relative magnitude :
$\E=c\B$ and$\E\perp\B$ in phase
-
EM waves in a non-conducting medium : velocity
$c_m=\frac{c}{\sqrt{\ep_r\mu_r}}$ (non-ferromagnetic : $c_m=\frac{c}{\sqrt{\ep_r}})$) with refractive index$n=\frac{c}{c_m}\approx\sqrt{\ep_r}$ - Cherenkov radiation :
$\ep_r>1$ means$c_m< c$ but particles can have speed larger as$c_m$ -
EM at boundaries of dielectrics :
$\nu_1=\nu_2$ ,$n_1\sin\theta_1=n_2\sin\theta_2$ - critical angle (in the split of materials) :
$\sin\theta_c=\frac{n_2}{n_1}$ - total internal reflection : same angle
- critical angle (in the split of materials) :
-
generation of electromagnetic waves :
$\frac{\p\E}{\p t}\not=0$ ,$\frac{\p\B}{\p t}\not=0$ only if$\frac{\p\v I}{\p t}\not=0$ - if
$Q=0$ ,$I=0$ : only propagation, no generation - if
$Q$ constant,$I=0$ : E-field is steady - if
$Q$ moves at constant speed,$I$ constant : B-field is steady
- if
- synchrotron radiation : changing direction is also an acceleration
-
dipole radiation and antenna : general rule of thumb
$length=\lambda/2$ -
Maxwell's equations
- Gauss's law :
$\nabla·\E=\frac{\rho}{\ep_0}$ or$\oiint_S\E·\d\v S=\frac{q}{\ep_0}$ - Gauss's law for B-field :
$\nabla·\B=0$ or$\oiint_S\B·\d\v S=0$ - Circuital law :
$\nabla\x\E=-\frac{\p\B}{\p t}$ or$\int_L\E·\d\v L=-\frac{\d}{\dt}\iint_S\B·\d\v S$ - Ampère's circuital law :
$\nabla\x\B=\mu_0\v I+\frac{1}{c^2}\frac{\p\E}{\p t}$ or$\oint_L\B·\d\v L=\mu_0 I$
- Gauss's law :
| Electricité | Magnétisme | |
|---|---|---|
| density | charge |
current |
| monopôle | none (due to |
|
| force | Coulomb |
Lorrentz |
| field | ||
| if variates, creates | ||
| potential | ||
| energy | ||
| medium ( |
permittivity |
permeability |
| Gauss's law | ||
| Ampère's circuital law |
|
|
| dipôle |
|
current |
| moment | ||
| torque |
|
|
| medium interface | ||
| modifier | polarization |
magnetisation |
| susceptibility |
|
|
| process | induction |
excitation |
| di/dia | insulator (can be slighlty polarized) |
|
| piezo | polarized under pressure | magnetized under pressure |
| para | polarization under E-field |
|
| ferro/ferri | spontaneous electric polarization |
|
$2\sin^2 t = 1-\cos 2t$ $2\cos^2 x = 1+\cos 2x$ $\sin 2x = 2\sin x \cos x$ $\sin(a+b)=\sin a\cos b+\cos a\sin b$ $\cos(a+b)=\cos a\cos b-\sin a\sin b$ $\tan^2 x +1=\frac{1}{\cos^2 x}$ $\cot^2 x +1=\frac{1}{\sin^2 x}$ $\sin 0 = \frac{1}{2}\sqrt{0} = \cos \pi/2 = 0$ -
$\sin \pi/6 = \frac{1}{2}\sqrt{1} = \cos \pi/3 = \frac{1}{2}$ $\sin \pi/4 =\frac{1}{2}\sqrt{2} = \cos \pi/4 =\frac{\sqrt{2}}{2}$
$\sin \pi/3 = \frac{1}{2}\sqrt{3} = \cos \pi/6=\frac{\sqrt{3}}{2}$ $\sin \pi/2 = \frac{1}{2}\sqrt{4} = \cos 0 = 1$ $\cosh z =\frac{e^z+e^{-z}}{2}$ $\sinh z =\frac{e^z-e^{-z}}{2}$ $\cos z =\frac{e^{iz}+e^{-iz}}{2}=\cosh iz$ $\sin z =\frac{e^{iz}-e^{-iz}}{2i}=\frac{\sinh iz}{i}$ $e^z=\cosh z +\sinh z=\cos z+i\sin z$ $e^{-z}=\cosh z -\sinh z$ $\cosh^2 z -\sinh^2 z = 1$ $\sin mx\cos nx=\frac{1}{2}[\sin (m+n)x + \sin (m-n)x]$ $\cos mx\cos nx=\frac{1}{2}[\cos (m+n)x + \cos (m-n)x]$ $\sin mx\sin nx=\frac{1}{2}[\sin (m-n)x - \cos (m-n)x]$