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Inglis-Teller limit |
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<div style="position: absolute; top: 4em; right: 5cm;"><font size="20" face="impact"><i>[[User_talk:DJIndica|My talk]]</i></font></div> |
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The linear [[Stark shift]] is given by |
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User:Iridescent |
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:<math>\Delta E_\text{S} = \frac{3}{2}er_nF,</math> |
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<area shape=circle coords="0,0,50"> |
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where e is the [[elementary charge]], radius of the ''n''<sup>th</sup> electron orbit in the [[Bohr model]] of the [[hydrogen atom]] and ''F'' is the component of the local [[electric field]] parallel with the electric dipole. The energy of the ''n''<sup>th</sup> electron orbit is simply the [[Coulomb potential energy]] with ''r'' = ''r<sub>n</sub>'' |
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<AREA |
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HREF="" ALT="Products" TITLE="Products" |
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SHAPE=CIRCLE COORDS="251,143, 47"> |
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:<math>U_n = -\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r_n} = -\left(\dfrac{e^2}{4\pi\varepsilon_0\hbar}\right)^2\dfrac{m_e}{n^2},</math> |
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{|<!--class="infobox" -->cellpadding="0" style="border: 1px solid #aaaaaa; font-size: 90%; float: right;" |
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|[[Image:Solenoid.svg|center|100px]] |
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!style="background:#dfdfdf; text-align: center;"|[[Electromagnetism]] |
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|style="text-align: center;"|[[Electricity]] · [[Magnetism]] |
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|- |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Electrostatics]]</div> |
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<div class="NavContent" style="{{#if: {{{estat|}}}| | display: none;}}"><!--if estat=true, then expand NavContent (default), else collapse NavContent (display: none;)--> |
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{|align=center |
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|- |
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|[[Electric charge]] |
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|- |
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|[[Coulomb's law]] |
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|- |
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|[[Electric field]] |
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|- |
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|[[Gauss's law]] |
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|- |
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|[[Electric potential]] |
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|- |
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|[[Electrical dipole moment|Electric dipole moment]] |
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|} |
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</div></div></div> |
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|- |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Magnetostatics]]</div> |
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<div class="NavContent" style="{{#if: {{{mstat|}}}| | display: none;}}"> |
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{|align=center |
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|- |
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|[[Ampère's circuital law]] |
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|- |
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|[[Magnetic field]] |
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|- |
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|[[Magnetic flux]] |
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|- |
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|[[Biot-Savart law]] |
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|[[Magnetic moment|Magnetic dipole moment]] |
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|} |
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</div></div></div> |
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|- |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Classical electromagnetism|Electrodynamics]]</div> |
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<div class="NavContent" style="{{#if: {{{edyn|}}}| | display: none;}}"> |
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{|align=center |
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|- |
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|[[Electric current]] |
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|- |
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|[[Lorentz force|Lorentz force law]] |
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|- |
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|[[Electromotive force]] |
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|- |
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|(EM) [[Electromagnetic induction]] |
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|- |
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|[[Faraday's law of induction|Faraday-Lenz law]] |
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|- |
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|[[Displacement current]] |
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|- |
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|[[Maxwell's equations]] |
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|- |
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|(EMF) [[Electromagnetic field]] |
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|- |
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|(EM) [[Electromagnetic radiation]] |
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|- |
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|} |
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</div></div></div> |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Electrical_network | Electrical Network]]</div> |
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<div class="NavContent" style="{{#if: {{{enet|}}}| | display: none;}}"> |
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{|align=center |
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|- |
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|[[Electrical conduction|Conduction]] |
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|- |
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|[[Electrical resistance|Resistance]] |
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|[[Capacitance]] |
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|- |
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|[[Inductance]] |
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|- |
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|[[Electrical impedance|Impedance]] |
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|- |
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|[[Resonator|Resonant cavities]] |
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|[[Waveguide_%28electromagnetism%29|Waveguides]] |
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|} |
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</div></div></div> |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Theory of relativity| Tensors in Relativity]]</div> |
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<div class="NavContent" style="width: 188px;{{#if: {{{rel|}}}| | display: none;}}"><!--Increased width to accommodate "em stress-energy tensor"--> |
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{|align=center |
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|- |
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|[[Electromagnetic tensor]] |
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|[[Electromagnetic stress-energy tensor]] |
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|} |
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</div></div></div> |
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|- |
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|style="text-align: center;"|{{Tnavbar|Electromagnetism3}} |
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|} |
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giving an energy level spacing |
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:<math>\Delta U_n = \left(\dfrac{e^2}{4\pi\varepsilon_0\hbar}\right)^2\dfrac{2m_e}{n^2}.</math> |
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The Inglis-Teller limit is defined as the field strength at which the Stark states with |(''n''<sub>1</sub> - ''n''<sub>2</sub>)| = n are shifted by half of this spacing, Δ''E''<sub>S</sub> = Δ''U''<sub>''n''</sub>/2, |
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:<math>\rightarrow F = </math> |
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{| border="3" |
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|+ The table's caption |
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! Column heading 1 !! Column heading 2 !! Column heading 3 |
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! Row heading 1 |
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| Cell 2 || Cell 3 |
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! Row heading A |
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|Cell B |
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|Cell C |
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{| class="collapsible collapsed" style="text-align:left; border:ridge; margin-top:0em; background-color:thistle;width:194px;" |
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! title1 |
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| style="border:solid 1px silver; padding:0px; background-color:white;" | |
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<div class="boilerplate metadata afc" style="background-color:#10A0B2; margin:2em 0 0 0; padding:0 10px 0 10px; border:1px solid #AAAAAA;> |
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{| border="0" |
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|num1 |
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RABI FREQUENCY |
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The '''Rabi frequency''' is an important quantity in [[First quantization|semi-classical]] [[atomic physics]]. For a given atomic transition in a given light field, the Rabi frequency represents the coupling strength between the (classical) light and the (quantum) transition. [[Rabi flopping]] between the levels of a 2-level system illuminated with resonant light, will occur at the Rabi frequency. |
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[[Image:Sine cosine plot.svg|thumb|right|290px|Rabi flopping in a 2-level system [[Media:Sine cosine plot.svg|Actual size]]]] |
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== Definition == |
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:<math> \chi_{ij} = {\vec{d}_{ij}.\vec{E}_0 \over \hbar}</math> |
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where |
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:<math>\vec{d}_{i,j}</math> is the [[transition dipole moment]] for the <math>i \rightarrow j</math> transition. |
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:<math>\vec{E}_0 = \hat{\epsilon}E_0</math> is the [[vector]] [[electric field]] [[amplitude]] which includes the [[polarization]]. |
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The numerator has dimensions of energy, dividing by <math>\hbar</math> gives an [[angular frequency]]. As <math>\vec{d}_{ij} = \vec{d}_{ji}^*</math>, we have <math>\chi_{ij} = \chi_{ji}^*</math>. |
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By analogy with a [[Classical physics|classical]] [[dipole]], it is clear that an atom with a large dipole moment will be more susceptible to perturbation by [[Electric field|electric]] and [[Magnetic field|magnetic]] fields. The [[dot product]] includes a factor of <math>\cos\theta</math>, where <math>\theta</math> is the angle between the polarization of the light and the transition dipole moment. When they are parallel or antiparallel the interaction is strongest, when they are perpendicular there is no interaction at all. The vector electric field amplitude defines both the intensity and the polarization of the light. |
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== Generalized Rabi frequency == |
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For light that is off resonance with a transition, it is common to define the ''generalized Rabi frequency'' <math>\Omega_{i,j}</math>. [[Rabi flopping]] actually occurs at the generalized Rabi frequency. |
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:<math>\Omega_{i,j} = \sqrt{|\chi_{i,j}|^2 + \Delta^2}</math> |
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where <math>\Delta = \omega_{light} - \omega_{transition}</math> is the ''detuning'', a measure of how far the light is off resonance with the transition. |
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== Derivation == |
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We start with the [[Hamiltonian]] for a 2-level system with frequency splitting <math>\omega_0</math>, illuminated by a light field with frequency <math>\omega</math> |
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:<math>\hat{H} = \begin{pmatrix} |
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{\hbar \omega_0 \over 2} & \chi_{21}e^{i\omega t/2}\\ |
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\chi_{12}e^{-i\omega t/2} & -{\hbar \omega_0 \over 2} |
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\end{pmatrix} |
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</math> |
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with a wavefunction for the system (in the [[interaction picture]]) given by |
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:<math>|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle</math> |
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We can write the [[Schrödinger equation|time-dependent Schrödinger equation]] in the form |
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:<math>\dot{c}(t) = {1 \over i\hbar}\hat{H}c(t)</math> |
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And solve using the [[rotating wave approximation]], giving a probability of being in states <math>|1\rangle</math> and <math>|2\rangle</math> |
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:<math>P_1 = \left({\chi_{21} \over \Omega_{21}}\right)^2 \sin^2\left({\Omega_{21} \over 2}t\right)</math> |
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:<math>P_2 = 1 - \left({\chi_{21} \over \Omega_{21}}\right)^2 \sin^2\left({\Omega_{21} \over 2}t\right)</math> |
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== Template test == |
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{| class="infobox" style="float: right; text-align: center; font-size: 85%; clear:right; width:194px;" |
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|- |
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| [[Image:Solenoid.svg|100px]] |
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|- |
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| style="background:#dfdfdf;" colspan="2" |<font size=2>'''[[Electromagnetism]]'''</font> |
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|- |
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|[[Electricity]] · [[Magnetism]] |
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|- |
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! style="background:#dfdfdf;"| [[User:DJIndica/Sandbox|Electrostatics]] |
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|- |
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|{{#if: {{{estat|}}} |
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[[Electric charge]]<br/> |
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[[Coulomb's law]]<br/> |
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[[Electric field]]<br/> |
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[[Gauss's law]]<br/> |
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[[Electric potential]]<br/> |
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[[Electrical dipole moment|Electric dipole moment]] |
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}} |
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|- |
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! style="background:#dfdfdf;"| [[User:DJIndica/Sandbox2|Magnetostatics]] |
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|- |
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|{{#if: {{{mstat|}}} |
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[[Ampère's circuital law]]<br/> |
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[[Magnetic field]]<br/> |
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[[Magnetic flux]]<br/> |
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[[Biot-Savart law]]<br/> |
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[[Magnetic moment|Magnetic dipole moment]] |
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}} |
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|- |
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! style="background:#dfdfdf;"| [[Classical electromagnetism|Electrodynamics]] |
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|- |
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|{{#if: {{{edyn|}}}| |
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[[Electrical current]]<br/> |
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[[Lorentz force|Lorentz force law]]<br/> |
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[[Electromotive force]]<br/> |
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(EM) [[Electromagnetic induction]]<br/> |
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[[Faraday's law of induction|Faraday-Lenz law]]<br/> |
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[[Displacement current]]<br/> |
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[[Maxwell's equations]]<br/> |
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(EMF) [[Electromagnetic field]]<br/> |
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(EM) [[Electromagnetic radiation]] |
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}} |
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|- |
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! style="background:#dfdfdf;"| [[Electrical_network | Electrical Network]] |
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|- |
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|{{#if: {{{enet|}}}| |
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[[Electrical_conduction|Electrical conduction]]<br/> |
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[[Electrical resistance]]<br/> |
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[[Capacitance]]<br/> |
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[[Inductance]]<br/> |
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[[Electrical impedance|Impedance]]<br/> |
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[[Resonator|Resonant cavities]]<br/> |
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[[Waveguide_%28electromagnetism%29|Waveguides]] |
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}} |
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|- |
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! style="background:#dfdfdf;"| [[Theory of relativity| Tensors in Relativity]] |
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|- |
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|{{#if: {{{rel|}}}| |
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[[Electromagnetic tensor]]<br/> |
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[[Electromagnetic stress-energy tensor]] |
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}} |
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|- |
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|{{Tnavbar|Electromagnetism2}} |
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|} |
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{| class="infobox" style="float: right; text-align: center; font-size: 85%; clear:right;" |
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|- |
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| [[Image:Solenoid.svg|100px]] |
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|- |
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| style="background:#dfdfdf;" colspan="2" |<font size=2>'''[[Electromagnetism]]'''</font> |
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|- |
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|[[Electricity]] · [[Magnetism]] |
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|- style="background:#dfdfdf;" |
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! [[Electrostatics]] |
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|- |
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|[[Electric charge]] |
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|- |
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|[[Coulomb's law]] |
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|- |
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|[[Electric field]] |
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|- |
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|[[Gauss's law]] |
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|- |
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|[[Electric potential]] |
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|- |
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|[[Electrical dipole moment|Electric dipole moment]] |
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|- style="background:#dfdfdf;" |
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! [[Magnetostatics]] |
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|- |
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|[[Ampère's circuital law]] |
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|- |
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|[[Magnetic field]] |
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|- |
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|[[Magnetic flux]] |
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|- |
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|[[Biot-Savart law]] |
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|- |
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|[[Magnetic moment|Magnetic dipole moment]] |
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|- style="background:#dfdfdf;" |
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! [[Classical electromagnetism|Electrodynamics]] |
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|- |
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|[[Electrical current]] |
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|- |
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|[[Lorentz force|Lorentz force law]] |
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|- |
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|[[Electromotive force]] |
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|- |
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|(EM) [[Electromagnetic induction]] |
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|- |
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|[[Faraday's law of induction|Faraday-Lenz law]] |
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|- |
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|[[Displacement current]] |
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|- |
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|[[Maxwell's equations]] |
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|- |
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|(EMF) [[Electromagnetic field]] |
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|- |
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|(EM) [[Electromagnetic radiation]] |
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|- style="background:#dfdfdf;" |
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! [[Electrical_network | Electrical Network]] |
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|- |
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|[[Electrical_conduction|Electrical conduction]] |
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|- |
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|[[Electrical resistance]] |
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|- |
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|[[Capacitance]] |
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|- |
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|[[Inductance]] |
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|- |
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|[[Electrical impedance|Impedance]] |
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|- |
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|[[Resonator|Resonant cavities]] |
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|- |
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|[[Waveguide_%28electromagnetism%29|Waveguides]] |
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|- style="background:#dfdfdf;" |
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! [[Theory of relativity| Tensors in Relativity]] |
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|- |
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|[[Electromagnetic tensor]] |
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|- |
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|[[Electromagnetic stress-energy tensor]] |
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|- |
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|{{Tnavbar|Electromagnetism}} |
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|} |
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== Testing == |
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{{hidden|headercss=width: 50%;|contentcss=text-align: left;|header=hello, what's your name?|content = {{User:DJIndica}}}} |
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== Family tree == |
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{{familytree/start}} |
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{{familytree |PATGFATHER|~|y|~|PATGMOTHER|MATGFATHER|~|y|~|MATGMOTHER|PATGFATHER=Bernhard of Lippe|PATGMOTHER=Armgard of Sierstorpff-Cramm|MATGFATHER=[[Hendrik of Mecklenburg-Schwerin]]|MATGMOTHER=[[Wilhelmina of the Netherlands]]}} |
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{{familytree |||FATHER|~|y|~|MOTHER|||FATHER=[[Bernhard of Lippe-Biesterfeld]]|||MOTHER=[[Juliana of the Netherlands ]]}} |
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{{familytree |||||BEA|||||BEA=[[Beatrix of the Netherlands]]}} |
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{{familytree/end}} |
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{{cquote|large quotation marks}} |
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[[Help:Formula]] |
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[[Wikipedia:Conditional_tables]] |
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[[m:ParserFunctions|ParserFunctions]] |
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[[Help:Parser functions]] |
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== Electromagnetism2== |
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{| class="infobox" style="float: right; text-align: left; font-size: 85%; clear:right; width:194px;" |
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|- |
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|[[Image:Solenoid.svg|center|100px]] |
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|- |
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| style="background:#dfdfdf; text-align:center;" colspan="2" |<font-size=2>'''[[Electromagnetism]]'''</font> |
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|- style="text-align:center" |
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| [[Electricity]] · [[Magnetism]] |
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|- |
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! style="background:#dfdfdf; text-align:center;"| [[User:DJIndica/Sandbox|Electrostatics]] |
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|- |
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|{{#if: {{{estat|}}}| |
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:[[Electric charge]]<br/> |
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:[[Coulomb's law]]<br/> |
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:[[Electric field]]<br/> |
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:[[Gauss's law]]<br/> |
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:[[Electric potential]]<br/> |
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:[[Electrical dipole moment|Electric dipole moment]] |
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}} |
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|- |
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! style="background:#dfdfdf; text-align:center;"| [[User:DJIndica/Sandbox2|Magnetostatics]] |
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|- |
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|{{#if: {{{mstat|}}}| |
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:[[Ampère's circuital law]]<br/> |
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:[[Magnetic field]]<br/> |
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:[[Magnetic flux]]<br/> |
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:[[Biot-Savart law]]<br/> |
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:[[Magnetic moment|Magnetic dipole moment]] |
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}} |
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|- |
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! style="background:#dfdfdf; text-align:center;"| [[User:DJIndica/Euler-Cromer algorithm<!--Classical electromagnetism-->|Electrodynamics]] |
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|- |
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|{{#if: {{{edyn|}}}| |
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:[[Electrical current]]<br/> |
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:[[Lorentz force|Lorentz force law]]<br/> |
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:[[Electromotive force]]<br/> |
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:(EM) [[Electromagnetic induction]]<br/> |
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:[[Faraday's law of induction|Faraday-Lenz law]]<br/> |
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:[[Displacement current]]<br/> |
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:[[Maxwell's equations]]<br/> |
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:(EMF) [[Electromagnetic field]]<br/> |
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:(EM) [[Electromagnetic radiation]] |
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}} |
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|- |
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! style="background:#dfdfdf; text-align:center;"| [[User:DJIndica/Heavy_Rydberg_system<!--Electrical_network--> | Electrical Network]] |
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|- |
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|{{#if: {{{enet|}}}| |
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:[[Electrical_conduction|Electrical conduction]]<br/> |
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:[[Electrical resistance]]<br/> |
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:[[Capacitance]]<br/> |
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:[[Inductance]]<br/> |
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:[[Electrical impedance|Impedance]]<br/> |
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:[[Resonator|Resonant cavities]]<br/> |
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:[[Waveguide_%28electromagnetism%29|Waveguides]] |
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}} |
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|- |
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! style="background:#dfdfdf; text-align:center;"| [[User:DJIndica<!--Theory of relativity-->| Tensors in Relativity]] |
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|- |
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|{{#if: {{{rel|}}}| |
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:[[Electromagnetic tensor]]<br/> |
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:[[Electromagnetic stress-energy tensor]] |
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}} |
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|-style="text-align:center" |
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|{{Tnavbar|Electromagnetism2}} |
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|} |
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<!--<noinclude>[[ar:قالب:كهرومغناطيسية]] |
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[[zh:Template:电磁学]] |
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[[si:Template:Electromagnetism]] |
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</noinclude>--> |
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<!--{| class="infobox" style="float: right; text-align: left; font-size: 85%; clear:right; width:194px;" |
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|--> |
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{{electromagnetism2|estat=true}} |
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{|class="infobox" cellpadding="0" style="font-size: 90%;" |
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|- |
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|[[Image:Solenoid.svg|center|100px]] |
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|- |
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!style="background:#dfdfdf; text-align: center;"|[[Electromagnetism]] |
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|- |
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|style="text-align: center;"|[[Electricity]] · [[Magnetism]] |
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|- |
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| |
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<div class="NavFrame" style="width: 184px; border-style: none; text-align: center; padding: 0px;"> |
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<div class="NavHead" style="background: #dfdfdf; text-align: center; padding-top: 2px; padding-bottom: 3px;">[[Electrostatics]]</div> |
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<div class="NavContent" style="{{#if: estat| | display: none;}}"><!--if estat=true, then expand NavContent (default), else collapse NavContent (display: none;)--> |
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{|align=center |
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|- |
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|[[Electric charge]] |
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|- |
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|[[Coulomb's law]] |
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|- |
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|[[Electric field]] |
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|- |
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|[[Gauss's law]] |
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|- |
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|[[Electric potential]] |
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|- |
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|[[Electrical dipole moment|Electric dipole moment]] |
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|- |
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|} |
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</div></div></div> |
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Revision as of 06:32, 30 November 2009
Inglis-Teller limit
The linear Stark shift is given by
where e is the elementary charge, radius of the nth electron orbit in the Bohr model of the hydrogen atom and F is the component of the local electric field parallel with the electric dipole. The energy of the nth electron orbit is simply the Coulomb potential energy with r = rn
giving an energy level spacing
The Inglis-Teller limit is defined as the field strength at which the Stark states with |(n1 - n2)| = n are shifted by half of this spacing, ΔES = ΔUn/2,
