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< a href = "./index.html" class = "homepage-link" > Pre-Quantum Electrodynamics< / a >
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Table of contents
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< a href = "./in.html#in" > Introduction< / a > < span class = "headline-id" > in< / span >
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< a href = "./in_p.html#in_p" > Preface< / a > < span class = "headline-id" > in.p< / span >
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< a href = "./in_t.html#in_t" > Tips for the reader< / a > < span class = "headline-id" > in.t< / span >
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< a href = "./in_t_l.html#in_t_l" > Section and equation labelling< / a > < span class = "headline-id" > in.t.l< / span >
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< a href = "./in_t_c.html#in_t_c" > Contextual colors< / a > < span class = "headline-id" > in.t.c< / span >
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< a href = "./ems.html#ems" > Electromagnetostatics< / a > < span class = "headline-id" > ems< / span >
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< a href = "./ems_es.html#ems_es" > Electrostatics< / a > < span class = "headline-id" > ems.es< / span >
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< a href = "./ems_es_ec.html#ems_es_ec" > Electric Charge< / a > < span class = "headline-id" > ems.es.ec< / span >
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< a href = "./ems_es_ec_b.html#ems_es_ec_b" > Basics< / a > < span class = "headline-id" > ems.es.ec.b< / span >
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< a href = "./ems_es_ec_c.html#ems_es_ec_c" > Conservation< / a > < span class = "headline-id" > ems.es.ec.c< / span >
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< a href = "./ems_es_ec_q.html#ems_es_ec_q" > Quantization< / a > < span class = "headline-id" > ems.es.ec.q< / span >
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< a href = "./ems_es_ec_s.html#ems_es_ec_s" > Structure< / a > < span class = "headline-id" > ems.es.ec.s< / span >
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< a href = "./ems_es_efo.html#ems_es_efo" > Electric Force and Energy< / a > < span class = "headline-id" > ems.es.efo< / span >
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< a href = "./ems_es_efo_cl.html#ems_es_efo_cl" > Coulomb's Law< / a > < span class = "headline-id" > ems.es.efo.cl< / span >
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< a href = "./ems_es_efo_ps.html#ems_es_efo_ps" > Principle of Superposition< / a > < span class = "headline-id" > ems.es.efo.ps< / span >
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< a href = "./ems_es_efo_exp.html#ems_es_efo_exp" > Experimental Investigations< / a > < span class = "headline-id" > ems.es.efo.exp< / span >
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< a href = "./ems_es_efo_e.html#ems_es_efo_e" > Energy in Systems of Point Charges< / a > < span class = "headline-id" > ems.es.efo.e< / span >
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< a href = "./ems_es_ef.html#ems_es_ef" > Electrostatic Fields< / a > < span class = "headline-id" > ems.es.ef< / span >
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< a href = "./ems_es_ef_pc.html#ems_es_ef_pc" > Electrostatic Field of Point Charges< / a > < span class = "headline-id" > ems.es.ef.pc< / span >
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< a href = "./ems_es_ef_ccd.html#ems_es_ef_ccd" > Electrostatic Field of Continuous Charge Distributions< / a > < span class = "headline-id" > ems.es.ef.ccd< / span >
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< a href = "./ems_es_ef_cE.html#ems_es_ef_cE" > The Curl of \({\bf E}\)< / a > < span class = "headline-id" > ems.es.ef.cE< / span >
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< a href = "./ems_es_ef_Gl.html#ems_es_ef_Gl" > Gauss's Law: the divergence of \({\bf E}\)< / a > < span class = "headline-id" > ems.es.ef.Gl< / span >
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2022-02-08 06:07:41 +00:00
< a href = "./ems_es_ep.html#ems_es_ep" > The Electrostatic Potential< / a > < span class = "headline-id" > ems.es.ep< / span >
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< a href = "./ems_es_ep_d.html#ems_es_ep_d" > Definition< / a > < span class = "headline-id" > ems.es.ep.d< / span >
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< a href = "./ems_es_ep_fp.html#ems_es_ep_fp" > Field in terms of the potential< / a > < span class = "headline-id" > ems.es.ep.fp< / span >
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< a href = "./ems_es_ep_ex.html#ems_es_ep_ex" > Example calculations for the potential< / a > < span class = "headline-id" > ems.es.ep.ex< / span >
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2022-02-09 21:41:42 +00:00
< a href = "./ems_es_ep_PL.html#ems_es_ep_PL" > Poisson's and Laplace's Equations< / a > < span class = "headline-id" > ems.es.ep.PL< / span >
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< a href = "./ems_es_ep_bc.html#ems_es_ep_bc" > Electrostatic Boundary Conditions< / a > < span class = "headline-id" > ems.es.ep.bc< / span >
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2022-02-08 06:07:41 +00:00
< a href = "./ems_es_e.html#ems_es_e" > Electrostatic Energy from the Potential< / a > < span class = "headline-id" > ems.es.e< / span >
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< a href = "./ems_es_c.html#ems_es_c" > Conductors< / a > < span class = "headline-id" > ems.es.c< / span >
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< a href = "./ems_es_c_p.html#ems_es_c_p" > Properties< / a > < span class = "headline-id" > ems.es.c.p< / span >
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< a href = "./ems_es_c_ic.html#ems_es_c_ic" > Induced Charges< / a > < span class = "headline-id" > ems.es.c.ic< / span >
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< a href = "./ems_es_c_sc.html#ems_es_c_sc" > Surface Charge and the Force on a Conductor< / a > < span class = "headline-id" > ems.es.c.sc< / span >
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< a href = "./ems_es_c_cap.html#ems_es_c_cap" > Capacitors< / a > < span class = "headline-id" > ems.es.c.cap< / span >
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< a href = "./ems_ca.html#ems_ca" > Calculating or Approximating the Electrostatic Potential< / a > < span class = "headline-id" > ems.ca< / span >
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< a href = "./ems_ca_fe.html#ems_ca_fe" > Fundamental Equations for the Electrostatic Potential< / a > < span class = "headline-id" > ems.ca.fe< / span >
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< a href = "./ems_ca_fe_L.html#ems_ca_fe_L" > The Laplace Equation< / a > < span class = "headline-id" > ems.ca.fe.L< / span >
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< a href = "./ems_ca_fe_g.html#ems_ca_fe_g" > Green's Identities< / a > < span class = "headline-id" > ems.ca.fe.g< / span >
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< a href = "./ems_ca_fe_uP.html#ems_ca_fe_uP" > Uniqueness of Solution to Poisson's Equation< / a > < span class = "headline-id" > ems.ca.fe.uP< / span >
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< a href = "./ems_ca_mi_fe.html#ems_ca_mi_fe" > Force and Energy< / a > < span class = "headline-id" > ems.ca.mi.fe< / span >
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< a href = "./ems_ca_sv_car.html#ems_ca_sv_car" > Cartesian Coordinates< / a > < span class = "headline-id" > ems.ca.sv.car< / span >
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< a href = "./ems_ca_sv_cyl.html#ems_ca_sv_cyl" > Cylindrical Coordinates< / a > < span class = "headline-id" > ems.ca.sv.cyl< / span >
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< a href = "./ems_ca_sv_sph.html#ems_ca_sv_sph" > Spherical Coordinates< / a > < span class = "headline-id" > ems.ca.sv.sph< / span >
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< a href = "./ems_ca_me.html#ems_ca_me" > The Multipole Expansion< / a > < span class = "headline-id" > ems.ca.me< / span >
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< a href = "./ems_ca_me_a.html#ems_ca_me_a" > Approximate Potential at Large Distance< / a > < span class = "headline-id" > ems.ca.me.a< / span >
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< a href = "./ems_ca_me_md.html#ems_ca_me_md" > Monopole and Dipole Terms< / a > < span class = "headline-id" > ems.ca.me.md< / span >
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< a href = "./ems_ca_me_h.html#ems_ca_me_h" > Higher Moments< / a > < span class = "headline-id" > ems.ca.me.h< / span >
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< a href = "./ems_ca_me_Ed.html#ems_ca_me_Ed" > The Electric Field of a Dipole< / a > < span class = "headline-id" > ems.ca.me.Ed< / span >
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< a href = "./ems_ca_me_Eq.html#ems_ca_me_Eq" > The Electric Field of a Quadrupole< / a > < span class = "headline-id" > ems.ca.me.Eq< / span >
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< a href = "./ems_ms.html#ems_ms" > Magnetostatics< / a > < span class = "headline-id" > ems.ms< / span >
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< a href = "./ems_ms_lf.html#ems_ms_lf" > Charges in Motion: the Lorentz Force Law< / a > < span class = "headline-id" > ems.ms.lf< / span >
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2022-02-21 09:35:02 +00:00
< a href = "./ems_ms_lf_pc.html#ems_ms_lf_pc" > Point Charges< / a > < span class = "headline-id" > ems.ms.lf.pc< / span >
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< a href = "./ems_ms_lf_sc.html#ems_ms_lf_sc" > Steady Currents< / a > < span class = "headline-id" > ems.ms.lf.sc< / span >
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< a href = "./ems_ms_ce.html#ems_ms_ce" > Charge Conservation and the Continuity Equation< / a > < span class = "headline-id" > ems.ms.ce< / span >
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< a href = "./ems_ms_BS.html#ems_ms_BS" > Steady Currents: the Biot-Savart Law< / a > < span class = "headline-id" > ems.ms.BS< / span >
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< li >
< a href = "./c_m_dd_3d.html#c_m_dd_3d" > The Three-Dimensional Delta Function< / a > < span class = "headline-id" > c.m.dd.3d< / span >
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< a href = "./c_m_vf_helm.html#c_m_vf_helm" > The Helmholtz Theorem< / a > < span class = "headline-id" > c.m.vf.helm< / span >
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< ul class = "breadcrumbs" > < li > < a class = "breadcrumb-link" href = "ems.html" > Electromagnetostatics< / a > < / li > < li > < a class = "breadcrumb-link" href = "ems_ca.html" > Calculating or Approximating the Electrostatic Potential< / a > < / li > < li > < a class = "breadcrumb-link" href = "ems_ca_sv.html" > Separation of Variables< / a > < / li > < li > Cartesian Coordinates< / li > < / ul > < ul class = "navigation-links" > < li > Prev: < a href = "ems_ca_sv.html" > Separation of Variables  < small > [ems.ca.sv]< / small > < / a > < / li > < li > Next: < a href = "ems_ca_sv_cyl.html" > Cylindrical Coordinates  < small > [ems.ca.sv.cyl]< / small > < / a > < / li > < li > Up: < a href = "ems_ca_sv.html" > Separation of Variables  < small > [ems.ca.sv]< / small > < / a > < / li > < / ul > < div id = "outline-container-ems_ca_sv_car" class = "outline-5" >
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< h5 id = "ems_ca_sv_car" > Cartesian Coordinates< a class = "headline-permalink" href = "./ems_ca_sv_car.html#ems_ca_sv_car" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
< path d = "M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z" / >
< path d = "M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z" / >
< / svg > < / a > < span class = "headline-id" > ems.ca.sv.car< / span > < / h5 >
< div class = "outline-text-5" id = "text-ems_ca_sv_car" >
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< div class = "example div" id = "org6fa8ae7" >
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< p >
< / p >
< p >
< b > Example: infinite grounded metal plates< / b >
< / p >
< p >
Consider two infinite grounded metal plates positioned parallel to the
\(xz\) plane, one at \(y = 0\) and the other at \(y = a\).
< / p >
< p >
At \(x = 0\), the setup is closed off with an infinite insulated strip maintained
at potential \(\phi_0 (y)\).
< / p >
< p >
< b > Question< / b > : find the potential inside the slot.
< / p >
< p >
< b > Solution< / b > :
< / p >
< p >
By translational symmetry along \(z\), the potential must be independent of \(z\).
This thus falls back onto a 2d problem. We need to solve the 2d Laplace equation
< a href = "./ems_ca_fe_L.html#Lap_2d" > Lap_2d< / a >
< / p >
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< p >
\[
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\frac{\partial^2 \phi}{\partial^2 x} + \frac{\partial^2 \phi}{\partial^2 y} = 0
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\]
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< / p >
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\begin{align}
& (i) ~\phi(x, y=0) = 0, \hspace{5mm} & (ii) ~\phi(x, y=a) = 0, \nonumber \\
& (iii) ~\phi(x=0, y) = \phi_0 (y), & (iv) ~\phi (x \rightarrow \infty, y) \rightarrow 0.
\end{align}
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< p >
Let us look for solutions in the form
< / p >
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< div class = "eqlabel" id = "orga36e788" >
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< p >
< a id = "Lap_sv_car" > < / a > < a href = "./ems_ca_sv_car.html#Lap_sv_car" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
< path d = "M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z" / >
< path d = "M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z" / >
< / svg > < / a >
< / p >
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< div class = "alteqlabels" id = "orge29141d" >
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< ul class = "org-ul" >
< li > Gr (3.23)< / li >
< / ul >
< / div >
< / div >
< p >
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\[
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\phi(x,y) = X(x) Y(y), \hspace{1cm}
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\frac{1}{X} \frac{d^2 X}{dx^2} + \frac{1}{Y} \frac{d^2 Y}{dy^2} = 0
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\tag{Lap_sv_car}\label{Lap_sv_car}
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\]
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Since the \(x\) and \(y\) dependencies are separated, the only possibility is that each
individual term in the Laplace equation equals a constant,
and that these constants add up to zero. We can thus put (the sign choice anticipates
the solution somewhat)
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< / p >
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< div class = "eqlabel" id = "orgc324eae" >
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< p >
< a id = "Lap_sv_car_sep" > < / a > < a href = "./ems_ca_sv_car.html#Lap_sv_car_sep" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
< path d = "M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z" / >
< path d = "M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z" / >
< / svg > < / a >
< / p >
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< div class = "alteqlabels" id = "org653d57a" >
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< ul class = "org-ul" >
< li > Gr (3.26)< / li >
< / ul >
< / div >
< / div >
< p >
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\[
\frac{d^2 X_n}{dx^2} = k_n^2 X_n, \hspace{1cm} \frac{d^2Y_n}{dy^2} = -k_n^2 Y_n
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\tag{Lap_sv_car_sep}\label{Lap_sv_car_sep}
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\]
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where \(k_n\) is some real number.
Since < a href = "./ems_ca_fe_L.html#Lap_2d" > Lap_2d< / a > is a linear equation for \(\phi\),
we can linearly combine solutions of < a href = "./ems_ca_sv_car.html#Lap_sv_car_sep" > Lap_sv_car_sep< / a >
for different \(k_n\) and still get a solution to < a href = "./ems_ca_sv_car.html#Lap_sv_car" > Lap_sv_car< / a > .
< / p >
< p >
Let's look first of all at the solutions of < a href = "./ems_ca_sv_car.html#Lap_sv_car_sep" > Lap_sv_car_sep< / a > for a given \(k_n\).
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Since this is a second-order linear differential equation, there are two linearly
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independent solutions. The most general solution for \(X\) and \(Y\) can be written
< / p >
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< div class = "eqlabel" id = "orgb25d4e1" >
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< p >
< a id = "Lap_sv_car_solXY" > < / a > < a href = "./ems_ca_sv_car.html#Lap_sv_car_solXY" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
< path d = "M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z" / >
< path d = "M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z" / >
< / svg > < / a >
< / p >
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< div class = "alteqlabels" id = "org2316d74" >
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< ul class = "org-ul" >
< li > Gr (3.27)< / li >
< / ul >
< / div >
< / div >
\begin{align}
X_n(x) & = Ae^{k_nx} + Be^{-k_nx}, \nonumber \\
Y_n(y) & = C \sin k_ny + D \cos k_ny
\tag{Lap_sv_car_solXY}\label{Lap_sv_car_solXY}
\end{align}
< p >
The constants can be fixed by fitting the boundary conditions.
From \((iv)\), we get \(A = 0\). From \((i)\), we get \(D = 0\). We are thus left with
< / p >
< p >
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\[
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\phi(x,y) = C_n e^{-k_nx} \sin k_n y
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\]
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Going further, \((ii)\) requires the momenta \(k_n\) to be quantized according to
< / p >
< p >
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\[
k_n = \frac{n\pi}{a}, \hspace{1cm} n = 1, 2, 3, ...
\]
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Since we can use as solution any linear combination of the functions defined
by these momenta, we obtain the solution in the form of a Fourier series,
< / p >
< p >
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\[
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\phi(x,y) = \sum_{n=1}^{\infty} C_n e^{-n\pi x/a} \sin (n\pi y/a)
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\]
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with yet-to-be-determined coefficients \(C_n\), which have to be chosen to fit
boundary condition \((iii)\). Using the orthogonality relation
< / p >
< p >
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\[
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\int_0^a dy ~\sin(n\pi y/a) \sin (n' \pi y/a) = \delta_{n n'} \frac{a}{2}
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\]
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we thus get that the \(C_n\) are Fourier coefficients of the boundary function \(\phi_0(y)\):
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< / p >
< p >
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\[
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C_n = \frac{2}{a} \int_0^a dy ~\phi_0(y) \sin(n\pi y/a)
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\]
< / p >
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< p >
< b > Specific example< / b > : say that \(\phi_0(y) = \phi_0\), < i > i.e.< / i > just a constant. Then,
< / p >
2022-03-15 09:07:27 +00:00
< div class = "eqlabel" id = "org6d2dab3" >
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< p >
< a id = "Cn" > < / a > < a href = "./ems_ca_sv_car.html#Cn" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
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< div class = "alteqlabels" id = "org58a2ab8" >
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< ul class = "org-ul" >
< li > Gr (3.35)< / li >
< / ul >
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< / div >
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< / div >
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< p >
\[
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C_n = \frac{2\phi_0}{a} \int_0^a dy \sin(n\pi y/a) = \frac{2\phi_0}{n\pi} (1 - \cos n\pi) = \frac{4\phi_0}{n\pi} \delta_{n, odd}
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\tag{Cn}\label{Cn}
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\]
< / p >
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< / div >
< p >
The logic of separation of variables exploited two characteristics of
the basis functions in the Fourier decomposition, namely
< b > completeness< / b > and < b > orthogonality< / b > .
< / p >
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< p >
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Completeness simply means that any differentiable function can be expressed
in terms of the basis functions:
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< / p >
< p >
\[
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f(x) = \sum_{n=1}^{\infty} C_n f_n (y), \hspace{1cm} \forall ~f \in C^{\infty}
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\]
< / p >
< p >
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Orthogonality is a convenient choice made to keep things simple: each function,
under the chosen integration domain and weighing, has no overlap with others, < i > i.e.< / i >
< / p >
< p >
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\[
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\int_0^a dx ~f_n (x) f_{n'} (x) = 0, \hspace{1cm} n \neq n'
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\]
< / p >
< p >
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The solution for the specific case \(\phi_0 (y) = \phi_0\) is thus
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\[
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\phi(x,y) = \frac{4\phi_0}{\pi}\sum_{n=1, 3, 5, ...}^{\infty} \frac{1}{n} e^{-n\pi x/a} \sin (n\pi x/a)
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\]
< / p >
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< div class = "example div" id = "org1ba9ed2" >
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< p >
< b > Example: rectangular pipe< / b >
< / p >
< p >
Take two infinitely long grounded plates at \(y = 0\) and \(y=a\),
cut at \(x = \pm b\) and joined to metal strips maintained at constant \(\phi = \phi_0\).
< / p >
< p >
< b > Question< / b > : find the potential in the resulting rectangular pipe.
< / p >
< p >
< b > Solution< / b > : by translational invariance along \(z\), the potential must be independent of \(z\).
We thus need to solve the 2d Laplace equation < a href = "./ems_ca_fe_L.html#Lap_2d" > Lap_2d< / a > ,
\[
\partial^2 \phi/\partial x^2 + \partial^2 \phi/\partial y^2 = 0,
\]
with boundary conditions
< / p >
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\begin{align}
& (i) ~\phi (x, y = 0) = 0, \hspace{5mm} & (ii) ~\phi (x, y = a) = 0,
\nonumber \\
& (iii) ~\phi(x = b, y) = \phi_0, & (iv) ~\phi(x = -b, y) = \phi_0.
\end{align}
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< p >
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The general solution is obtained from < a href = "./ems_ca_sv_car.html#Lap_sv_car_solXY" > Lap_sv_car_XY< / a > ,
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\[
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\phi(x,y) = (Ae^{kx} + Be^{-kx}) (C\sin ky + D\cos ky).
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\]
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By symmetry, \(\phi(x,y) = \phi(-x,y)\) so \(A = B\).
Since \(e^{kx} + e^{-kx} = 2\cosh kx\), the generic solution becomes
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(redefining \(C\) and \(D\))
\[
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\phi(x,y) = \cosh kx ~(C\sin ky + D\cos ky).
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\]
Boundary conditions \((i)\) and \((ii)\) require \(D = 0\), \(k = n\pi/a\) so
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< / p >
< p >
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\[
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\phi(x,y) = C \cosh(n\pi x/a) \sin(n\pi y/a)
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\]
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with \((iv)\) already satisfied if \((iii)\) is.
< / p >
< p >
The full solution is then a linear combination of complete set of functions,
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\[
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\phi(x,y) = \sum_{n=1}^{\infty} C_n \cosh(n\pi x/a) \sin(n\pi y/a).
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\]
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The coefficients must be chosen such that \((iii)\) is fulfilled, \(\phi(b,y) = \phi_0\).
This simple case of a constant value \(\phi_0\) gives us the same relation as < a href = "./ems_ca_sv_car.html#Cn" > Cn< / a > , so
< / p >
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< div class = "eqlabel" id = "org3d51969" >
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< p >
< a id = "p_recpipe" > < / a > < a href = "./ems_ca_sv_car.html#p_recpipe" > < svg xmlns = "http://www.w3.org/2000/svg" width = "16" height = "16" fill = "currentColor" class = "bi bi-link" viewBox = "0 0 16 16" >
< path d = "M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z" / >
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< / svg > < / a >
< / p >
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< div class = "alteqlabels" id = "orgc8e3689" >
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< ul class = "org-ul" >
< li > Gr (3.42)< / li >
< / ul >
< / div >
< / div >
< p >
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\[
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\phi(x,y) = \frac{4\phi_0}{\pi} \sum_{n = 1, 3, 5, ...} \frac{1}{n} \frac{\cosh (n\pi x/a)}{\cosh(n\pi b/a)} \sin(n\pi y/a).
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\]
< / p >
< / div >
< / div >
< / div >
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< br > < ul class = "navigation-links" > < li > Prev: < a href = "ems_ca_sv.html" > Separation of Variables  < small > [ems.ca.sv]< / small > < / a > < / li > < li > Next: < a href = "ems_ca_sv_cyl.html" > Cylindrical Coordinates  < small > [ems.ca.sv.cyl]< / small > < / a > < / li > < li > Up: < a href = "ems_ca_sv.html" > Separation of Variables  < small > [ems.ca.sv]< / small > < / a > < / li > < / ul >
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< div id = "postamble" class = "status" >
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< p class = "author" > Author: Jean-Sébastien Caux< / p >
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< p class = "date" > Created: 2022-03-15 Tue 08:10< / p >
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