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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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2022-02-08 16:21:33 +00:00
< 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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< a href = "./ems_ms_lf_pc.html#ems_ms_lf_pc" > Point Charge< / a > < span class = "headline-id" > ems.ms.lf.pc< / span >
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< a href = "./ems_ms_lf_c.html#ems_ms_lf_c" > Currents< / a > < span class = "headline-id" > ems.ms.lf.c< / 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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< a href = "./ems_ms_BS_sc.html#ems_ms_BS_sc" > The Magnetic Field issuing from a Steady Current< / a > < span class = "headline-id" > ems.ms.BS.sc< / span >
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< a href = "./ems_ms_dcB.html#ems_ms_dcB" > Divergence and Curl of \({\bf B}\)< / a > < span class = "headline-id" > ems.ms.dcB< / span >
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2022-02-08 16:21:33 +00:00
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2022-02-07 14:11:58 +00:00
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< div class = "example div" id = "org355aed7" >
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< p >
{\bf Example: separation of variables (Cartesian coordinates)}< br >
Two infinite grounded metal plates parallel to the \(xz\) plane, one at \(y = 0\) and the
other at \(y = a\). End at \(x = 0\) closed off with infinite insulated strip maintained
at potential \(V_0 (y)\). Find potential inside the slot.
\paragraph{Solution:} indep of \(z\), so 2d problem. Solve
\[
\frac{\partial^2 V}{\partial^2 x} + \frac{\partial^2 V}{\partial^2 y} = 0
\label{Gr(3.20)}
\]
\[
(i) V(x, y = 0) = 0, \hspace{5mm} (ii)V(x, y = a) = 0, \hspace{5mm} (iii) V(0, y) = V_0 (y), \hspace{5mm} (iv) V (x \rightarrow \infty) \rightarrow 0.
\label{Gr(3.21)}
\]
Look for solutions of form
\[
V(x,y) = X(x) Y(y), \hspace{1cm}
\frac{1}{X} \frac{d^2 X}{dx^2} + \frac{1}{Y} \frac{d^2 Y}{dy^2} = 0
\label{Gr(3.23)}
\]
Choose
\[
\frac{d^2 X_n}{dx^2} = k_n^2 X_n, \hspace{1cm} \frac{d^2Y_n}{dy^2} = -k_n^2 Y_n
\label{Gr(3.26)}
\]
where \(k_n\) is some real number. We can linearly combine solutions of (\ref{Gr(3.26)})
for different \(k_n\) and still get a solution to (\ref{Gr(3.23)}).
Let's look first of all at the solutions of (\ref{Gr(3.26)}) for a given \(k_n\).
Since this is a second-order linear differential equation, there are two linearly
independent solutions. Most general solution:
\[
X_n(x) = Ae^{k_nx} + Be^{-k_nx}, \hspace{1cm} Y_n(y) = C \sin k_ny + D \cos k_ny
\label{Gr(3.27)}
\]
Fix constants: from \((iv)\), \(A = 0\). From \((i)\), D = 0. Left with
\[
V(x,y) = C_n e^{-k_nx} \sin k_n y
\label{Gr(3.28)}
\]
Then, \((ii)\) requires
\[
k_n = \frac{n\pi}{a}, \hspace{1cm} n = 1, 2, 3, ...
\label{Gr(3.29)}
\]
But we can use as solution any linear combination of the functions defined
by these momenta. Fix coefficients with Fourier series:
\[
V(x,y) = \sum_{n=1}^{\infty} C_n e^{-n\pi x/a} \sin (n\pi x/a)
\label{Gr(3.30)}
\]
Needed:
\[
\int_0^a dy \sin(n\pi y/a) \sin (n' \pi y/a) = \delta_{n n'} \frac{a}{2}
\label{Gr(3.33)}
\]
so
\[
C_n = \frac{2}{a} \int_0^a dy V_0(y) \sin(n\pi y/a)
\label{Gr(3.34)}
\]
< / p >
< / div >
< p >
\paragraph{Specific example:} say that \(V_0(y) = V_0\), some constant. Then,
\[
C_n = \frac{2V_0}{a} \int_0^a dy \sin(n\pi y/a) = \frac{2V_0}{n\pi} (1 - \cos n\pi) = \frac{4V_0}{n\pi} \delta_{n, odd}
\label{Gr(3.35)}
\]
< / p >
< p >
Applicable provided {\bf completeness} and {\bf orthogonality}:
< / p >
< p >
Completeness:
\[
f(x) = \sum_{n=1}^{\infty} C_n f_n (y), \hspace{1cm} \forall f \in C^{\infty}
\label{Gr(3.38)}
\]
< / p >
< p >
Orthogonality:
\[
\int_0^a dx f_n (x) f_{n'} (x) = 0, \hspace{1cm} n \neq n'
\label{Gr(3.39)}
\]
< / p >
< p >
The solution for the specific case \(V_0 (y) = V_0\) therefore is
\[
V(x,y) = \frac{4V_0}{\pi}\sum_{n=1, 3, 5, ...}^{\infty} \frac{1}{n} e^{-n\pi x/a} \sin (n\pi x/a)
\]
< / p >
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< div class = "example div" id = "org87e64a1" >
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< p >
{\bf Example: rectangular pipe}< br >
Two infinitely long grounded plates at \(y = 0,a\) are connected at \(x = \pm b\)
to metal strips maintained at constant \(V = V_0\). Find the potential in the resulting rectangular pipe.
\paragraph{Solution:} indep of \(z\). Laplace: \(\partial^2 V/\partial x^2 + \partial^2 V/\partial y^2 = 0\),
boundary conditions
\[
(i) V (y = 0) = 0, \hspace{5mm} (ii) V (y = a) = 0, \hspace{5mm} (iii) V(x = b) = 0, \hspace{5mm} (iv) V(x = -b) = 0.
\label{Gr(3.40)}
\]
Generic solution: as (\ref{Gr(3.27)}),
\[
V(x,y) = (Ae^{kx} + Be^{-kx}) (C\sin ky + D\cos ky).
\]
By symmetry, \(V(x,y) = V(-x,y)\) so \(A = B\). Now \(e^{kx} + e^{-kx} = 2\cosh kx\). Generic solution becomes
(redefining \(C\) and \(D\))
\[
V(x,y) = \cosh kx (C\sin ky + D\cos ky).
\]
Boundary conditions \((i)\) and \((ii)\) require \(D = 0\), \(k = n\pi/a\) so
\[
V(x,y) = C \cosh(n\pi x/a) \sin(n\pi y/a)
\label{Gr(3.41)}
\]
with \((iv)\) already satisfied if \((iii)\) is. Full solution is linear combination of complete set of functions,
\[
V(x,y) = \sum_{n=1}^{\infty} C_n \cosh(n\pi x/a) \sin(n\pi y/a).
\]
Coefficients: chosen such that \((iii)\) is fulfilled, \(V(b,y) = V_0\). From (\ref{Gr(3.35)}):
\[
V(x,y) = \frac{4V_0}{\pi} \sum_{n = 1, 3, 5, ...} \frac{1}{n} \frac{\cosh (n\pi x/a)}{\cosh(n\pi b/a)} \sin(n\pi y/a).
\label{Gr(3.42)}
\]
< / p >
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< div id = "postamble" class = "status" >
2022-02-07 14:11:58 +00:00
< p class = "author" > Author: Jean-Sébastien Caux< / p >
2022-02-10 07:34:34 +00:00
< p class = "date" > Created: 2022-02-10 Thu 08:32< / p >
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