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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_c.html#ems_es_ep_c" > Comments on the Electrostatic Potential< / a > < span class = "headline-id" > ems.es.ep.c< / 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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< a href = "./ems_es_ep_PL.html#ems_es_ep_PL" > The Poisson Equation and the Laplace Equation< / 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_e_pcd.html#ems_es_e_pcd" > The Energy of a Point Charge Distribution< / a > < span class = "headline-id" > ems.es.e.pcd< / span >
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< a href = "./ems_es_e_ccd.html#ems_es_e_ccd" > The Energy of a Continuous Charge Distribution< / a > < span class = "headline-id" > ems.es.e.ccd< / span >
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< a href = "./ems_es_e_c.html#ems_es_e_c" > Comments on Electrostatic Energy< / a > < span class = "headline-id" > ems.es.e.c< / 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_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_isc.html#ems_ca_mi_isc" > Induced Surface Charges< / a > < span class = "headline-id" > ems.ca.mi.isc< / 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_mi_o.html#ems_ca_mi_o" > Other Image Problems< / a > < span class = "headline-id" > ems.ca.mi.o< / span >
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< a href = "./ems_ca_sv.html#ems_ca_sv" > Separation of Variables< / a > < span class = "headline-id" > ems.ca.sv< / 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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2022-02-08 16:21:33 +00:00
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2022-02-07 14:11:58 +00:00
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< p >
Suppose now that we have two solutions to the Poisson equation, \(V_1 ({\bf r})\) and \(V_2 ({\bf r})\).
Defining \(U = V_1 - V_2\), we see that \(U\) manifestly obeys Laplace
within \({\cal V}\), \({\boldsymbol \nabla}^2 U = 0\).
We can now use Green's first identity (\ref{eq:GreensFirstIdentity}) to shed some light on the boundary
problem for the electrostatic potential. Namely, put \(\phi = \psi = U\). This yields
\[
\int_{\cal V} d\tau \left( U {\boldsymbol \nabla}^2 U + {\boldsymbol \nabla} U \cdot {\boldsymbol \nabla} U \right)
= \oint_{\cal S} da ~U \frac{\partial U}{\partial n}.
\]
The first term on the left-hand side vanishes since \(U\) satisfies Laplace.
The right-hand side can be made to vanish if \(U\) obeys either
< / p >
\begin{align}
& U|_{\cal S} = 0 & \mbox{({\bf Dirichlet})} \label{eq:Dirichlet}\\
\mbox{or}& & \nonumber \\
& \frac{\partial U}{\partial n}|_{\cal S} = 0 & \mbox{({\bf Neumann})} \label{eq:Newmann}
\end{align}
< p >
boundary conditions on each individual boundary surface. In those cases, we are left with
\[
\int_{\cal V} d\tau \left|{\boldsymbol \nabla} U \right|^2 = 0, \longrightarrow {\boldsymbol \nabla} U = 0.
\]
\(U\) is thus constant. For Dirichlet, \(U = 0\) throughout \({\cal V}\), and thus \(V_2 = V_1\) and the solution
is unique. For Neumann, the solution is unique apart from an unimportant constant.
< / p >
< p >
We can thus finally state the
< / p >
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< p >
< b > Uniqueness Theorem< / b >
< / p >
< p >
The solution to Poisson's equation \({\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}\) inside a volume \({\cal V}\)
bounded by a (in general disconnected) surface \({\cal S}\) is uniquely defined provided either Dirichlet \(V |_{{\cal S}_i}\) or Neumann
\(\frac{\partial V}{\partial n} |_{{\cal S}_i}\) boundary conditions are used on each individual surface.
< / p >
< / div >
< p >
Note that these types of boundary conditions can be mixed, < i > i.e.< / i > Dirichlet on some surfaces,
Neumann on others).
< / p >
< p >
< b > Existence of solutions< / b > : this is another matter.
Intuitively, from our first case:
the solution always exists for Dirichlet boundary conditions.
< / p >
< p >
< b > Link to earlier cases< / b > : the 'second case' above, in which the potential is specified on the
boundaries, is the case of Dirichlet boundary conditions.
The 'first case corollary', where the normal derivative of the potential is given, is a subcase involving
Neumann boundary conditions (subcase, because we could imagine other charges living outside volume \({\cal V}\),
whereas our first case corollary involved only surface charges).
< / p >
< p >
\paragraph{Note on Griffiths' presentation of uniqueness theorem(s):}
we have used Green's identity to provide a general statement on uniqueness. Griffiths
might mislead you into thinking that there are numerous cases and corollaries.
Here (in italics) is his (confused) way of thinking about it (see Comment/warning below):
< / p >
< p >
\subsubsection*{\it Boundary Conditions and Uniqueness Theorem}
\paragraph{\it First uniqueness theorem:} {\it The solution to Laplace's equation in some volume
\({\cal V}\) is uniquely determined if \(V\) is specified on the boundary surface \({\cal S}\).
{\bf Corollary:} the potential in a volume \({\cal V}\) is uniquely determined if
a) the charge density throughout the region and b) the value of \(V\) on all boundaries
are specified.}
< / p >
< p >
\subsubsection*{\it Conductors and the Second Uniqueness Theorem}
\paragraph{\it Second uniqueness theorem:} {\it In a volume \({\cal V}\) surrounded by conductors
and containing a specified charge density \(\rho\), the electric field is uniquely determined
if the total charge on each conductor is given.}
< / p >
< p >
{\it Comments: this is the same uniqueness as before, in view of the fact that conductors are
equipotentials, and capacitance relates the charge to the potential.
}
< / p >
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< p >
{\bf Comment/warning: {\color{blue} uniqueness theorem on uniqueness theorems}}< br >
Do not be misled by Griffiths: there is a {\it unique} uniqueness theorem for the
solution of Poisson's equation, namely the one we have stated starting from Green's first identity.
< / p >
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< div id = "postamble" class = "status" >
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< p class = "author" > Author: Jean-Sébastien Caux< / p >
2022-02-09 06:44:58 +00:00
< p class = "date" > Created: 2022-02-09 Wed 07:31< / p >
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