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Observe that the calculated potentials in the region between
the plates are no different than you would expect,
and even the fringe regions immediately near the plate openings
bear resemblance to that his response an infinite ground sphere surrounding
the system. The Laplace and Poisson equations are discussed in much detail (with derivation, applications and problems) here in EML-5, 6, 7 and 8.
The boundary conditions on the Laplace equation typically take the
form of specifying the values of potential (or some other constraint
such as the values of the derivatives of potential)
that are known over the surface enclosing that volume (the boundary surface).

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However, do the Neumann boundary conditions represent physical
reality?
Probably they don’t except to the extent that as you approach
infinite distance, not only might potential approach zero,
but the normal derivative of potential would
approach zero too, so the Neumann boundaries on the PA edges
approximate the slope of potential at infinite distance
except that this constraint has just been moved closer, to
the PA edges, which is an approximation. e. g. (ii) from Neumann condition: so on the surfaces, thus Φ is constant (Φ = Φ1 Φ2 = constant) and so Φ1 and Φ2 differ by an additive constant.
The proof of this proposition proceeds along the usual lines.

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Suppose
that the magnetic field is not uniquely determined. (i) from Dirichlet condition: Φ1 = Φ2 so Φ = 0, and Φ must be zero everywhere in R and that means the two solutions satisfying click reference same boundary value condition must themselves be same (i. Lets consider: Φ = Φ1 Φ2.
The potentials within the gaps must vary continuously
from +10 V to -10 V along a small distance,
so this fairly well constrains what the potentials are in this
region regardless of what is outside.
Observe Figure 1c, which is the PA in Figure 1a but has been tightly
cropped so that no surrounding space has been included in the array.

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document. B
For

g

{\displaystyle g}

and

h

{\displaystyle h}

sequences whose non-zero duration is less than or equal to

N
,

{\displaystyle N,}

a final simplification is:
(Eq. 1 SIMION User Manual). We will derive the equation for the convolution of two discrete-time signals.

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There are other slides on different topics at that account of mine onslideshare.
Thus,

Suppose that some volume contains a number of conductors.  We have a detailed post on the difference between linear and circular convolutions. Other versions of the convolution theorem are applicable to various Fourier-related transforms.
Fortunately this might be a region we don’t care much about, but it
still might leak through the gaps to some degree.

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Lets say Φ1 and Φ2 are two solutions to the Laplace equation so that they satisfy the same boundary conditions on the surfaces: S1, S2, , Sn. 1a)Applying the inverse Fourier transform

F

1

{\displaystyle {\mathcal {F}}^{-1}}

, produces the corollary:2eqs. .