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2. Problem 1
Prob. 2.16.5 A planar region, shown in Table 2.16.1, is filled by an inhomogeneous
dielectric, with a permittivity that depends on x:
The free charge density is zero.
(a) Show that the potential distribution is
where
(b) Show that the transfer relations are
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3. Problem 2
Prob. 4.3.3 The moving member of an EQS device takes the form of a sheet, supporting
the surface charge of and moving in the z direction, as shown in Fig. P4.3.3. Electrodes
on the adjacent walls constrain the potentials there.
Courtesy of MIT Press. Used with permission. Problem 4.3.3 in Melcher, James R.
Continuum Electromechanics. Cambridge, MA: MIT Press, 1981, p. 4.57. ISBN:
9780262131650.
Problem 3
A general right-handed orthogonal curvilinear coordinate system is described by
variables (u, v, w), where
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4. Since the incremental coordinate quantities du, dv, and dul do not necessarily have
units of length, the differential length elements must be multiplied by coefficients that
generally are a function of u, v, and w:
(a) What are the h coefficients for the Cartesian, cylindrical, and spherical coordinate
systems?
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5. (b) What is the gradient of any function f(u, v, w)?
(c) What is the area of each surface and the volume of a differential size volume
element in the (u, v, w) space?
(d) What are the curl and divergence of the vector
(e) What is the scalar Laplacian
(f) Check your results of (b)-(e) for the three basic coordinate systems.
1. DIFFERENTIAL OPERATORS IN CYLINDRICAL AND SPHERICAL
COORDINATES
If r, φ, and z are circular cylindrical coordinates and are unit vectors
in the directions of increasing values of the corresponding coordinates,
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6. If r, θ, and φ are spherical coordinates and
the directions of increasing values of the corresponding coordinates,
are unit vectors in
If r, θ, and φ are spherical coordinates
the directions of increasing values of the corresponding coordinates,
are unit vectors in
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7. 2. SOLUTIONS OF LAPLACE’S EQUATIONS
A. Rectangular coordinates, two dimensions (independent of z):
B. Cylindrical coordinates, two dimensions (independent of z):
C. Spherical coordinates, two dimensions (independent of φ):
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8. Problem 1
Prob. 2.16.5
Gauss’ law and E = requires that if there is no free charge
For the given exponential dependence of the permittivity, the x dependence of the
coefficients in this expression factors out and it again reduces to a constant coefficient
expression
In terms of the complex amplitude forms from Table 2.16.1, Eq. 2 requires that
Thus, solutions have the form exp(px) where p =
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9. The linear combination of these that satisfies the conditions that
on the upper and lower surfaces respectively is as given in the problem. The
displacement vector is then evaluated as
Evaluation of this expression at the respective surfaces then gives the transfer
relations summarized in the problem.
Problem 2
Prob. 4.3.3
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10. Figure 1: Force per unit area on middle charged surface (c/d) is found using the
Maxwell stress tensor (Image by MIT OpenCourseWare.)
With positions as designated in the sketch, the total force per unit area is
With the understanding that the surface charge on the sheet is a given quantity,
boundary conditions reflecting the continuity of tangential electric field at the three
surfaces and that Gauss’ law be satisfied through the sheet are
Bulk relations are given by Table 2.16.1. In the upper region
and in the lower
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11. In view of Eq. 6, Eq. 5 becomes
so what is now required is the amplitude The surface charge, given by Eq. 6, as the
Difference follows in terms of the potentials from taking the difference of Eqs. 7
(second row) and 8 (first row). The resulting expression is solved for
Substituted into Eq. 9 (where the self terms in are imaginary and can therefore
be dropped) the force is expressed in terms of the given excitations
b)
Translation of the given excitations into complex amplitudes gives
Thus, with the even excitation, where
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12. and with the odd excitation, < fz >z = 0.
c) This is a specific case from part (b) with ω = 0 and δ = λ/4. Thus,
The sign is consistent with the sketch of charge distribution on the sheet and electric
field due to the potentials on the walls sketched.
Figure 2: Potentials = −V0 cos kz shown with surface charge σf = σ0 sin kz
(Image by MIT OpenCourseWare.)
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13. Problem 3
a)
Cartesian Cylindrical Spherical
hx = 1 hr = 1 hr = 1
hy = 1 hφ = r hθ = r
hz = 1 hz = 1 hφ = r sin θ
b)
c)
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14. d)
Divergence
Figure 3: Contour for determining (Image by MIT OpenCourseWare.)
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15. Curl
Similarly
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