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Bipolar charging and discharging of a perfectly conducting sphere in a lossy medium stressed by a uniform electric field
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10.1063/1.3563074
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Affiliations:
1 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 10-174, Cambridge, Massachusetts 02139 USA
2 ABB Corporate Research S-72178 Västerås, Sweden
J. Appl. Phys. 109, 084331 (2011)
/content/aip/journal/jap/109/8/10.1063/1.3563074
http://aip.metastore.ingenta.com/content/aip/journal/jap/109/8/10.1063/1.3563074
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## Figures

FIG. 1.

Electric field lines around a perfectly conducting sphere with radius R surrounded by a dielectric fluid with a permittivity of ε and conductivity σ stressed by a uniform z-directed electric field, turned on at t = 0.

FIG. 2.

(Color online) Unipolar charging of a perfectly conducting sphere versus time for (a) positive mobile charge (ρ μ  = 0 for various values of τ s +) and for (b) negative mobile charge (ρ + μ + = 0 for various values of τ s ). Note that the (a) positive charge plot has Q(t) > 0 for all time with time constant τ given by Eq. (25a), while the (b) negative charge plot has Q(t) < 0 for all time with time constant τ given by Eq. (29a).

FIG. 3.

(Color online) Electric field lines for various times after a uniform z-directed electric field is turned on at t = 0 around a perfectly conducting sphere of radius R surrounded by a lossless dielectric with permittivity ε, conductivity σ = 0, and free mobile positive charge with uniform positive charge density ρ + and mobility μ +, and zero negative charge current such that ρ μ  = 0. The thick electric field lines terminate on the particle at r = R and θ = θ c , where Er (r = R) = 0, and separate field lines that terminate on the sphere from field lines that go around the sphere. The cylindrical radius Rb (t) of Eq. (13) of the separation field line at z → − ∞ defines the positive mobile charge current I(t) in Eq. (23) with ρ μ  = 0 and σ = 0. The cylindrical radius Ra (t) of Eq. (12) defines the separation field line at z → + ∞. The electric field lines in this figure were plotted using Mathematica StreamPlot (Ref. 16). (a) ; (b) ; (c) ; and (d) .

FIG. 4.

(Color online) The charge Q(t)/Qs and critical charging polar angle θ c (t) of a perfectly conducting sphere for bipolar charging versus time for various values of Σ from Eq. ((18b)). (a), (b) 0 ≤ Σ ≤ 1 (i.e., ρ + μ + > − ρ μ ) for positive charging and (c), (d) − 1 ≤ Σ ≤ 0 (i.e., − ρ μ  > ρ + μ +) for negative charging.

FIG. 5.

(Color online) Electric field lines for Σ = 0.95 and various times after a uniform z-directed electric field is turned on at t = 0 around a perfectly conducting sphere of radius R surrounded by a dielectric with permittivity ε, conductivity σ, and free mobile positive charge with uniform positive charge density ρ + and mobility μ +, and free mobile negative charge with uniform negative charge density ρ and mobility μ . The thick electric field lines terminate on the particle at r = R and θ = θ c where Er (r = R) = 0 and separate field lines that terminate on the sphere from field lines that go around the sphere. The cylindrical radius Ra (t) of Eq. (12) describes the separation field line at z → + ∞ and defines the region where negative mobile charge current charges the sphere. The cylindrical radius Rb (t) of Eq. (13) describes the separation field line at z → − ∞ and defines the region where positive mobile charge current charges the sphere. (a) ; (b) ; (c) ; and (d) .

FIG. 6.

The perfectly conducting sphere’s (a) saturation charge, Q a (t → ∞)/Q s and (b) saturation critical angle as a function of Σ, θ c (t → ∞). Note for non-unipolar cases (i.e., Σ ≠ ± 1) the sphere does not fully charge to the saturation charge Qs . Consequently, the critical angle of the charging area will be 0 < θ c  < π.

FIG. 7.

(Color online) Electric field lines for t → ∞ and opposite polarity values of Σ when a uniform z-directed electric field is applied around a perfectly conducting sphere of radius R surrounded by a dielectric with permittivity ε, conductivity σ, and free mobile positive charge with uniform positive charge density ρ + and mobility μ +, and free mobile negative charge with uniform negative charge density ρ and mobility μ . The thick electric field lines terminate on the particle at r = R and θ = θ c where Er (r = R) = 0 and separate field lines that terminate on the sphere from field lines that go around the sphere. (a) ; and (b) .

FIG. 8.

(Color online) The (a) upper Ra and (b) lower Rb cylindrical radii of the charging window at z → ± ∞ that charges a perfectly conducting sphere for bipolar charging versus time for various values of 0 ≤ Σ ≤ 1 (i.e., ρ + μ +> − ρ μ ). Note that for − 1 ≤ Σ ≤ 0 (i.e., − ρ μ  > ρ + μ +) the upper Ra and lower Rb cylindrical radii are identical to (a) and (b) except that they are interchanged, such that Ra decreases while Rb increases with time and larger .

/content/aip/journal/jap/109/8/10.1063/1.3563074
2011-04-22
2014-04-19

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