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Journal of Microwave Power and Electromagnetic Energy, 48 (4), 2014, pp. 221-232.
A Publication of the International Microwave Power Institute
Analytical Expression of the Electric Field
Induced by Uniformly Charged 3D Plates
Jérémy Kerneis
PLOSC Brest, 38 rue Yves Collet, 29200 Brest, France
Received: July 8,2014
Accepted: October 17, 2014
ABSTRACT
This paper establishes the analytical expression of the electric field induced by a uniformly
charged plate (parallelepiped), which is useful for describing more complex configurations by
addition of uniformly charged parallelepiped. Examples of applications involve capacitors
(electrical engineering) and electrochemotherapy (medicine). The asymptotic behavior, at
the origin and far away from the plates, is firstly discussed, followed by a comparison with an
existing software, and finally the electric field (vector field and magnitude) is shown for two
3D orthogonal plates with equal or opposite charge density.
KEYWORDS: Electric field, uniformly charged plate.
INTRODUCTION
The electric field between two uniformly charged thin plates constituting a capacitor
is generally estimated using the relation E=q/(εA) where q is the electric charge, ε is the
electric permittivity of the material between the plates, and A the surface of each plate. This
expression is a simplification that can be easily deduced by studying the electric field between
two infinite plates in cylindrical coordinates. Other methods have been developed to estimate
the parameters of capacitors, possibly in different configurations: energetic models [Al Jaber,
2000], fictitious currents [Ravaud, 2010], finite elements [Nishiyama, 1994] and [CatalanIzquierdo, 2009], calculations in hall plates [Raman, 2013], non parallel plates [BuenoBarrachina, 2011]. In this context, the present paper provides the possibility to analytically
calculate the magnitude and direction of the electric field induced by a parallelepiped, i.e.
an inclined capacitor can be easily described as the sum of two uniform plates.
Another domain interested by electric field calculation (also induced by thin plates)
is biomedical engineering. As explained by Corovic, [2007] electropermeabilization is a
phenomenon, which permits the membrane of a cell to become permeable under an electric
field of magnitude 200-400 V/cm. This phenomenon is very important in novel promising
treatments such as electrochemotherapy and gene electrotransfers. More details on this
subject can be found in Zimmerman [1982] and Weaver [1996].
EXPRESSION OF THE ELECTRIC FIELD
We consider a uniformly charged parallelepiped with charge density ρ and dimensions
2a, 2b, 2c such that the total charge of the object is:
Q = 8 a b c ρ (1)
The following equations for the electric field are relative to the reference frame where O is located at the center of mass of the object (assuming that the mass
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