Concept explainers
(a) Use Maxwell’s first equation. đ�›�. D=Pv, to describe the variation of the electric field intensity with x in a region in which no charge density exists and in which a nonhomogeneous dielectric has a permittivity that increases exponentially with x, The field has an x component on1y; (b) repeat part a, but with a radially directed electric field (spherical coordinates) in which again pv = 0, but in which the permittivity decrease exponentially with r.
(a)
Electric field intensity for the given conditions.
Answer to Problem 3.30P
Electric field intensity is
Explanation of Solution
Given Information:
There is no charge density in the region. The nonhomogeneous dielectric permittivity increases exponentially with x.
Concept used:
The expression for permittivity
Using Maxwell's first equation,
General solution for first order non-homogeneous function is
Calculation:
The expression for permittivity
Using Maxwell's first equation,
Differentiating the expression with respect to x we have,
General solution for first order non-homogeneous function is
Using the expression to find the expression for electric field intensity,
Here
Thus, the electric field intensity is
(b)
Electric field intensity for the given conditions.
Answer to Problem 3.30P
Explanation of Solution
Given Information:
Permittivity decreases exponentially with r
General solution for first order non-homogeneous function is
Concept used:
The expression for permittivity in spherical coordinates is,
Expression for
Calculation:
Expression for
Simplifying the expression to find the expression for electric field intensity,
General solution for first order non-homogeneous function is
Using this expression to find the expression for electric field intensity
Conclusion:
Thus, electric field intensity in spherical coordinates is
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Chapter 3 Solutions
Engineering Electromagnetics
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