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Solved Problem-1

Problem-1
Two extensive homogeneous isotropic dielectrics meet on the plane z = 0. For z  0, Description: 160514.png and for z ≤ 0, Description: 160505.png A uniform electric field, Description: 160498.png kV/m exists for z  0. Find
(a) Description: 160490.png for z ≤ 0
(b) The angles between electric field intensity and the normal to the boundary surface in both media
(c) The energy densities in J/m3 in both dielectrics
(d) The energy within a cube of 2 m side centred at (3, 4, -5)
 
Solution
(a) Since, Description: 160483.png is the normal to the boundary plane, the normal component is,
Description: 160476.png
 
Description: 160466.png
Arrangement of dielectrics 
 
By boundary conditions,
(1) Description: 160457.png
 
and,
 
(2) Description: 160450.png
 
So, the field in the second medium is given as,
 
Description: 160442.png

(b) Let Description: 160431.png and Description: 160422.png be the angles Description: 160413.png and Description: 160406.png make with the interface while Description: 160399.png and Description: 160392.png are the angles they make with the normal to the interface.
 
Description: 160384.png
 
Description: 160373.png
 
 
Description: 160366.png
 
Similarly,
Description: 160359.png
 
Description: 160350.png
 
Hence, the angles between electric field intensity and the normal to the boundary surface in both media are given as,
Description: 219416.png
 
Note: The relation Description: 162465.png is satisfied.

(c) The energy densities are given as,
Description: 160306.png
 
(d) At the centre (3, 4, -5) of the cube of side 2 m, z = -5 < 0; i.e., the cube is in the region 2 with 2 ≤ x ≤ 4, 3 ≤ y ≤ 5, -6 ≤ z ≤ -4.
 
Hence, the energy within the cube is,
Description: 160297.png
 





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