Boundary Value Problems With Linear Dielectrics We [PDF]

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BOUNDARY VALUE PROBLEMS WITH LINEAR DIELECTRICS We have shown that the bound volume charge density bound is proportional to the free charge volume density



What happens at the boundary / interface between two Linear dielectrics?



We have the boundary condition



boundary condition



for linear dielectrics



And



(eq. 2.34)



Example



Consider a hemispherical linear dielectric of radius R placed in between two infinite conducting parallel plates



far away from the hemisphere (r>>R)



We want to know/determine the following quantities: Inside the dielectric (r < R): Outside the dielectric (r > R):



Since there is NO volume free charge density inside the dielectric, therefore



However, at r=R: i.e. a bound surface charge density will exist on/at the surface of the hemispherical dielectric. 1. Since



, then



2. Note also that this problem has azimuthal / axial symmetry, therefore V, E, D, P have NO ϕ -dependence Therefore, the general solution can be represented in terms of Legendré polynomials



boundary conditions



since



Now, we can solve this problem directly.



Example 4.7 „ A dielectric sphere is placed in a Uniform electric field, find the electric field inside the sphere.



„ Boundary conditions:



Example 4.7 (conti.) „ Solution of Laplace’s equation



„ Therefore



BC3



Example 4.7 (conti.) BC1



BC2



Example 4.7 (conti.) „ Therefore



Energy in dielectric systems „ As ρf is increased by an amount Δρf, the work done is



„ Since



„ integrating by parts



By divergence theorem, vanishes as →∞



Energy in dielectric systems (conti.) „ Therefore „ For linear dielectric material



„ compare



Forces on dielectrics w



L dielectric



x



Assume Q=constant



Forces on dielectrics (conti.) „ Therefore



„ In this case,



( Check it by yourself)