Spherical Void in Dielectric
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 46
11-8-2016
1589
Spherical Void in Dielectric
Suppose there is a spherical void of radius R in an otherwise homogeneous material of dielectric constant ε (see Figure 1.1). At the center of the void is a point dipole p. Solve for the electric field everywhere.
Figure 1.1
SOLUTION
We expect the dipole to induce some charge in the dielectric which would create a constant electric field inside the void, proportional to the dipole moment p. Therefore the field inside is due to the dipole field plus (presumably) a constant field. The field outside is the “screened” dipole field, which goes to zero at infinity. We look for a solution in the form
(1)
(2)
where n is normal to the surface of the void (see Figure 1.2). Use the
Figure 1.2
boundary conditions to find the coefficients α and β
and D = εE, so
Write (1) and (2) at some point P on the surface of the void
(3)
(4)
where θ is the angle between p and the normal to the surface of the void,
(5)
(6)
We solve for α and β in (5) and (6):
(7)
(8)
and find
(9)
(10)
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