3.93. A half-space filled with uniform isotropic dielectric with
permittivity a has the conducting boundary plane. Inside the dielectric, at a distance 1 from this plane, there is a small metal ball possessing a charge q. Find the surface density of the bound charges at
the boundary plane as a function of distance r from the ball.
3.94. A plate of thickness h made of uniform statically polarized
dielectric is placed inside a capacitor whose parallel plates are interconnected by a conductor. The polarization of the dielectric is equal
C
/7
Fig. 3.15.
to P (Fig. 3.15). The separation between the capacitor plates is d.
Find the strength and induction vectors for the electric field both
inside and outside the plates.
3.95. A long round dielectric cylinder is polarized so that the
vector P = ar, where a is a positive constant and r is the distance
from the axis. Find the space density p' of bound charges as a function
of distance r from the axis.
3.96. A dielectric ball is polarized uniformly and statically. Its
polarization equals P. Taking into account that a ball polarized in
this way may be represented as a result of a small shift of all positive
charges of the dielectric relative to all negative charges,
(a) find the electric field strength E inside the ball;
(b) demonstrate that the field outside the ball is that of a dipole
located at the centre of the ball, the potential of that field being
equal to q = por/4a co, where Po is the electric moment of the ball,
and r is the distance from its centre.
3.97. Utilizing the solution of the foregoing problem, find the electric field strength E0 in a spherical cavity in an infinite statically polarized uniform dielectric if the dielectric's polarization is P, and far from
the cavity the field strength is E.
3.98. A uniform dielectric ball is placed in a uniform electric
field of strength E0. Under these conditions the dielectric becomes
polarized uniformly. Find the electric field strength E inside the ball
and the polarization P of the dielectric whose permittivity equals e.
Make use of the result obtained in Problem 3.96.
3.99. An infinitely long round dielectric cylinder is polarized
uniformly and statically, the polarization P being perpendicular
to the axis of the cylinder. Find the electric field strength E inside the
dielectric.
3.100. A long round cylinder made of uniform dielectric is placed
in a uniform electric field of strength E0. The axis of the
cylinder is perpendicular to vector E0. Under these conditions
the dielectric becomes polarized uniformly. Making use of the result
117
permittivity a has the conducting boundary plane. Inside the dielectric, at a distance 1 from this plane, there is a small metal ball possessing a charge q. Find the surface density of the bound charges at
the boundary plane as a function of distance r from the ball.
3.94. A plate of thickness h made of uniform statically polarized
dielectric is placed inside a capacitor whose parallel plates are interconnected by a conductor. The polarization of the dielectric is equal
C
/7
Fig. 3.15.
to P (Fig. 3.15). The separation between the capacitor plates is d.
Find the strength and induction vectors for the electric field both
inside and outside the plates.
3.95. A long round dielectric cylinder is polarized so that the
vector P = ar, where a is a positive constant and r is the distance
from the axis. Find the space density p' of bound charges as a function
of distance r from the axis.
3.96. A dielectric ball is polarized uniformly and statically. Its
polarization equals P. Taking into account that a ball polarized in
this way may be represented as a result of a small shift of all positive
charges of the dielectric relative to all negative charges,
(a) find the electric field strength E inside the ball;
(b) demonstrate that the field outside the ball is that of a dipole
located at the centre of the ball, the potential of that field being
equal to q = por/4a co, where Po is the electric moment of the ball,
and r is the distance from its centre.
3.97. Utilizing the solution of the foregoing problem, find the electric field strength E0 in a spherical cavity in an infinite statically polarized uniform dielectric if the dielectric's polarization is P, and far from
the cavity the field strength is E.
3.98. A uniform dielectric ball is placed in a uniform electric
field of strength E0. Under these conditions the dielectric becomes
polarized uniformly. Find the electric field strength E inside the ball
and the polarization P of the dielectric whose permittivity equals e.
Make use of the result obtained in Problem 3.96.
3.99. An infinitely long round dielectric cylinder is polarized
uniformly and statically, the polarization P being perpendicular
to the axis of the cylinder. Find the electric field strength E inside the
dielectric.
3.100. A long round cylinder made of uniform dielectric is placed
in a uniform electric field of strength E0. The axis of the
cylinder is perpendicular to vector E0. Under these conditions
the dielectric becomes polarized uniformly. Making use of the result
117