3.157. Two metal balls of the same radius a are located in a homogeneous poorly conducting medium with resistivity p. Find the
resistance of the medium between the balls provided that the separation between them is much greater than the radius of the ball.
3.158. A metal ball of radius a is located at a distance 1 from an
infinite ideally conducting plane. The space around the ball is filled
with a homogeneous poorly conducting medium with resistivity p.
In the case of a <1 find:
(a) the current density at the conducting plane as a function of
distance r from the ball if the potential difference between the ball
and the plane is equal to V;
(b) the electric resistance of the medium between the ball and
the plane.
3.159. Two long parallel wires are located in a poorly conducting
medium with resistivity p. The distance between the axes of the
wires is equal to 1, the cross-section radius of each wire equals a.
In the case a <1 find:
(a) the current density at the point equally removed from the axes
of the wires by a distance r if the potential difference between the
wires is equal to V;
(b) the electric resistance of the medium per unit length of the
wires.
3.160. The gap between the plates of a parallel-plate capacitor
is filled with glass of resistivity p = 100 GQ•m. The capacitance
of the capacitor equals C = 4.0 nF. Find the leakage current of the
capacitor when a voltage V = 2.0 kV is applied to it.
3.161. Two conductors of arbitrary shape are embedded into an
infinite homogeneous poorly conducting medium with resistivity
p and permittivity e. Find the value of a product RG for this system,
where R is the resistance of the medium between the conductors,
and C is the mutual capacitance of the wires in the presence of the
medium.
3.162. A conductor with resistivity p bounds on a dielectric with
permittivity a. At a certain point A at the conductor's surface the
electric displacement equals D, the vector D being directed away
from the conductor and forming an angle a with the normal of the
surface. Find the surface density of charges on the conductor at the
point A and the current density in the conductor in the vicinity of
the same point.
3.163. The gap between the plates of a parallel-plate capacitor
is filled up with an inhomogeneous poorly conducting medium whose
conductivity varies linearly in the direction perpendicular to the
plates from o = 1.0 pS/m to o-, = 2.0 pS/m. Each plate has an
area S = 230 cm2, and the separation between the plates is d =
= 2.0 mm. Find the current flowing through the capacitor due to
a voltage V = 300 V.
3.164. Demonstrate that the law of refraction of direct current
lines at the boundary between two conducting media has the form
resistance of the medium between the balls provided that the separation between them is much greater than the radius of the ball.
3.158. A metal ball of radius a is located at a distance 1 from an
infinite ideally conducting plane. The space around the ball is filled
with a homogeneous poorly conducting medium with resistivity p.
In the case of a <1 find:
(a) the current density at the conducting plane as a function of
distance r from the ball if the potential difference between the ball
and the plane is equal to V;
(b) the electric resistance of the medium between the ball and
the plane.
3.159. Two long parallel wires are located in a poorly conducting
medium with resistivity p. The distance between the axes of the
wires is equal to 1, the cross-section radius of each wire equals a.
In the case a <1 find:
(a) the current density at the point equally removed from the axes
of the wires by a distance r if the potential difference between the
wires is equal to V;
(b) the electric resistance of the medium per unit length of the
wires.
3.160. The gap between the plates of a parallel-plate capacitor
is filled with glass of resistivity p = 100 GQ•m. The capacitance
of the capacitor equals C = 4.0 nF. Find the leakage current of the
capacitor when a voltage V = 2.0 kV is applied to it.
3.161. Two conductors of arbitrary shape are embedded into an
infinite homogeneous poorly conducting medium with resistivity
p and permittivity e. Find the value of a product RG for this system,
where R is the resistance of the medium between the conductors,
and C is the mutual capacitance of the wires in the presence of the
medium.
3.162. A conductor with resistivity p bounds on a dielectric with
permittivity a. At a certain point A at the conductor's surface the
electric displacement equals D, the vector D being directed away
from the conductor and forming an angle a with the normal of the
surface. Find the surface density of charges on the conductor at the
point A and the current density in the conductor in the vicinity of
the same point.
3.163. The gap between the plates of a parallel-plate capacitor
is filled up with an inhomogeneous poorly conducting medium whose
conductivity varies linearly in the direction perpendicular to the
plates from o = 1.0 pS/m to o-, = 2.0 pS/m. Each plate has an
area S = 230 cm2, and the separation between the plates is d =
= 2.0 mm. Find the current flowing through the capacitor due to
a voltage V = 300 V.
3.164. Demonstrate that the law of refraction of direct current
lines at the boundary between two conducting media has the form