3.31. There is an infinitely long straight thread carrying a charge
with linear density X, = 0.40 RC/m. Calculate the potential difference
between points
1
and
2
if point 2 is removed it = 2.0 times farther
from the thread than point
I.
3.32. Find the electric field potential and strength at the centre
of a hemisphere of radius
R
charged uniformly with the surface
density a.
3.33. A very thin round plate of radius
R
carrying a uniform sur-
face charge density a is located in vacuum. Find the electric field
potential and strength along the plate's axis as a function of a dis-
tance
1
from its centre. Investigate the obtained expression at
1--4-
0
and / »
R.
3.34. Find the potential
p
at the edge of a thin disc of radius
R
carrying the uniformly distributed charge with surface densi-
ty a.
3.35. Find the electric field strength vector if the potential of
this field has the form p = ar, where a is a constant vector, and r
is the radius vector of a point of the field.
3.36. Determine the electric field strength vector if the potential
of this field depends on
x,
y
coordinates as
a) cp = a (x
2
— y
2
); (b) q =
axy,
where a is a constant. Draw the approximate shape of these fields
.using lines of force (in the
x,
y plane).
3.37. The potential of a certain electrostatic field has the
form
cp
=
a (x
2
+ y
2
) +
bz
2
, where a and
b
are constants. Find the mag-
nitude and direction of the electric field strength vector. What shape
have the equipotential surfaces in the following cases:
(a) a > 0,
b>
0; (b)
a >
0,
b <
0?
3.38. A charge
q
is uniformly distributed over the volume of
a sphere of radius
R.
Assuming the permittivity to be equal to unity
throughout, find the potential
(a)
at the centre of the sphere;
(b)
inside the sphere as a function of the distance
r
from its centre.
3.39. Demonstrate that the potential of the field generated by
a dipole with the electric moment p (Fig. 3.4) may be represented as
pr/4ns
o
r
3
, where r is the radius vector.
Using this expression, find the magnitude of the
electric field strength vector as a function of r
z
and 0.
9
3.40. A point dipole with an electric moment
p
oriented in the positive direction of the z axis is
located at the origin of coordinates. Find the
p
projections
E
z
and E
1
of the electric field strength
vector (on the plane perpendicular to the z axis at
Fig. 3.4.
the point
S
(see Fig. 3.4)). At which points is
E
perpendicular to p?
3.41. A point electric dipole with a moment p is placed in the
external uniform electric field whose strength equals E
0
, with
