3.49. Find the potential cp (x, y) of an electrostatic field E
2axyi a (x 2 — y 2 ) j, where a is a constant, i and j are the unit
vectors of the x and y axes.
3.50. Determine the potential cp (x, y, z) of an electrostatic field
E = ayi (ax bz) j byk, where a and b are constants, i, j, k
are the unit vectors of the axes x, y, z.
3.51. The field potential in a certain region of space depends only
on the x coordinate as cp = — ax 3 b, where a and b are constants.
Find the distribution of the space charge p (x).
3.52. A uniformly distributed space charge fills up the space be-
tween two large parallel plates separated by a distance d. The poten-
tial difference between the plates is equal to Ay. At what value of
charge density p is the field strength in the vicinity of one of the
plates equal to zero? What will then be the field strength near
the other plate?
3.53. The field potential inside a charged ball depends only on
the distance from its centre as cp = are b, where a and b are cons-
tants. Find the space charge distribution p (r) inside the ball.
3.2. Conductors and Dielectrics in an Electric Field
- Electric field strength near the surface of a conductor in vacuum:
En = crieo. (3.2a) - Flux of polarization P across a closed surface:
dS = —q', (3.2b)
where q' is the algebraic sum of bound charges enclosed by this surface.- Vector D and Gauss's theorem for it:
D = e,) E P, (11) dS = q,^ (3.2c)where q is the algebraic sum of extraneous charges inside a closed surface.
- Relations at the boundary between two dielectrics:
Pan—Pin=—a', D271 = a, E2T = Err,^ (3.2d)
where a' and a are the surface densities of bound and extraneous charges, and
the unit vector n of the normal is directed from medium 1 to medium 2. - In isotropic dielectrics:
P = xe 0 E, D = ce 0 E, e = 1 x. (3.2e) - In the case of an isotropic uniform dielectric filling up all the space
between the equipotential surfaces:
E = Ede.^ (3.2f)
3.54. A small ball is suspended over an infinite horizontal con-
ducting plane by means of an insulating elastic thread of stiffness k.
As soon as the ball was charged, it descended by x cm and its sepa-
ration Horn the plane became equal to 1. Find the charge of the
ball.
HI