3.49. Find the potential cp (x, y) of an electrostatic field E 2axyi a (x2 — y2) 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 = — ax3 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 between two large parallel plates separated by a distance d. The potential 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 constants. 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: (3.2d) Pan — Pin= — a', D271 = a, E2T = Err, 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: x. P = xe0 E, D = ce0 E, e = 1 (3.2e) In the case of an isotropic uniform dielectric filling up all the space • between the equipotential surfaces: (3.2f) E = Ede.
3.54. A small ball is suspended over an infinite horizontal conducting 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 separation Horn the plane became equal to 1. Find the charge of the ball. HI