Irodov – Problems in General Physics

PART THREE. ELECTRODYNAMICS

3.1. Constant Electric Field in Vacuum

• Strength and potential of the field of a point charge q:
  1 q
  E=— r, 4nso r 3

1 q

ID 4ns, F.- (3.1a)

• Relation between field strength and potential:
  E = —VT,^ (3.1b)
  i.e. field strength is equal to the antigradient of the potential.


• Gauss's theorem and circulation of the vector E:

IS)E dS = q/so, 1;;,E dr = 0.^ (3.1c)

• Potential and strength of the field of a point dipole with electric mo-
  ment p:
  1 pr
  4ns, r 3

1

eo

p

E = 4n (^) r 3 171+3cos 2 0, (3.1d)
where 0 is the angle between the vectors r and p.

• Energy W of the dipole p in an external electric field, and the moment
  N of forces acting on the dipole:
  W = —pE, N = [pE].^ (3.1e)


• Force F acting on a dipole, and its projection Fx:
  DE

F=p-- 0/ (^) ' (3.1f)
where alai is the derivative of the vector E with respect to the dipole direction,
VE is the gradient of the function E x.
3.1. Calculate the ratio of the electrostatic to gravitational inter-
action forces between two electrons, between two protons. At what
value of the specific charge qlm of a particle would these forces be-
come equal (in their absolute values) in the case of interaction of
identical particles?
3.2. What would be the interaction force between two copper
spheres, each of mass 1 g, separated by the distance 1 m, if the total
electronic charge in them differed from the total charge of the nuclei
by one per cent?
3.3. Two small equally charged spheres, each of mass m, are
suspended from the same point by silk threads of length 1. The
distance between the spheres x << 1. Find the rate dqldt with which