Problems in General Physics → Electrodynamics → Conductors and Dielectrics in an Electric Field 
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 from the plane became equal to l. Find the charge of the ball.
3.55. A point charge q is located at a distance l from the infinite conducting plane. What amount of work has to be performed in order to slowly remove this charge very far from the plane.
3.56. Two point charges, q and q, are separated by a distance l, both being located at a distance l/2 from the infinite conducting plane. Find:
3.57. A point charge q is located between two mutually perpendicular conducting halfplanes. Its distance from each halfplane is equal to l. Find the modulus of the vector of the force acting on the charge.
3.58. A point dipole with an electric moment p is located at a distance l from an infinite conducting plane. Find the modulus of the vector of the force acting on the dipole if the vector p is perpendicular to the plane.
3.59. A point charge q is located at a distance l from an infinite conducting plane. Determine the surface density of charges induced on the plane as a function of separation r from the base of the perpendicular drawn to the plane from the charge.
3.60. A thin infinitely long thread carrying a charge λ per unit length is oriented parallel to the infinite conducting plane. The distance between the thread and the plane is equal to l. Find:
3.61. A very long straight thread is oriented at right angles to an infinite conducting plane; its end is separated from the plane by a distance l. The thread carries a uniform charge of linear density λ. Suppose the point O is the trace of the thread on the plane. Find the surface density of the induced charge on the plane
3.62. A thin wire ring of radius R carries a charge q. The ring is oriented parallel to an infinite conducting plane and is separated by a distance l from it. Find:
3.63. Find the potential φ of an uncharged conducting sphere outside of which a point charge q is located at a distance l from the sphere's centre.
3.64. A point charge q is located at a distance r from the centre O of an uncharged conducting spherical layer whose inside and outside radii are equal to R_{1} and R_{2} respectively. Find the potential at the point O if r < R_{1}.
3.66. Four large metal plates are located at a small distance d from one another as shown in Fig. 3.8. The extreme plates are interconnected by means of a conductor while a potential difference Δφ is applied to internal plates. Find:
3.68. Find the electric force experienced by a charge reduced to a unit area of an arbitrary conductor if the surface density of the charge equals σ.
3.69. A metal ball of radius R = 1.5 cm has a charge q = 10 μC. Find the modulus of the vector of the resultant force acting on a charge located on one half of the ball.
3.72. A nonpolar molecule with polarizability β is located at a great distance l from a polar molecule with electric moment p. Find the magnitude of the interaction force between the molecules if the vector p is oriented along a straight line passing through both molecules.
3.73. A nonpolar molecule is located at the axis of a thin uniformly charged ring of radius R. At what distance x from the ring's centre is the magnitude of the force F acting on the given molecule
3.74. A point charge q is located at the centre of a ball made of uniform isotropic dielectric with permittivity ε. Find the polarization P as a function of the radius vector r relative to the centre of the system, as well as the charge q' inside a sphere whose radius is less than the radius of the ball.
3.75. Demonstrate that at a dielectricconductor interface the surface density of the dielectric's bound charge σ' = σ(ε  1)/ε, where ε is the permittivity, σ is the surface density of the charge on the conductor.
3.76. A conductor of arbitrary shape, carrying a charge q, is surrounded with uniform dielectric of permittivity ε (Fig. 3.9). Find the total bound charges at the inner and outer surfaces of the dielectric.
3.77. A uniform isotropic dielectric is shaped as a spherical layer with radii a and b. Draw the approximate plots of the electric field strength E and the potential φ vs the distance r from the centre of the layer if the dielectric has a certain positive extraneous charge distributed uniformly:
3.78. Near the point A (Fig. 3.10) lying on the boundary between glass and vacuum the electric field strength in vacuum is equal to E_{0} = 10.0 V/m, the angle between the vector E_{0} and the normal n of the boundary line being equal to α_{0} = 30°. Find the field strength E in glass near the point A, the angle α between the vector E and n, as well as the surface density of the bound charges at the point A.
3.79. Near the plane surface of a uniform isotropic dielectric with permittivity ε the electric field strength in vacuum is equal to E_{0}, the vector E_{0} forming an angle θ with the normal of the dielectric's surface (Fig. 3.11). Assuming the field to be uniform both inside and outside the dielectric, find:
3.80. An infinite plane of uniform dielectric with permittivity ε is uniformly charged with extraneous charge of space density ρ. The thickness of the plate is equal to 2d. Find:
3.81. Extraneous charges are uniformly distributed with space density ρ > 0 over a ball of radius R made of uniform isotropic dielectric with permittivity ε. Find:
3.82. A round dielectric disc of radius R and thickness d is statically polarized so that it gains the uniform polarization P, with the vector P lying in the plane of the disc. Find the strength E of the electric field at the centre of the disc if d << R.
3.83. Under certain conditions the polarization of an infinite uncharged dielectric plate takes the form P = P_{0}(1  x^{2}/d^{2}), where P_{0} is a vector perpendicular to the plate, x is the distance from the middle of the plate, d is its halfthickness. Find the strength E of the electric field inside the plate and the potential difference between its surfaces.
3.84. Initially the space between the plates of the capacitor is filled with air, and the field strength in the gap is equal
to E_{0}. Then half the gap is filled with uniform isotropic dielectric with permittivity ε as shown in Fig. 3.12. Find the moduli of the vectors E and D in both parts of the gap (1 and 2) if the introduction of the dielectric
3.85. Solve the foregoing problem for the case when half the gap is filled with the dielectric in the way shown in Fig. 3.13.
3.87. Two small identical balls carrying the charges of the same sign are suspended from the same point by insulating threads of equal length. When the surrounding space was filled with kerosene the divergence angle between the threads remained constant. What is the density of the material of which the balls are made?
3.89. A point charge q is located in vacuum at a distance l from the plane surface of a uniform isotropic dielectric filling up all the halfspace. The permittivity of the dielectric equals ε. Find:
3.91. A point charge q is located on the plane dividing vacuum and infinite uniform isotropic dielectric with permittivity ε. Find the moduli of the vectors D and E as well as the potential φ as functions of distance r from the charge q.
3.93. A halfspace filled with uniform isotropic dielectric with permittivity ε has the conducting boundary plane. Inside the dielectric, at a distance l from this plane, there is a small metal ball possessing a charge q. Find the surface density of the bound charges at the boundary plane as a function of distance r from the ball.
3.95. A long round dielectric cylinder is polarized so that the vector P = αr, where α is a positive constant and r is the distance from the axis. Find the space density ρ' of bound charges as a function of distance r from the axis.
3.96. A dielectric ball is polarized uniformly and statically. Its polarization equals P. Taking into account that a ball polarized in this way may be represented as a result of a small shift of all positive charges of the dielectric relative to all negative charges,
3.99. An infinitely long round dielectric cylinder is polarized uniformly and statically, the polarization P being perpendicular to the axis of the cylinder. Find the electric field strength E inside the dielectric.
