# 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. Free solution >> 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. Free solution >> 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: (a) the modulus of the vector of the electric force acting on each charge; (b) the magnitude of the electric field strength vector at the midpoint between these charges. Free solution >> 3.57. A point charge q is located between two mutually perpendicular conducting half-planes. Its distance from each half-plane is equal to l. Find the modulus of the vector of the force acting on the charge. Free solution >> 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. Free solution >> 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. Free solution >> 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: (a) the modulus of the vector of the force acting on a unit length of the thread; (b) the distribution of surface charge density σ(x) over the plane, where x is the distance from the plane perpendicular to the conducting surface and passing through the thread. Free solution >> 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 (a) at the point O; (b) as a function of a distance r from the point O. Free solution >> 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: (a) the surface charge density at the point of the plane symmetrical with respect to the ring; (b) the strength and the potential of the electric field at the centre of the ring. Free solution >> 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. Free solution >> 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 R1 and R2 respectively. Find the potential at the point O if r < R1. Free solution >> 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: (a) the values of the electric field strength between neighbouring plates; (b) the total charge per unit area of each plate. Free solution >> 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. Free solution >> 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. Free solution >> 3.72. A non-polar 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. Free solution >> 3.73. A non-polar 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 (a) equal to zero; (b) maximum? Draw the approximate plot Fx(x). Free solution >> 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. Free solution >> 3.75. Demonstrate that at a dielectric-conductor 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. Free solution >> 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. Free solution >> 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: (a) over the internal surface of the layer; (b) over the volume of the layer. Free solution >> 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 E0 = 10.0 V/m, the angle between the vector E0 and the normal n of the boundary line being equal to α0 = 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. Free solution >> 3.79. Near the plane surface of a uniform isotropic dielectric with permittivity ε the electric field strength in vacuum is equal to E0, the vector E0 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: (a) the flux of the vector E through a sphere of radius R with centre located at the surface of the dielectric; (b) the circulation of the vector D around the closed path Γ of length l (see Fig. 3.11) whose plane is perpendicular to the surface of the dielectric and parallel to the vector E0. Free solution >> 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: (a) the magnitude of the electric field strength and the potential as functions of distance l from the middle point of the plane (where the potential is assumed to be equal to zero); having chosen the x coordinate axis perpendicular to the plate, draw the approximate plots of the projection Ex(x) of the vector E and the potential φ(x); (b) the surface and space densities of the bound charge. Free solution >> 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: (a) the magnitude of the electric field strength as a function of distance r from the centre of the ball; draw the approximate plots E(r) and φ(r); (b) the space and surface densities of the bound charges. Free solution >> 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. Free solution >> 3.83. Under certain conditions the polarization of an infinite uncharged dielectric plate takes the form P = P0(1 - x2/d2), where P0 is a vector perpendicular to the plate, x is the distance from the middle of the plate, d is its half-thickness. Find the strength E of the electric field inside the plate and the potential difference between its surfaces. Free solution >> 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 E0. 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 (a) does not change the voltage across the plates; (b) leaves the charges at the plates constant. Free solution >> 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. Free solution >> 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? Free solution >> 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 half-space. The permittivity of the dielectric equals ε. Find: (a) the surface density of the bound charges as a function of distance r from the point charge q; analyse the obtained result at l → 0; (b) the total bound charge on the surface of the dielectric. Free solution >> 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. Free solution >> 3.93. A half-space 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. Free solution >> 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. Free solution >> 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, (a) find the electric field strength E inside the ball; (b) demonstrate that the field outside the ball is that of a dipole located at the centre of the ball, the potential of that field being equal to φ = p0r/4πε0, where p0 is the electric moment of the ball, and r is the distance from its centre. Free solution >> 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. Free solution >>