– Electrie feld strength sear the surface of a cenducter in vacuum:
\[
\boldsymbol{E}_{\mathrm{n}}=\boldsymbol{a} / \mathrm{t}_{\mathbf{6}} \text {. }
\]
– Fhux of pelariation $\mathbf{P}$ across = closed surface:
\[
\oint \mathbf{P} \mathbf{s}=-r^{\prime} \text {. }
\]
shere $f^{\prime}$ is the alpobraic sum of bound eharges enclosed by this narface.
– Vecter D and Gause’s theoren for it:
\[
\mathrm{D}=\epsilon_{4} \mathrm{E}+\boldsymbol{r}, \quad \oint \mathrm{D} \text { as }=\mathrm{s} .
\]
stere of is the algalraic sus of exinanewa charges inside s closed surface.
– Relations at the boundary Setween two dielectrics:
\[
P_{1 n}-P_{1 n}=-\sigma^{\prime}, \quad D_{n n}-D_{1 n}=0, \quad E_{n}=E_{1 n} .
\]
shere $\sigma^{\prime}$ and $\sigma$ are the murlace densities of bound and extraneoss charges, and the unit vector of the normal is directed from medium 1 to medium 2 .
– In imotropie dielectrics:
\[
\mathbf{P}=\mathrm{xt}, \mathbf{E}, \quad \mathrm{D}=\epsilon, \mathrm{E}, \quad \epsilon=1+\mathrm{x} .
\]
– In the case of an isotropie uniform dielectric filling up all the space het seen the equipotential sarlaces:
\[
\mathbf{E}=\mathbf{E} \cdot \mathbf{t} \text {. }
\]
3.54. A small ball is suspended over an infinite herizontal conducting plane by means of an insulating elastic thread of stiffness $k$. As soon as the ball was charged, it descended by $x \mathrm{~cm}$ and its separation thom 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 $/ / 2$ 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.
3.57. A point charge $q$ is located between two mutually perpendieular conducting half-planes. Its distance from each hall-plane 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 $\mathbf{p}$ 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 earrying a eharge $\lambda$ per unit length is oriented parallel to the infinite conducting plahe. The distance between the thread and the plane is equal to $L$. Find:
(a) the modulus of the vector of the force acting on anit length of the thread;
(b) the distribution of surface change density $\sigma(x)$ over the plane, where $x$ is the distance from the plane perpendicular to the conducting surface and passing through the thread.
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 eharge of linear density $\lambda$. Suppose the point $O$ is the trace of the thread on the plane. Find the surface density of the induced eharge on the plane
(a) at the point $O$;
(b) as a function of a distance $r$ from the poist $O$.
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 $I$ 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 electrie field at the centre of the ring.
3.63. Find the potential $s$ of an uncharged conducting sphere outside of which a point eharge $q$ is located at a distance Ifrom 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 $n_{h}$ and $h_{4}$ respectively. Find the potential at the point $O$ if $r<h_{1}$.
3.65. A system consists of two concentric condueting spheres, with the inside sphere of radius a carrying a positive charge q.: What charge $\%_{1}$ has to be deposited on the outside sphere of radius $b$ to reduce the potential of the inside sphere to rero? How does the potential 8 depend in this case on a distance $r$ from the centre of the system? Draw the approsimate plot of this dependence.
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 inter-
Fis. 3.8.
connected by means of a conductor while a potential difference $\Delta \varphi$ is applied to internal plates. Find:
(a) the values of the electrie field strength between neighbouring plates;
(b) the total charge per unit area of each plate.
3.67. Two infinite condueting plates 1 and 2 are separated by – distance 1 . A point charge $q$ is located between the plates at a distance $x$ from plate 1. Find the eharges induced on each plate.
3.68. Find the electric lorce experienced by a charge reduced to a unit area of an arbitrary conductor if the surface density of the charge equals $\sigma$.
3.69. A metal ball of radius $R=1.5 \mathrm{~cm}$ has a charge $q=10 \mu \mathrm{C}$. Find the modulas of the vector of the resultant force acting on a charge located on one half of the ball.
3.70. When an uncharged conducting ball of radius $R$ is placed in an external uniform electric field, a surface charge density $\theta=$ $=\sigma_{0} \cos \theta$ is induced on the ball’s surface (here $\sigma_{0}$ is a constant, $\theta$ is a polar angle). Find the magnitude of the resultant electric force acting on an induced charge of the same sign.
3.71. An electrie field of strength $E=1.0 \mathrm{kV} / \mathrm{cm}$ produces polarisation in water equivalent to the correct orientation of only one out of $\boldsymbol{N}$ molecules. Find $\boldsymbol{N}$. The electric moment of a water molecule equals $P=0.62 \cdot 10^{-10} \mathrm{C} \cdot \mathrm{m}$.
3.72. A non-polar molecule with polarizability $\beta$ is located at – great distance $l$ from a polar molecule with electrie 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 non-polar molecule is located at the axis of a thin uniformly charged ring of radius $\boldsymbol{R}$. 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 $F_{x}(x)$.
3.74. A point charge $q$ is located at the centre of a ball made of uniform isotropie dielectric with permittivity $\varepsilon$. Find the polarization $\mathrm{P}$ as a function of the radius vector $\mathrm{r}$ relative to the centre of the system, as well as the charge $q^{\prime}$ inside a sphere whose radius is less than the radius of the ball.
3.75. Demonstrate that at a dielectric-conductor interface the surface density of the dielectric’s bound charge $\sigma^{\prime}=-\sigma(\varepsilon-1) / \varepsilon$, where $\varepsilon$ is the permittivity, $\sigma$ 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 $\&$ (Fig. 3.9).
Fie. 3.9 .
Fie. 3.10.

Find the total bound charges at the inser 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 s vs the distance $r$ from the centre of the layer if the dielectric has a certain positive extraneous charge distributed uniformly:
(a) ever the internal surface of the layer; (b) over the volume of the layer.
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 $E_{0}=10.0 \mathrm{~V} / \mathrm{m}$, the angle between the vector $\mathbf{E}_{\text {, }}$, and the normal $n$ of the boundary line being equal to $\alpha_{4}=30^{\circ}$. Find the field strength $E$ in glass near the point $A$, the angle $a$ between the vector $\mathbf{E}$ and $\mathrm{n}$, as well as the surface density of the bound charges at the point $\boldsymbol{A}$.
3.79. Near the plane surlace of a uniform isotropic dielectric with permittivity $\&$ the electric field strength in vacuum is equal to $E_{6}$, the vector $\mathrm{E}_{4}$ forming an angle $\theta$ with the normal of the dielectrie’s surface (Fig. 3.11). Assuming the field to be uniform both inside and outside the dielectric, find:
(a) the fux of the vector $\mathbf{E}$ through a sphere of radius $R$ with centre located at the surface of the dielectric;
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(b) the circulation of the vector $\mathbf{D}$ around the closed path $\mathbf{\Gamma}$ of length $l$ (see Fig. 3.11) whose plane is perpendicular to the surface of the dielectric and parallel to the vector $\mathbf{E}_{8}$.
Fie. 3.11.
3.80. An infinite plane of uniform dielectric with permittivity e is uniformly charged with extraneous charge of space density $\rho$. The thickness of the plate is equal to $2 d$. Find:
(a) the magnitude of the electrie 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 $E_{N}(x)$ of the vector $\mathrm{E}$ and the potential $q(x)$;
(b) the surface and space densities of the bound charge.
3.81. Extraneous charges are uniformly distributed with space density $\rho>0$ over a ball of radius $\boldsymbol{n}$ made of uailorm isotropic dielectric with permittivity e. 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 $\varphi(r)$ :
(b) the space and surface densities of the bound charges.
3.82. A round dielectric disc of radius $R$ and thickness $d$ is statically polarized so that it gains the uniform polarization $\mathbf{P}$, with the vector $\mathbf{P}$ lying in the plane of the dise. Find the strength $\mathbf{E}$ of the electric field at the centre of the dise if $d<R$.
3.83. Under certain conditions the polarization of an infinite uncharged dielectric plate takes the form $\mathbf{P}=\mathbf{P}_{0}\left(1-x^{2} / d^{7}\right)$, where $\mathbf{P}_{\text {, is a }}$ 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 surtaces.
3.84. Initially the space between the plates of the capacitor is flled with air, and the field strength in the gap is equal to $E_{\text {, }}$. Then half the gap is filled with
Fis. 3.12. uniform isotropic dielectric with permittivity $\varepsilon$ as shown in Fig. 3.12. Find the moduli of the vectors $\mathbf{E}$ and $\mathbf{D}$ in both parts of the gap ( $I$ and 2 ) if the introduction of the dielectrie
(a) does not change the voltage across the plates;
(b) leaves the charges at the plates constant.
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.
คie. 3.13.
Fig. 3.14.
3.86. Half the space between two concentric electrodes of a spherical capacitor is filled, as shown in Fig. 3.14, with uniform isotropic dielectrie with permittivity $\boldsymbol{\epsilon}$. The eharge of the capacitor is 4 . Find the magnitude of the electrie field strength between the electrodes as a function of distance $r$ from the eurvature centre of the electrodes.
3.87. Two small identical balls carrying the eharges of the same sign are suspended from the same point by insulating threads of equal length. When the surrounding space was flled with kerosene the divergence angle between the threads remained constant. What is the density of the material of which the balls are made?
3.88. A uniform electric field of strength $E=100 \mathrm{~V} / \mathrm{m}$ is generated inside a ball made of uniform isotropic dielectrie with permittivity $\varepsilon=5.00$. The radius of the ball is $R=3.0 \mathrm{~cm}$. Find the maximum surface density of the bound charges and the total bound charge of one sign.
3.89. A point eharge $q$ 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 $\mathrm{e}$. 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 $t \rightarrow 0$;
(b) the total bound charge on the surface of the dielectric.
3.90. Making use of the formulation and the solution of the foregoing problem, find the magnitude of the force exerted by the charges bound on the surface of the dielectric on the point change q.
3.91. A point charge $q$ is located on the plane dividing vacuum and infinite uniform isotropie dielectrie with permittivity e. Find the moduli of the vectors $\mathbf{D}$ and $\mathbf{E}$ as well as the potential $\varphi$ as functions of distance $r$ from the charge $q$.
3.92. A small conducting ball carrying a charge qq is located in a uniform isotropie dielectrie with permittivity o at a distance $t$ from an infinite boundary plane between the dielectric and vacuum. Find the surface density of the bound charges on the boundary plane as a function of distance $r$ from the ball. Analyse the obtained result for $t \rightarrow 0$.
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3.93. A half-space filled with uniform isotropic dielectrie with permittivity $\varepsilon$ has the conducting boundary plane. Inside the dieleciric, at a distance $l$ from this plane, there is a small metal ball possessing a charge \&. Find the surface density of the bound charges at the boundary plane as a function of distance $r$ from the ball.
3.94. A plate of thickness $h$ made of uniform statically polarized dielectric is placed inside a capaciter whose parallel plates are interconnected by a conductor. The polarization of the dielectric is equal
Fie. 3.15.
to $\mathbf{P}$ (Fig. 3.15). The separation between the capacitor plates is $d$. Find the strength and induction vectors for the electric field both inside and outside the plates.
3.95. A long round dielectric cylinder is polarized so that the vector $\mathbf{P}=a r$, where $a$ is a positive constant and $r$ is the distance from the axis. Find the space density $p^{\prime}$ of bound changes as a function of distance $r$ trom the axis.
3.96. A dielectrie ball is polarined uniformly and statically. It 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 eharges of the dielectrie relative to all negative eharges,
(a) find the electric field strength $\mathbf{E}$ 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 $\varphi_{0}=\mathrm{P}_{0} \mathbf{r} / 4 \pi \varepsilon_{6}$. where $\mathrm{P}_{4}$ is the electric moment of the ball, and $r$ is the distance from its centre.
3.97. Utilizing the solution of the foregoing problem, find the electric field strength $\mathbf{E}_{8}$ in a spherical cavity is an infinite statically polarized uniform dielectric if the dielectric’s polarization is $\mathbf{P}$, and far from the cavity the field strength is $\mathbf{E}$.
3.98. A uniform dielectne ball is placed in a uniform electric field of strength $\mathbf{E}_{\text {, }}$. Under these conditions the dielectric becomes polarized uniformly. Find the electric field strength $\mathbf{E}$ inside the ball and the polarization $\mathbf{P}$ of the dielectric whose permittivity equals $\varepsilon$. Make use of the result ebtained in Problem 3.96.
3.99. An infinitely long round diclectric cylinder is polarized úniformly and statically, the polarization $\mathbf{P}$ being perpendicular to the axis of the eylinder. Find the electric field strength $\mathbf{E}$ inside the dielectric.
3.100. A long round cylinder made of uniform dielectric is placed in a uniform electric field of strength $\mathbf{E}_{8}$. The axis of the cylinder is perpendicular to vector $\mathbf{E}_{5}$. Under these conditions the dielectric becomes polarized uniformly. Making use of the result obtained in the foregoing problem, find the electric field strength $\mathbf{E}$ in the cylinder and the polarization $\mathbf{P}$ of the dielectric whose permittivity is equal to $\mathrm{c}$.