– Strength and potential of the field of a point charge $q$ :
\[
\mathbf{E}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r^{3}} \mathbf{r}, \quad \varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r} .
\]
– Relation between field strength and potential:
\[
\mathbf{E}=-\boldsymbol{
abla} \varphi \text {, }
\]
i.e. field strength is equal to the antigradient of the potential.
– Gauss’s theorem and circulation of the vector $\mathbf{E}$ :
\[
\oint \mathbf{E} d \mathbf{S}=q / \varepsilon_{0}, \quad \oint \mathbf{E} d \mathbf{r}=0 .
\]
– Potential and strength of the field of a point dipole with electric moment p:
\[
\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{pr}}{r^{3}}, \quad E=\frac{1}{4 \pi \varepsilon_{0}} \frac{p}{r^{3}} \sqrt{1+3 \cos ^{2} \theta},
\]
where $\theta$ is the angle between the vectors $\mathbf{r}$ and $\mathbf{p}$.
– Energy $W$ of the dipole $\mathbf{p}$ in an external electric field, and the moment $\mathbf{N}$ of forces acting on the dipole:
\[
W=-\mathrm{pE}, \quad \mathrm{N}=[\mathrm{pE}] .
\]
– Force $\mathbf{F}$ acting on a dipole, and its projection $F_{x}$ :
\[
\mathbf{F}=p \frac{\partial \mathrm{E}}{\partial l}, \quad F_{x}=\mathbf{p} \cdot \boldsymbol{
abla} E_{x},
\]
where $\partial \mathbf{E} / \partial l$ is the derivative of the vector $\mathbf{E}$ with respect to the dipole direction, $
abla E_{x}$ is the gradient of the function $E_{x}$.
3.1. Calculate the ratio of the electrostatic to gravitational interaction forces between two electrons, between two protons. At what value of the specific charge $q / m$ of a particle would these forces become 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 $1 \mathrm{~g}$, separated by the distance $1 \mathrm{~m}$, 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 $l$. The distance between the spheres $x \ll l$. Find the rate $d q / d t$ with which the charge leaks off each sphere if their approach velocity varies as $v=a / V \bar{x}$, where $a$ is a constant.
3.4. Two positive charges $q_{1}$ and $q_{1}$ are located at the points with radius vectors $r_{1}$ and $r_{8}$. Find a negative charge $f_{0}$ and a radius vector $r$, of the point at which it has to be placed for the force acting on each of the three charges to be equal to zero.
3.5. A thin wire ring of radius, thas an electrie charge q. What will be the increment of the force stretching the wire if a point charge 4. is placed at the ring’s centme?
3.6. A positive point charge $50 \mu \mathrm{C}$ is lecated in the plane $x y$ at the point with radius vector $\mathbf{r}_{0}=2 \mathbf{i}+3 \mathbf{j}$, where $\mathbf{i}$ and $j$ are the unit vectors of the $x$ and $y$ axes. Find the vector of the electric field strength $\mathbf{E}$ and its magnitude at the point with radius vector $\mathbf{r}=8 \mathbf{i}-5 \mathbf{j}$. Here $r_{0}$ and $r$ are expressed in metres.
3.7. Point eharges $q$ and $-q$ are located at the vertices of a square with diagonals $2 l$ as shown in Fig. 3.1. Find the magnitude of the electric field strength at a point located symmetrically with respect to the vertios of the square at a distance $x$ from its centre.
3.8. A thin hall-ring of radius $A=20 \mathrm{~cm}$ is unilormly charged with a total eharge $q=0.70 \mathrm{sC}$. Find the magnitude of the electrie field strength at the curvature centre of this hall-ring.
3.9. A thin wire ring of radius $r$ carries a charge q. Find the magaitude of the electric field strength on the axis of the ring as a function of distance $l$ from its centre. Investigate the obtained function at $l \gg r$. Find the maximum strength magnitude and the corresponding distance $L$. Draw the approximate plot of the function $E(l)$.
3.10. A point charge $q$ is located at the centre of a thin ring of radius $\boldsymbol{A}$ with uniformly distributed charge -4 . Find the magnitude of the electric field strength vector at the point lying on the axis of the ring at a distance $x$ from its centre, if $x>R$.
3.11. A system consists of a thin charged wine ring of radius $\boldsymbol{R}$ and a very long uniformly charged thread oriented along the axis of the ring, with one of its ends coinciding with the centre of the ring. The total ehange of the ring is equal to $q$. The charge of the thread (per unit length) is equal to $\lambda$. Find the interaction force between the ring and the thread.
3.12. A thin nonconducting ring of radius $\boldsymbol{A}$ has a linear charge density $\lambda=\lambda_{0} \cos \varphi$, where $\lambda_{0}$ is a constant, $\varphi$ is the azimuthal angle. Find the magnitude of the electric field strength
(a) at the centre of the ring:
(b) on the axis of the ring as a function of the distance $x$ from its centre. Investigate the obtained function at $x>R$.
3.13. A thin straight rod of length $2 a$ carrying a uniformly distributed charge $q$ is located in vacuum. Find the magnitude of the
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electric field strength as a function of the distance $r$ from the rod’s centre along the straight line
(a) perpendicular to the rod and passing through its centre;
(b) coinciding with the rod’s direction (at the points lying outside the rod).
Investigate the obtained expressions at $r>a$.
3.14. A very long straight uniformly charged thread carries a charge $\lambda$ per unit length. Find the magnitude and direction of the electric field strength at a point which is at a distance $y$ from the thread and lies on the perpendicular passing through one of the thread’s ends.
3.15. A thread carrying a aniform charge $\lambda$ per unit length has the configurations shows in Fig. 3.2 a and b. Assuming a curvature
Fie. 3.2.
Fit. 3.2.
radius $\boldsymbol{n}$ to be considerably less than the length of the thread, find the magnitude of the electric field strength at the point $O$.
3.16. A sphere of radius $r$ carries a surface change of density $\sigma=$ – ar, where a is a constant vector, and $r$ is the radius vector of a point of the sphere rlative to its centre. Find the electric field streagth vector at the centre of the sphere.
3.17. Suppose the surface charge density over a sphere of radius $A$ depends on a polar angle $\theta$ as $\sigma=\sigma_{0} \cos \theta$, where $\sigma_{0}$ is a positive constant. Show that such a charge distribution can be represented as a result of a small relative shift of two uniformly charged balls of radius $\boldsymbol{R}$ whose charges are equal in magnitude and opposite in sign. Resorting to this representation, find the electric field strength vector inside the given sphere.
3.18. Find the electric field strength vector at the centre of a ball of radius $\boldsymbol{R}$ with volume charge density $p-$ ar, where a is a constant vector, and $r$ is a radius vector drawn from the ball’s centre.
3.19. A very long uniformly charged thread oriented along the axis of a circle of radius $R$ rests on its centre with one of the ends. The charge of the thread per unit length is equal to $\lambda$. Find the fux of the vector $\mathbf{E}$ across the circle area.
3.20. Two point eharges $q$ and $-q$ are separated by the distance 21 (Fig. 3.3). Find the fux of the electrie field strength vector across a circle of radius $\boldsymbol{R}$.
3.21. A ball of radius $R$ is uniformly eharged with the volume density $p$. Find the flux of the electric field strength vector across the ball’s section formed by the plane located at a distance $r_{0}<\boldsymbol{R}$ from the centre of the ball.
3.22. Each of the two long parallet threads carries a uniform charge $\lambda$ per unit length. The threads are separated by a distance $l$. Find the maximum magnitude of the electric field strength in the symmetry plane of this system located between the threads.
3.23. An infinitely long eylindrical surface of circular cross. section is uniformly charged lengthwise with the surface density $\sigma=\sigma_{\theta} \cos \varphi$, where $\varphi$ is the polar angle of the cylindrical coordinate system whose $z$ axis coincides with the axis of the given surface. Find the magnitude and direction of the electrie field strength vector on the $a$ axis.
3.24. The electrie field strength depends only on the $x$ and $y$ coordinates according to the law $\mathbf{E}=a(x i+y j) /\left(x^{3}+y^{2}\right)$, where a is a constant, $i$ and $j$ are the unit vectors of the $x$ and $y$ axes. Find the flux of the vector $\mathbf{E}$ through a sphere of radius $\boldsymbol{A}$ with its centre at the origin of coordinates.
3.25. A ball of radius $A$ carries a positive charge whose volume density depends only on a separation $r$ from the ball’s centre as $p=\rho_{0}(1-r / \boldsymbol{I})$, where $\rho_{0}$ is a constant. Assuming the permittivities of the ball and the environment to be equal to unity, find:
(a) the magnitude of the electric field strength as a function of the distance $r$ both inside and outside the ball:
(b) the maximum intensity $E_{\text {mor }}$ and the corresponding distance $r_{\text {m }}$.
3.26. A system consists of a ball of radius $R$ carrying a spherically symmetric charge and the surrounding space filled with a charge of volume density $p=a / r$, where $a$ is a constant. $r$ is the distance from the centre of the ball. Find the ball’s charge at which the maznitude of the electric field strength vector is independent of $r$ outside the ball. How high is this strength? The permittivities of the ball and the surrounding space are assumed to be equal to unity.
3.27. A space is filled up with a charge with volume density $\rho=\rho_{e} e^{-a r}$, where $\rho_{0}$ and $a$ are positive constants, $r$ is the distance from the centre of this system. Find the magnitude of the electric field strength vector as a function of $r$. Investigate the obtained expression for the small and large values of $r$, i.e. at $\alpha r^{2}<1$ and $\alpha r^{3} \gg 1$.
3.28. Inside a ball charged uniformly with volume density $\rho$ there is a spherical cavity. The centre of the cavity is displaced with respect to the centre of the ball by a distance a. Find the field strength $\mathbf{E}$ inside the cavity, assuming the permittivity equal to unity.
3.29. Inside an infinitely long eircular cylinder charged uniformly with volume density $\rho$ there is a circular cylindrical cavity. The distance between the axes of the cylinder and the cavity is equal to a. Find the electric field strength $\mathbf{E}$ inside the cavity. The permittivity is assumed to be equal to unity.
3.30. There are two thin wire rings, each of radius $\boldsymbol{R}$, whose axes coincide. The charges of the rings are $q$ and $-q$. Find the potential difference between the centres of the rings separated by a distance a.
3.31. There is an infinitely long straight thread carrying a charge with linear density $\lambda=0.40 \mu \mathrm{C} / \mathrm{m}$. Calculate the potential difference between polnts $t$ and 2 if polnt 2 is removed $\eta=2.0$ times farther from the thread than point $\boldsymbol{I}$.
3.32. Find the electric field potential and strength at the centre of a hemisphere of radius $R$ charged uniformly with the surface density $\sigma$.
3.33. A very thin round plate of radius $\boldsymbol{R}$ carrying a uniform surface charge density $\sigma$ is located in vacuum. Find the electric field potential and strength along the plate’s axis as a function of a distance $l$ from its centre. Investigate the obtained expression at $l \rightarrow 0$ and $l>R$.
3.34. Find the potential $\varphi$ at the edge of a thin dise of radius $\boldsymbol{R}$ carrying the uniformly distributed charge with surface density e.
3.35. Find the electrie field strength vector if the potential of this field has the form $q=$ ar, where $a$ is a constant vector, and $r$ is the radius vector of a point of the field.
3.36. Determine the electric field strength vector if the potential of this field depends on $x, y$ coordinates as
a) $\varphi=a\left(x^{2}-y^{2}\right)$ (b) $\varphi=a x y$. where $a$ is a constant. Draw the approximate shape of these fields wusing lines of force (in the $x, y$ plane).
3.37. The potential of a certain electrostatic field has the form $\varphi=a\left(x^{2}+y^{2}\right)+b a^{2}$, where $a$ and $b$ are constants. Find the magaitude and direction of the electric field streagth vector. What shape have the equipotential surfaces in the following cases:
(a) $a>0, b>0$; (b) $a>0, b<0$ ?
3.38. A charge $g$ is uniformly distributed over the volume of a sphere of radius $\boldsymbol{A}$. Assuming the permittivity to be equal to unity throughout, find the potential
(a) at the centre of the sphere;
(b) inside the sphere as a function of the distance $r$ from its centre.
3.39. Demonstrate that the potential of the field generated by a dipole with the electrie moment p (Fig. 3.4) may be represented as $q=p r / 4 \pi r_{0} r^{2}$, where $r$ is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of $r$ and $\theta$.
3.40. A point dipole with an electric moment $p$ oriented in the positive direction of the $z$ axis is located at the origin of coordinates. Find the projections $E_{2}$ and $E_{2}$ of the electric field strength vector (on the plane perpendicular to the $z$ axis at the point $S$ (see Fig, 3.4)). At which points is $\mathbf{E}$
Fie. 3.4. perpendicular to $\mathbf{p}$ ?
3.41. A point electric dipole with a moment $p$ is placed in the external uniform electric field whose strength equals $\mathrm{E}_{8}$. with
p $t \uparrow \mathbf{E}_{\text {e. }}$. In this case one of the equipotential surfaces enclosing the dipole forms a sphere. Find the radius of this sphere.
3.42. Two thin parallet threads carry a uniform charge with linear densities $\lambda$ and $-\lambda$. The distance between the threads is equal to $L$. Find the potential of the electric field and the magnitude of its strength vector at the distance $r \gg l$ at the angle $\theta$ to the vector 1 (Fig. 3.5).
3.43. Two coaxial ring, each of radius $R$, made of thin wire are separated by a small distance $l(l<m)$ and carry the charges $q$ and $-q$. Find the electric field potential and strength at the axis of the
nis. 2.5 .
Fie. 3.7 .
system as a function of the $x$ coordinate (Fig. 3.6). Show in the same drawing the approximate plots of the functions obtained. Investigate these functions at $|x|>R$.
3.44. Two infinite planes separated by a distance $l$ carry a uniform surlace charge of densities $\sigma$ and $-\sigma$ (Fig. 3.7). The planes have round coasial holes of radius $\boldsymbol{R}$, with $l<R$. Taking the origin $O$ and the $x$ coondinate axis as shown in the figure, find the potential of the electrie field and the projection of its strength vector $E_{\text {, on }}$, the axes of the system as functions of the $x$ coordinate. Draw the approximate plot $\varphi(x)$.
3.45. An electric capacitor consists of thin round parallel plates, each of radius $R$, separated by a distance $l(l<m)$ and unilormly charged with surface densities $\sigma$ and $-\sigma$. Find the potential of the electric field and the magnitude of its strength vector at the axes of the capacitor as functions of a distance $x$ from the plates if $x>L$. Investigate the obtained expressions at $x \geqslant R$.
3.46. A dipole with an electrie moment $p$ is located at a distance $r$ from a long thread charged uniformly with a linear density $\lambda$. Find the force $\boldsymbol{F}$ acting on the dipole if the vector $\mathbf{p}$ is oriented
(a) along the thread;
(b) along the radius vector $r$;
(c) at right angles to the thread and the radius vector $\mathbf{r}$.
3.47. Find the interaction force between two water molecules separated by a distance $l=10 \mathrm{~nm}$ if their electric moments are oriented along the same straight line. The moment of each molecule equals $P^{2}=0.62 \cdot 10^{-10} \mathrm{C} \cdot \mathrm{m}$.
3.48. Find the potential $\varphi(x, y)$ of an electrostatie field $\mathbf{E}=$ $=a(y i+x j)$, where $a$ is a constant, $i$ and $i$ are the unit vectors of the $x$ and $y$ axes.
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3.49. Find the potential $\varphi(x, y)$ of an electrostatie field $\mathbf{E}=$ $\left.=2 a x y i+a\left(x^{2}-y^{2}\right)\right)$, where $a$ is a constant, $i$ and $j$ are the un it vectors of the $x$ and $y$ axes.
3.50. Determine the potential $y(x, y, z)$ of an electrostatic field $\mathbf{E}=a y \mathbf{i}+(a x+b z) \mathbf{j}+b y \mathbf{k}$, where $a$ and $b$ are constants, $\mathbf{i}, \mathbf{j} . \mathbf{k}$ 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 $5=-a x^{3}+b$, where $a$ and $b$ are constants. Find the distribution of the space charge $p(x)$.
3.52. A uniformly distributed space change fills up the space between two large parallel plates separated by a distance $d$. The potential difference between the plates is equal to $\Delta \boldsymbol{y}$. At what value of charge density $\rho$ is the field strength in the vicinity of one of the plates equal to rero? 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 $\varphi=a r^{2}+b$, where $a$ and $b$ are constants. Find the space charge distribution $\rho(r)$ inside the ball.