# Problems in General Physics → Electrodynamics → Constant Electric Field in Vacuum

 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? Free solution >> 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? Free solution >> 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 << l. Find the rate dq/dt with which the charge leaks off each sphere if their approach velocity varies as v = a/sqrt(x), where a is a constant. Free solution >> 3.4. Two positive charges q1 and q2 are located at the points with radius vectors r1 and r2. Find a negative charge q3 and a radius vector r3 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. Free solution >> 3.5. A thin wire ring of radius r has an electric charge q. What will be the increment of the force stretching the wire if a point charge q0 is placed at the ring's centre? Free solution >> 3.6. A positive point charge 50 μC is located in the plane xy at the point with radius vector r0 = 2i + 3j, where i and j are the unit vectors of the x and y axes. Find the vector of the electric field strength E and its magnitude at the point with radius vector r = 8i - 5j. Here r0 and r are expressed in metres. Free solution >> 3.7. Point charges q and -q are located at the vertices of a square with diagonals 2l as shown in Fig. 3.1. Find the magnitude of the electric field strength at a point located symmetrically with respect to the vertices of the square at a distance x from its centre. Free solution >> 3.8. A thin half-ring of radius R = 20 cm is uniformly charged with a total charge q = 0.70 nC. Find the magnitude of the electric field strength at the curvature centre of this half-ring. Free solution >> 3.9. A thin wire ring of radius r carries a charge q. Find the magnitude 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 >> r. Find the maximum strength magnitude and the corresponding distance l. Draw the approximate plot of the function E(l). Free solution >> 3.10. A point charge q is located at the centre of a thin ring of radius R with uniformly distributed charge -q. 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. Free solution >> 3.11. A system consists of a thin charged wire ring of radius 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 charge of the ring is equal to q. The charge of the thread (per unit length) is equal to λ. Find the interaction force between the ring and the thread. Free solution >> 3.12. A thin nonconducting ring of radius R has a linear charge density λ = λ0 cos φ, where λ0 is a constant, φ 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. Free solution >> 3.13. A thin straight rod of length 2a carrying a uniformly distributed charge q is located in vacuum. Find the magnitude of the 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. Free solution >> 3.14. A very long straight uniformly charged thread carries a charge λ 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. Free solution >> 3.15. A thread carrying a uniform charge λ per unit length has the configurations shown in Fig. 3.2 a and b. Assuming a curvature radius R to be considerably less than the length of the thread, find the magnitude of the electric field strength at the point O. Free solution >> 3.16. A sphere of radius r carries a surface charge of density σ = ar, where a is a constant vector, and r is the radius vector of a point of the sphere relative to its centre. Find the electric field strength vector at the centre of the sphere. Free solution >> 3.17. Suppose the surface charge density over a sphere of radius R depends on a polar angle θ as σ = σ0 cos θ, where σ0 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 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. Free solution >> 3.18. Find the electric field strength vector at the centre of a ball of radius R with volume charge density ρ = ar, where a is a constant vector, and r is a radius vector drawn from the ball's centre. Free solution >> 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 λ. Find the flux of the vector E across the circle area. Free solution >> 3.20. Two point charges q and -q are separated by the distance 2l (Fig. 3.3). Find the flux of the electric field strength vector across a circle of radius R. Free solution >> 3.21. A ball of radius R is uniformly charged with the volume density ρ. Find the flux of the electric field strength vector across the ball's section formed by the plane located at a distance r0 < R from the centre of the ball. Free solution >> 3.22. Each of the two long parallel threads carries a uniform charge λ 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. Free solution >> 3.24. The electric field strength depends only on the x and y coordinates according to the law E = a(xi + yj)/(x2 + y2), where a is a constant, i and j are the unit vectors of the x and y axes. Find the flux of the vector E through a sphere of radius R with its centre at the origin of coordinates. Free solution >> 3.25. A ball of radius R carries a positive charge whose volume density depends only on a separation r from the ball's centre as ρ = ρ0(1 - r/R), where ρ0 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 Emax and the corresponding distance rm. Free solution >> 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 ρ = α/r, where α is a constant, r is the distance from the centre of the ball. Find the ball's charge at which the magnitude 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. Free solution >> 3.27. A space is filled up with a charge with volume density ρ = ρ0e-αr3, where ρ0 and α 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 αr3 << 1 and αr3 >> 1. Free solution >> 3.28. Inside a ball charged uniformly with volume density ρ 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 E inside the cavity, assuming the permittivity equal to unity. Free solution >> 3.30. There are two thin wire rings, each of radius 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. Free solution >> 3.31. There is an infinitely long straight thread carrying a charge with linear density λ = 0.40 μC/m. Calculate the potential difference between points 1 and 2 if point 2 is removed η = 2.0 times farther from the thread than point 1. Free solution >> 3.32. Find the electric field potential and strength at the centre of a hemisphere of radius R charged uniformly with the surface density σ. Free solution >> 3.33. A very thin round plate of radius R carrying a uniform surface charge density σ 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 → 0 and l >> R. Free solution >> 3.34. Find the potential φ at the edge of a thin disc of radius R carrying the uniformly distributed charge with surface density σ. Free solution >> 3.35. Find the electric field strength vector if the potential of this field has the form φ = ar, where a is a constant vector, and r is the radius vector of a point of the field. Free solution >> 3.36. Determine the electric field strength vector if the potential of this field depends on x, y coordinates as (a) φ = a(x2 - y2); (b) φ = axy, where a is a constant. Draw the approximate shape of these fields using lines of force (in the x, y plane). Free solution >> 3.37. The potential of a certain electrostatic field has the form φ = a(x2 + y2) + bz2, where a and b are constants. Find the magnitude and direction of the electric field strength vector. What shape have the equipotential surfaces in the following cases: (a) a > 0, b > 0; (b) a > 0, b < 0? Free solution >> 3.38. A charge q is uniformly distributed over the volume of a sphere of radius R. 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. Free solution >> 3.39. Demonstrate that the potential of the field generated by a dipole with the electric moment p (Fig. 3.4) may be represented as φ = pr/4πε0r3, where r is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of r and θ. Free solution >> 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 Ez and E⊥ 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 E perpendicular to p? Free solution >> 3.41. A point electric dipole with a moment p is placed in the external uniform electric field whose strength equals E0, with p ↑↑ E0. In this case one of the equipotential surfaces enclosing the dipole forms a sphere. Find the radius of this sphere. Free solution >> 3.42. Two thin parallel threads carry a uniform charge with linear densities λ and -λ. 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 >> l at the angle θ to the vector l (Fig. 3.5). Free solution >> 3.43. Two coaxial rings, each of radius R, made of thin wire are separated by a small distance l (l << R) and carry the charges q and -q. Find the electric field potential and strength at the axis of the 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. Free solution >> 3.45. An electric capacitor consists of thin round parallel plates, each of radius R, separated by a distance l (l << R) and uniformly charged with surface densities σ and -σ. 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 >> R. Free solution >> 3.46. A dipole with an electric moment p is located at a distance r from a long thread charged uniformly with a linear density λ. Find the force F acting on the dipole if the vector p is oriented (a) along the thread; (b) along the radius vector r; (c) at right angles to the thread and the radius vector r. Free solution >> 3.47. Find the interaction force between two water molecules separated by a distance l = 10 nm if their electric moments are oriented along the same straight line. The moment of each molecule equals p = 0.62*10-29 C*m. Free solution >> 3.48. Find the potential φ (x, y) of an electrostatic field E = a(yi + xj), where a is a constant, i and j are the unit vectors of the x and y axes. 3.49. (joined) Find the potential φ (x, y) of an electrostatic field E = 2axyi + a(x2 - y2)j, where a is a constant, i and j are the unit vectors of the x and y axes. 3.50. (joined) Determine the potential φ (x, y, z) of an electrostatic field E = ayi + (ax + bz)j + byk, where a and b are constants, i, j, k are the unit vectors of the axes x, y, z. Free solution >> 3.51. The field potential in a certain region of space depends only on the x coordinate as φ = -ax3 + b, where a and b are constants. Find the distribution of the space charge ρ(x). Free solution >> 3.52. A uniformly distributed space charge fills up the space between two large parallel plates separated by a distance d. The potential difference between the plates is equal to Δφ. At what value of charge density ρ is the field strength in the vicinity of one of the plates equal to zero? What will then be the field strength near the other plate? Free solution >> 3.53. The field potential inside a charged ball depends only on the distance from its centre as φ = ar2 + b, where a and b are constants. Find the space charge distribution ρ(r) inside the ball. Free solution >>