| № |
Condition |
free/or 0.5$ |
| 681 | Determine the electric field strength vector if the potential of this field depends on x, y coordinates as (a) φ = a(x^2 - y^2); (b) φ = axy, where a is a constant. Draw the approximate shape of these fields using lines of force (in the x, y plane). |
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| 682 | 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? |
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| 683 | 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. |
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| 684 | 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. |
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| 685 | 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? |
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| 686 | 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. |
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| 687 | 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). |
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| 688 | 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. |
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| 689 | Two infinite planes separated by a distance L carry a uniform surface charge of densities s and —s (Fig. 3.7). The planes have round coaxial holes of radius R, with L < R. Taking the origin O and the x coordinate axis as shown in the figure, find the potential of the electric field and the projection of its strength vector Ex on the axes of the system as functions of the x coordinate. Draw the approximate plot f(x). |
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| 690 | 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. |
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| 691 | 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. |
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| 692 | 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. |
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| 693 | 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. |
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| 694 | 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. |
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| 695 | 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. |
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| 696 | 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). |
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| 697 | 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? |
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| 698 | 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. |
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| 699 | 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. |
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| 700 | 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. |
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